We consider profunctors between posets and introduce their graph and ascent. The profunctors \(\text {Pro}(P,Q)\) form themselves a poset, and we consider a partition \(\mathcal {I}\sqcup \mathcal {F}\) of this into a down-set \(\mathcal {I}\) and up-set \(\mathcal {F}\), called a cut. To elements of \(\mathcal {F}\) we associate their graphs, and to elements of \(\mathcal {I}\) we associate their ascents. Our basic results is that this, suitably refined, preserves being a cut: We get a cut in the Boolean lattice of subsets of the underlying set of \(Q \times P\). Cuts in finite Booleans lattices correspond precisely to finite simplicial complexes. We apply this in commutative algebra where these give classes of Alexander dual square-free monomial ideals giving the full and natural generalized setting of isotonian ideals and letterplace ideals for posets. We study \(\text {Pro}({\mathbb N}, {\mathbb N})\). Such profunctors identify as order preserving maps \(f: {\mathbb N}\rightarrow {\mathbb N}\cup \{\infty \}\). For our applications when P and Q are infinite, we also introduce a topology on \(\text {Pro}(P,Q)\), in particular on profunctors \(\text {Pro}({\mathbb N},{\mathbb N})\).
Notes
Communicated by M. M. Clementino.
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1 Introduction
This article advocates for general posets P and Q the notion of profunctor as more effective than the notion of isotone (order preserving) maps \(P \rightarrow Q\) between posets, especially for applications in algebra. When Q is totally ordered, these notions are practically the same, but when Q is not, profunctors seem to have a clear advantage for developing natural theory.
Let \(\textbf{2}\) be the two element boolean poset \(\{0 < 1 \}\). A profunctor is simply an isotone map \(P \times Q^\text {op}\,\rightarrow \textbf{2}\). If P and Q are sets (discrete posets), then this is simply a relation between P and Q. The notion of profunctor may generally be defined between categories or between categories enriched in a symmetric monoidal closed category (like \(\textbf{2}\)), see [2, 3], or, for a recent gentle introduction focusing on applications, [13, Section 4].
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The opposite \(P^\text {op}\,\) of a poset P, has the same elements but order relation reversed. The elements in the distributive lattice \(\widehat{P}\) associated to P identifies as pairs (I, F), called cuts, where I is a down-set in P and F the complement up-set. There is then a duality between \(\widehat{P}\) and \(\widehat{P^\text {op}\,}\) sending (I, F) to \((F^\text {op}\,, I^\text {op}\,)\). We call two such pairs dual or Alexander dual (as is common in combinatorial commutative algebra).
Denote by \(\text {Pro}(P,Q)\) the profunctors . This is again a a partially ordered set and the opposite of this poset is \(\text {Pro}(Q,P)\). The basic notions we introduce associated to a profunctor between posets are the notions of its graph\(\Gamma f\) and its ascent\(\Lambda f\). These are dual notions in the sense that if is the dual profunctor, the graph of f equals the ascent of g. Let UP and UQ denote the underlying sets of P and Q. In \(\text {Pro}(P,Q)\) let \(\mathcal {I}\) be a down-set and \(\mathcal {F}\) its complement up-set, so \((\mathcal {I},\mathcal {F})\) is a cut for \(\text {Pro}(P,Q)\). Let \(\mathcal {F}_\Lambda \) be the up-set in the Boolean lattice of all subsets of \(UQ \times UP^\text {op}\,\) generated by the ascents \(\Lambda f\) for \(f \in \mathcal {F}\). Let \(\mathcal {I}_\Gamma \) be the down-set in this Boolean lattice generated by the complements of the graphs \(\Gamma f\) for \(f \in \mathcal {I}\).
Example 0.1
In Fig. 1 the red discs give the graph of a profunctor . The blue circles give the ascent of this profunctor. The graphs and the ascents are then subsets of \(U[4] \times U[5]^{\text {op}\,}\). The up-set \(\mathcal {F}_\Lambda \) is the up-set of the Boolean lattice of subsets of \(U[4] \times U[5]^\text {op}\,\) generated by the ascents of \(f \in \mathcal {F}\). The down-set \(\mathcal {I}_\Gamma \) is the down-set of this Boolean lattice generated by the complements of graphs of \(f \in \mathcal {I}\).
Fig. 1
Profunctor from [5] to [4]
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Our main theorem states the following.
Theorem 3.7. (Preserving the cut) Let PandQbe well-founded posets, and\((\mathcal {I},\mathcal {F})\)a cut for\(\text {Pro}(P,Q)\). Then\((\mathcal {I}_\Gamma ,\mathcal {F}_\Lambda )\)is a cut for the Boolean lattice of subsets of\(UQ \times UP^\text {op}\,\).
Example 0.1 continued. This says that given any subset S of \(U[4] \times U[5]^\text {op}\,\), then either S contains an ascent \(\Lambda f\) for \(f \in \mathcal {F}\), or the complement \(S^c\) contains a graph \(\Gamma f\) for \(f \in \mathcal {I}\). These two cases are also mutually exclusive.
This theorem has alternative formulations in Theorem 3.9 asserting that two up-sets are Alexander dual, with applications to Stanley–Reisner theory, and in Theorem 3.10 asserting that the map sending the ideal \(\mathcal {I}\) to the ideal \(\mathcal {I}_\Gamma \) respects the duality on profunctors. In Theorem 7.1 we give a version with conditions on P and Q ensuring that \(\Gamma f\) and \(\Lambda f\) are always finite sets, suitable for applications to monomial ideals, Sect. 8. An open problem is if a more functorial formulation is possible, Problem 3.11.
Although we develop a general theory here, our original motivation came from applications related to commutative algebra.
Applications to Stanley–Reisner theory. When P and Q are finite posets we get general constructions, Sect. 4, of Alexander dual squarefree monomial ideals, generalizing isotonian ideals and letterplace and co-letterplace ideals, [7, 8, 10, 16, 17]. In particular, when Q is a chain these constructions have given very large classes of simplicial balls and spheres, [5, 11].
Applications to order preserving maps\(f: {\mathbb N}\rightarrow {\mathbb N}\cup \{\infty \}\). The profunctors from \({\mathbb N}\) to \({\mathbb N}\) identify as order preserving maps from \({\mathbb N}\) to the distributive lattice \(\widehat{{\mathbb N}}\), and the latter identifies as \({\mathbb N}\cup \{\infty \}\). Profunctors are the topic of many our examples. These benefit the reader with a quick access to many of our notions and results: 1.2, 1.4, 2.5, 2.12, 2.13, 3.3, 3.8, 5.10, 6.7, 6.8, 6.9. See also the end of Sect. 8.
Injective order preserving maps \(f: {\mathbb N}\rightarrow {\mathbb N}\) form the so called increasing monoid, which has gained recent interest. In [20] Nagel and Römer show that ideals in the infinite polynomial ring invariant for the increasing monoid, have an essentially finite Gröbner basis, thereby generalizing previous results for the symmetric group. In [14] Güntürkün and Snowden studies in depth the representation theory of the increasing monoid. Note that the injective order preserving maps \(g: {\mathbb N}\rightarrow {\mathbb N}\) are in bijection with order preserving maps \(f: {\mathbb N}\rightarrow {\mathbb N}\) by \(g = f + \text {{id}}- 1\). Order preserving maps \(f: {\mathbb N}\rightarrow {\mathbb N}\) also occur in the definition of the bicylic semi-group [6], a basic notion in inverse semi-group theory. Our main application is in [9], relating profunctors (i.e. order preserving maps \(f: {\mathbb N}\rightarrow {\mathbb N}\cup \{ \infty \}\)) to the duality theory of strongly stable ideals in the the infinite polynomial ring \(k[x_i]_{i \in {\mathbb N}}\).
In order for the substantial parts of our theory, related to graphs and ascents, to work well we must have certain conditions on the posets P and Q. Our weakest condition is that they are well-founded. For our applications to polynomial rings, we work in the class of natural posets, Sect. 6. These are posets where all anti-chains are finite and for every x in the poset the principal down-set \(\downarrow x\) is finite. This is a subclass (closer to natural numbers) of well partially ordered sets.
Another feature we introduce is a topology on \(\text {Pro}(P,Q)\), Sect. 5, in particular on \(\text {Pro}({\mathbb N}, {\mathbb N})\). This is needed for our applications to commutative algebra. For \(\text {Pro}({\mathbb N},{\mathbb N})\) a basis for the topology consists of intervals [f, g] where i) the image \(f({\mathbb N})\) is contained in a finite interval, and ii) \(g(p) = \infty \) for all but a finite set of p’s.
Organization of article:
1.
Preliminaries on posets. Notions for posets are recalled, most significantly cuts for posets and the associated distributive lattice. It relates to simplicial complexes, Alexander duality, and Stanley–Reisner rings.
2.
Profunctors between posets. We introduce these and develop basic theory.
3.
The graph, the ascent and preserving the cut. We give our main theorem on Alexander duality, together with variants.
4.
Applications to finite posets and Alexander duality. We connect to commutative algebra and get Alexander dual ideals in finite dimensional polynomial rings.
5.
Topology on \(\text {Pro}(P,Q)\). We define the topology and in particular look at interior open down-sets.
6.
Profunctors between natural posets. We consider natural posets P and Q and investigate the topology in this setting. We show an open down-set is also closed (clopen down-set), if and only if it is finitely generated.
7.
Natural posets and finite type cuts. We give the version of the main theorem for natural posets, suitable to get Alexander dual ideals in (infinite dimensional) polynomial rings.
8.
Monomial ideals. When \(Q = {\mathbb N}\) we get monomial ideals in the polynomial ring generated by \(x_p, p \in P\). We briefly indicate the applications to strongly stable ideals in [9] when \(P = Q = {\mathbb N}\).
Note. We let \({\mathbb N}= \{1,2,3, \ldots \}\). We only use the ordered structure on this so we could equally well have used \({\mathbb N}_0 = \{0,1,2,3, \cdots \}\). Only in the last Sect. 8 do we, in a somewhat different setting, use the commutative monoid structure and then we explicitly write \({\mathbb N}_0\).
2 Preliminaries on Posets
We give basic notions and constructions concerning posets: down-sets, up-sets, dualities, distributive lattices, simplicial complexes. We also recount the algebraic notions of Stanley–Reisner ideal and ring.
2.1 Down- and Up-Sets in P
Let P be a partially ordered set. The opposite poset\(P^{\text {op}\,}\) has the same underlying set as P but with order relation \(\le _\text {op}\,\) where \(p \le _\text {op}\,q\) if \(p \ge q\) in P.
A down-setI of P is a subset of P closed under taking smaller elements. An up-setF in P is a subset of P closed under taking larger elements. If I and F are complements of each other, we call (I, F) a cut for P. Since each of I and F determine each other, we sometimes denote this as \((-,F)\) if we focus on F, and similarly with \((I,-)\). Down-sets are sometimes called order ideals and up-sets order filters, whence the suggestive notation I and F. (The single term ideal is usually reserved for order ideals in lattices closed under joins.)
An element \(p \in P\) induces the principal up-set \(\uparrow p\) consisting of all \(p^\prime \) with \(p^\prime \ge p\), and a principal down-set \(\downarrow p\) consisting of all \(p^\prime \le p\).
Definition 1.1
The Alexander dual (or just dual) of the cut (I, F) for P is the cut \((F^\text {op}\,, I^\text {op}\,)\) for \(P^\text {op}\,\). The Alexander dual of the down-set I is the down-set \(J = F^\text {op}\,\) of \(P^\text {op}\,\), and the Alexander dual of the up-set F is the up-set \(G = I^\text {op}\,\).
2.2 The Distributive Lattice
If P and Q are two partially ordered sets, a map \(f: P \rightarrow Q\) is isotone if it is order-preserving, i.e. \(p_1 \le p_2\) implies \(f(p_1) \le f(p_2)\). We denote by \(\text {Hom}(P,Q)\) the set of all isotone (order-preserving) maps \(f: P \rightarrow Q\). It is itself a partially ordered set (an internal Hom) by \(f \le g\) if \(f(p) \le g(p)\) for every \(p \in P\). The opposite poset \(\text {Hom}(P,Q)^{\text {op}\,}\) naturally identifies as \(\text {Hom}(P^{\text {op}\,}, Q^{\text {op}\,})\). The category of posets forms a closed symmetric monoidal category and so for any three posets P, Q, R we have:
Denote by \(\textbf{2}\) the ordered set \(\{ 0 < 1 \}\). The (complete) distributive lattice associated to P is \(\widehat{P} = \text {Hom}(P^{\text {op}\,},\textbf{2})\). Given an \(f \in \widehat{P}\), the elements p in P such that \(p^\text {op}\,\) maps to \(1 \in \textbf{2}\), constitute a down-set I in P. The complement up-set F in P consists of those \(p \in P\) such that \(p^\text {op}\,\) maps to 0. An element f of \(\widehat{P}\) may thus be identified either with:
A down-set I of P,
An up-set F of P,
A cut (I, F) for P.
We shall usually identify elements of \(\widehat{P}\) with the down-sets \({\mathbb D} P\). In categorical terms \({\mathbb D} P\) is the free cocompletion of P. Thus an element \(i \in \widehat{P}\) is a down-set I of P. However we sometimes will consider the elements of \(\widehat{P}\) to be cuts (I, F) for P. We speak of a cut (I, F) in\(\widehat{P}\), or equivalently a cut (I, F) forP. The cuts for P are then ordered by
$$\begin{aligned} (I,F) \le (J,G) \text { if and only if } I \subseteq J \text { or equivalently } F \supseteq G. \end{aligned}$$
The distributive lattice \(\widehat{P}\) has a unique maximal element, denoted \(\infty \). It sends every \(p^{\text {op}\,}\) to 1, and corresponds to the cut \((I,F) = (P,\emptyset )\).
Example 1.2
\(P = [n] = \{ 1< 2< \cdots < n\}\) has cuts (I, F) where \(I = \{1,2,\ldots , i-1\}\) and \(F = \{i, \ldots , n\}\) for \(i = 1, \ldots , n+1\). Thus \(\widehat{P} \cong [n+1] = [n] \cup \{ \infty \}\) where \(\infty = n+1\).
Any set A may be considered a discrete poset (only relations are \(a \le a\)). Then \(\widehat{A}\) is the Boolean lattice on A. It consists of subsets \(S \subseteq A\). We identify such a subset with the cut \((S,S^c)\). (So for instance the cut \((-,T)\) identifies as the subset \(T^c\) in A.)
If \(P = {\mathbb N}\) the natural numbers, then \(\widehat{{\mathbb N}} = {\mathbb N}\cup \{ \infty \}\).
where if \(p^\prime \ge p\) we send \(p^{\prime \text {op}\,} \mapsto 0\) and all other elements of \(P^\text {op}\,\) to 1. It corresponds to the cut \(((\uparrow p)^c,\uparrow p)\) for P (where \(()^c\) denotes the complement set).
Fig. 2
The poset P and its cut \(((\uparrow p)^c, \uparrow p))\), giving \(\hat{p}\) in \(\hat{P}\)
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This gives a distinguished injective poset map
$$\begin{aligned} P \longrightarrow \widehat{P} = \text {Hom}(P^{\text {op}\,}, \textbf{2}), \quad p \mapsto \hat{p}, \text { corresponding to the cut } ((\uparrow p)^c,\uparrow p) \end{aligned}$$
(2)
see Fig. 2. Note that the image of the map (2) is in \(\widehat{P} \backslash \{ \infty \}\).
Remark 1.3
A poset may be considered a \(\textbf{2}\)-category for the symmetric monoidal closed category \(\textbf{2}\). Where
$$\begin{aligned} P \rightarrow \text {Hom}(P,\textbf{2})^\text {op}\,. \end{aligned}$$
(4)
This is the co-Yoneda embedding, [1]. Note that \(\textbf{2}^\text {op}\,\cong \textbf{2}\) by sending \(0^\text {op}\,\mapsto 1\) and \(1^\text {op}\,\mapsto 0\). So
In general however, for instance ordinary categories with \(\textbf{set}\) instead of \(\textbf{2}\), the Yoneda and co-Yoneda embedding map to different categories. Composing (4) above with (5) we get the embedding (2).
Example 1.4
Let \(P = [n]\). The map (2), the co-Yoneda embedding, is
$$\begin{aligned}{}[n] \longrightarrow \widehat{[n]} = [n+1], \quad i \mapsto i. \end{aligned}$$
$$\begin{aligned}{}[n] \longrightarrow \widehat{[n]} = [n+1], \quad i \rightarrow i+1. \end{aligned}$$
In particular the top element \(n \mapsto n+1 = \infty \).
In more abstract terms, the Yoneda embedding embeds a chain of length n into the upper part of a chain of length \(n+1\), while the co-Yoneda embedding embeds it into the lower part.
2.3 Down- and Up-Sets in \(\widehat{P}\)
Let \(\mathcal {I}\) be a down-set for \(\widehat{P}\), and \(\mathcal {F}\) the complement up-set of \(\widehat{P}\). So \(\mathcal {I}\) consists of cuts (I, F) closed under forming cuts with smaller I’s, and \(\mathcal {F}\) consists of cuts (I, F) closed under forming cuts with larger I’s (or equivalently smaller F’s). Then \((\mathcal {I},\mathcal {F})\) is a cut for\(\widehat{P}\) (note again terminology: (I, F) is a cut in\(\widehat{P}\)). Also \((\mathcal {I},\mathcal {F})\) is a cut in\(\widehat{\!\widehat{P}}\).
Lemma 1.5
A cut \((I^\prime , F^\prime )\) in \(\widehat{P}\) is in \(\mathcal {I}\) iff \(F^\prime \cap I \ne \emptyset \) for every (I, F) in \(\mathcal {F}\).
Proof
That \((I^\prime ,F^\prime )\) is in \(\mathcal {I}\) means that we cannot find any (I, F) in \(\mathcal {F}\) such that \(I \subseteq I^\prime \). Alternatively \(F^\prime \cap I \ne \emptyset \) for each (I, F) in \(\mathcal {F}\). \(\square \)
For later use we look closer at Alexander duality for \(\widehat{P}\) and \(\widehat{\!\widehat{P}}\). A cut (I, F) forP, i.e. a cut in \(\widehat{P}\), gives a dual cut \((F^{\text {op}\,},I^\text {op}\,)\) for \({P}^\text {op}\,\). A cut \((\mathcal {I},\mathcal {F})\)for\(\widehat{P}\), i.e. a cut in \(\widehat{\!\widehat{P}}\) gives a dual cut \((\mathcal {F}^\text {op}\,, \mathcal {I}^\text {op}\,)\) for \((\widehat{P})^\text {op}\,= \widehat{P^{\, {\text {op}\,}}}\) which we denote by \(P^{\, \widehat{\text {op}\,}}\). Thus \((\mathcal {F}^\textrm{op}, \mathcal {I}^\textrm{op})\) is a cut in \(P^{\widehat{\widehat{op}}}\). For \(u = (I,F) \in \mathcal {I}\subseteq \widehat{P}\), then \(u^\text {op}\,\in \mathcal {I}^\text {op}\,\) is
So when we take the dual of the cut \((\mathcal {I},\mathcal {F})\) for \(\widehat{P}\), we not only get a switch \((\mathcal {F}^\text {op}\,, \mathcal {I}^\text {op}\,)\) but also the elements \(u = (I,F)\) of \(\mathcal {I}\) or \(\mathcal {F}\) are switched, to \(u^\text {op}\,= (F^\text {op}\,, I^\text {op}\,)\).
2.4 Finite Type Cuts
The elements of the distributive lattice \(\widehat{P}\) identify as cuts (I, F).
Definition 1.6
\(\widehat{P}_{fin}\) is the sublattice of \(\widehat{P}\) consisting finite cuts: cuts (I, F) where I is finite.
\(\widehat{P}^{fin}\) is the sublattice of \(\widehat{P}\) consisting of cofinite cuts: cuts (I, F) where F is finite.
When P is an infinite discrete poset, an infinite set, such cuts (I, F) of P are a standard example of infinite Boolean algebras, called finite-cofinite algebras.
Definition 1.7
A finite type\(\widehat{P}\)-cut is a pair \((\mathcal {I},\mathcal {F})\) where \(\mathcal {F}\) is an up-set for \(\widehat{P}_{fin}\) and \(\mathcal {I}\) a down-set for \(\widehat{P}^{fin}\) such that the following holds.
1.
\( (I,F) \in \mathcal {I}\text { if and only if } F \cap J \ne \emptyset \text { for every } (J,G) \in \mathcal {F}. \)
2.
\( (J,G) \in \mathcal {F}\text { if and only if } F \cap J \ne \emptyset \text { for every } (I,F) \in \mathcal {I}. \)
Note that for P finite then 1 and 2 are equivalent. If P is infinite, one of the above will in general not imply the other. The point of having both fulfilled is that \(\mathcal {F}\) determines \(\mathcal {I}\) by 1, and vice versa by 2. If only 1 holds then \(\mathcal {F}\) determines \(\mathcal {I}\), but one may not be able to reconstruct \(\mathcal {F}\) from \(\mathcal {I}\).
In Sect. 7 we construct such finite type cuts for infinite posets.
2.5 Simplicial Complexes and Stanley–Reisner Rings
Let A be a set. A simplicial complex X on A is a set of subsets of A closed under taking smaller subsets, i.e. if \(I \in X\) and \(J \subseteq I\), then \(J \in X\).
The set A may be considered as a poset with the discrete poset structure, i.e. the only comparable elements are \(a \le a\) for \(a \in A\). Then \(\widehat{A}\) identifies as the Boolean lattice on A (see Example 1.2), consisting of all subsets of A. A cut \((\mathcal {I},\mathcal {F})\) for \(\widehat{A}\) corresponds precisely to a simplicial complex X. The elements I in X give the cuts \((I,I^c)\) in \(\mathcal {I}\).
The Alexander dual simplicial complex Y of X consists of all the complements \(I^c\) of subsets \(I \subseteq A\) such that I is not in X. The Alexander dual cut \((\mathcal {F}^\text {op}\,, \mathcal {I}^\text {op}\,)\) for \(\widehat{A}^{\text {op}\,} \cong \widehat{A}\) then corresponds to Y: The cuts \((F^\text {op}\,, I^\text {op}\,)\) in \(\mathcal {F}^\text {op}\,\) give precisely the elements \(F^\text {op}\,= (I^\text {op}\,)^c\) in Y.
Denote by \(k[x_A]\) the polynomial ring in the variables \(x_a\) for \(a \in A\). When A is finite, to the simplicial complex X corresponding to the cut \((\mathcal {I},\mathcal {F})\), we associate a monomial ideal \(I_X\) in \(k[x_A]\), the Stanley–Reisner ideal of X. It is generated by monomials \(x_I = \prod _{i \in I}x_i\) for \((I,F) \in \mathcal {F}\). These are the subset I of A such that I is not in the simplicial complex X. The monomials in the Alexander dual Stanley–Reisner ideal \(I_Y\) are then precisely those monomials which have non-trivial common divisor with every monomial in \(I_X\), by the characterization of Lemma 1.5.
When A is infinite, we still have a polynomial ring \(k[x_A]\). But the construction above does not give meaning if \(\mathcal {F}\) is non-empty, since there would be (I, F) in \(\mathcal {F}\) with I infinite. However if \((\mathcal {I},\mathcal {F})\) is a finite type \(\widehat{A}-\)cut, it gives rise to a Stanley–Reisner ideal \(I_X\) of \(k[x_A]\), as above.
3 Profunctors Between Posets
We introduce profunctors between posets. Such a profunctor has a dual and we investigate this correspondence. For an introduction to profunctors, see [13, Chap.4]. See also [3, Section 7] and [2] where they are called distributors.
3.1 Duality on Profunctors
A profunctor is simply a poset homomorphism \(P \rightarrow \widehat{Q}\). By the adjunction
Thus a profunctor is equivalently an isotone map \(P \times Q^\text {op}\,\rightarrow \textbf{2}\) and this is often taken as the definition. It is also equivalently an element of the distributive lattice \((Q \times P^\text {op}\,)^{\widehat{}}\), and so corresponds to a cut (I, F) for \(Q \times P^\text {op}\,\). (Our convention differs somewhat from [13], since there a profunctor corresponds to an isotone map \(P^\text {op}\,\times Q \rightarrow \textbf{2}\).)
In particular if \(Q = B\) and \(P = A\) are sets, this is simply a subset of \(B \times A^\text {op}\,\) or a relation between the sets A and B. Profunctors are also called distributors or bimodules in the literature. We denote the set of profunctors as \(\text {Pro}(P,Q) (= \text {Hom}(P,\widehat{Q}))\). It is itself a partially ordered set, in fact a distributive lattice.
Remark 2.1
Profunctors can be taken to be the morphisms between two ordered sets. They may be composed and form a category. R.Rosebrugh and R.J.Wood show in [21] that this category is equivalent to the category of totally algebraic lattices, those of the form \(\widehat{P}\) for some poset P, with supremum-preserving isotone maps. (There the term ideal is used for profunctor.)
Order theory may also be done in a more categorical setting, for objects in a topos as in [21], or even more general categories [4]. The result of Rosebrugh and Wood mentioned above also has a more general categorical formulation [22] in terms of a monad on a category where idempotents split.
When (I, F) is the cut of \(Q \times P^\text {op}\,\) corresponding to f, the down-set I may be considered as a relation defining the profunctor. We then write qfp when \((q,p^\text {op}\,) \in I\).
Lemma 2.2
Given a profunctor .
a.
The following are equivalent: i) \(q \in f(p)\) and ii) qfp.
b.
The following are equivalent: i) \(q \in f(p)^c\), ii) \(f(p) \le \hat{q}\), and iii) \(\lnot \, qfp\).
Proof
Let f correspond to \(P \rightarrow \widehat{Q} = \text {Hom}(Q^\text {op}\,, \textbf{2})\). That \(q \in f(p)\), the latter a down-set, means that \(q^\text {op}\,\mapsto 1\). This says \((p,q^\text {op}\,) \mapsto 1\), and so \((q,p^\text {op}\,)\) is in the down-set I corresponding to f.
The element \(\hat{q}\) corresponds to the cut \(((\uparrow q)^c, \uparrow q)\). That \(f(p)\le \widehat{q}\) then means that \(f(p) \subseteq (\uparrow q)^c\), or equivalently \(q \not \in f(p)\). \(\square \)
Since \(\text {Pro}(P,Q)\) identifies as \((Q \times P^\text {op}\,)^{\widehat{}}\), the following is seen to be natural, by taking opposites.
Lemma 2.3
Let P, Q be posets. There is a a natural isomorphism of posets
In particular \(f(p) = \infty \) if and only if \(g(q) \le \widehat{p}\) for every \(q \in Q\).
Proof
If f corresponds to the cut (I, F) of \(Q \times P^\text {op}\,\), the dual profunctor g corresponds to the cut \((F^\text {op}\,, I^\text {op}\,)\) of \(P \times Q^\text {op}\,\). The statements in a are equivalent to \((q,p^\text {op}\,) \in I\).
For part b, the condition \(q \in f(p)\) is equivalent to \(g(q) \le \widehat{p}\) by part a and Lemma 2.2. \(\square \)
Example 2.5
Let \(P = Q = {\mathbb N}= \{1,2,3,\ldots \}\) and consider a profunctor (see Example 1.2) which is an isotone map \(f: {\mathbb N}\rightarrow {\mathbb N}\cup \{\infty \}\). Let its values be
In Fig. 3 the graph of f are marked with red discs
. We fill in with blue circles
to make a connected “snake”, starting at position (1, 1). The graph of the dual map \(g = Df\) is given by the blue circles by considering the vertical axis as the argument for g. The values of g are
Observe that for a profunctor (which identifies as an isotone \(f: {\mathbb N}\rightarrow \widehat{{\mathbb N}}\)), then \(f(1) \ge 2\) iff the dual map \(g = Df\) has \(g(1) = 1\). Hence there are no self-dual maps f.
×
×
Fig. 3
Graph (red) and ascent (blue) of profunctor f in Example 2.5
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The profunctor f corresponds to the cut (I, F) for \({\mathbb N}\times {\mathbb N}^\text {op}\,\) (where \({\mathbb N}\) corresponds to the y-axis and \({\mathbb N}^\text {op}\,\) to the (reversed) x-axis) where the up-set F is given by filling in red discs vertically above those in the graph, see Fig. 4, and the ideal I is given by filling in blue circles to the right of those which are present in Fig. 3.
Fig. 4
Cut (I, F) (in resp. blue, red) corresponding to the profunctor f in Example 2.5
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3.2 Alexander Duality
Definition 2.6
Recall that a profunctor is an isotone \(f: P \rightarrow \widehat{Q}\).
For fixed \(q \in Q\), the down-set \(\text {Pro}_{\le \widehat{q}}(P,Q)\) is the set of profunctors f such that \(f(p) \le \widehat{q}\) for all p.
For fixed \(p \in P\), the down-set \(\text {Pro}^{\text {im}\,(p) < \infty }(P,Q)\) is the set of profunctors f such that \(f(p) < \infty \).
The down-set \(\text {Pro}^{< \infty }(P,Q)\) is the set of profunctors f such that \(f(p) < \infty \) for every \(p \in P\).
Lemma 2.7
The down-sets \(\text {Pro}_{\le \widehat{q}}(P,Q)\) and \(\text {Pro}^{\text {im}\,(q) < \infty }(Q,P)\) are Alexander duals.
Proof
That \(f(p) \le \widehat{q}\) is equivalent to, letting \(g = Df\), that \(p \in g(q)\). That this holds for all \(p \in P\) is equivalent to \(g(q) = \infty \). The Alexander dual down-set of \(\text {Pro}_{\le \widehat{q}}(P,Q)\) is then those maps gnot fulfilling \(g(q) = \infty \). \(\square \)
Corollary 2.8
If Q has a maximal element, then \(\text {Pro}^{< \infty }(P,Q)\) and \(\text {Pro}^{< \infty }(Q,P)\) are Alexander dual down-sets.
Note: By symmetry of the conclusion, this holds under the weaker assumption that P or Q has a maximal element.
Proof
If q is the maximal element of Q the down-sets of Lemma 2.7 are respectively \(\text {Pro}^{< \infty }(P,Q)\) and \(\text {Pro}^{< \infty }(Q,P)\). \(\square \)
3.3 Profiles and Co-profiles
Definition 2.9
The profile of a profunctor \(f: P \rightarrow Q\) is the cut (I, F) for P where the profile up-setF consists of all \(p \in P\) such that \(f(p) = \infty \), and the profile down-set\(I = F^c\) is the complement down-set.
The co-profile of f is the cut (J, G) for Q where J is the union of all f(p) (considered as a down-set of Q) for \(p \in P\), and G is the complement of J.
The following is immediate.
Lemma 2.10
Let be a profunctor.
a.
Its profile up-set \(F = \{ p \in P \, | \, qfp \text { for every } q \in Q \}\).
b.
Its co-profile up-set \(G = \{ q \in Q \, | \, \lnot qfp \text { for every } p \in P \}\).
If \(g = Df\) is the dual, the co-profile of f equals the profile of g (by Lemma 2.4a).
We identify the following subsets of \(\text {Pro}(P,Q)\).
\(\text {Pro}^L(P,Q)\) are the profunctors f whose profile down-set I is finite. These maps are called large. Then \(f(p) = \infty \) for all but a finite number of p’s.
\(\text {Pro}_S(P,Q)\) are the f whose coprofile down-set J is finite. These maps are called small. Then there is a finite down-set bounding the f(p), i.e. all \(f(p) \subseteq J\).
\(\text {Pro}^u(P,Q)\) are the f which are in neither the above, so both the profile down-set I and co-profile down-set J are infinite.
The duality D switches \(\text {Pro}^L(P,Q)\) and \(\text {Pro}_S(Q, P)\) and maps \(\text {Pro}^u(P,Q)\) to \(\text {Pro}^u(Q,P)\).
b.
If P is finite, then \(\text {Pro}^L(P,Q) = \text {Pro}(P,Q)\).
c.
If Q is finite, then \(\text {Pro}_S(P,Q) = \text {Pro}(P,Q)\).
Example 2.12
Consider profunctors . By Example 1.2 recall \(\widehat{{\mathbb N}} = {\mathbb N}\cup \{ \infty \}\). Such a map is large if \(f(n) = \infty \) for some n. It is small if f is eventually constant, so \(f(n) = c\) for all \(n \ge n_0\). The maps in \(\text {Pro}^u({\mathbb N}, {\mathbb N})\) are the maps \(f: {\mathbb N}\rightarrow {\mathbb N}\) which are not bounded, so \(\lim _{n \rightarrow \infty } f(n) = \infty \). We see the naturalness of considering \(\text {Pro}({\mathbb N}, {\mathbb N})\) instead of \(\text {Hom}({\mathbb N}, {\mathbb N})\): The latter is not self-dual while the former is. \(\text {Pro}({\mathbb N}, {\mathbb N})\) has two countable dual “shores” \(\text {Pro}^L({\mathbb N}, {\mathbb N})\) and \(\text {Pro}_S({\mathbb N},{\mathbb N})\) and between them an uncountable self-dual “ocean” \(\text {Pro}^u({\mathbb N},{\mathbb N})\).
Remark 2.13
The small profunctors \(\text {Pro}_S({\mathbb N},{\mathbb N})\), which are simply bounded maps \(f: {\mathbb N}\rightarrow {\mathbb N}\) are in bijection with the tame (small) increasing monoid of [14] by sending the bounded map f to \(f + \text {{id}}- 1\).
3.4 Adjunctions
Given an isotone map \(f: P \rightarrow Q\) it induces a pull-back map
where \(f(I)^\downarrow \) is the smallest down-set in Q containing f(I). There is also a right adjoint of \(f^*\):
where \(f(F)^\uparrow \) is the smallest up-set in Q containing f(F). All these maps are functorial in P and Q.
3.4.1 A Variation on the Setting
There is a variation for these maps as follows. There is a forgetful functor \(U: {\textbf{Poset}}\rightarrow {\textbf{Set}}\) by mapping a poset to the underlying set. Composing with the natural inclusion \({\textbf{Set}}\rightarrow {\textbf{Poset}}\) we get \(U: {\textbf{Poset}}\rightarrow {\textbf{Poset}}\). The inclusion \(i: UP \rightarrow P\) induces the dual map \(i^*: \widehat{P} \rightarrow (UP)^{\widehat{}}\).
Suppose we have an isotone map of posets
$$\begin{aligned} g: UP \rightarrow Q.\end{aligned}$$
(This instead of an isotone map \(f: P \rightarrow Q\). Note also that g is really just a map of sets \(UP \rightarrow UQ\).) We then get composites
(6)
Here sends a cut (I, F) of P to the cut (J, G) of Q where \(G = g(F)^\uparrow \) is the up-set of Q generated by the g(p) for \(p \in F\). Note that the above composite has a left adjoint \(i^! \circ g^*\) (although it will not play a role for us).
Also note the following
(7)
4 The Graph, the Ascent, and Preserving the Cut
We define the two significant notions of this article, the graph and ascent of a profunctor , or equivalently the right and left boundaries of the cut (I, F) for \(Q \times P^\text {op}\,\) corresponding to this profunctor. Then we state several versions of the main theorem of this article, Theorem 3.7, on preserving cuts.
4.1 The Graph and Ascent
Definition 3.1
Denote by \({\textbf{0}}\) the minimal element in \(\text {Pro}(P,Q)\). It sends every \(p \mapsto \emptyset \), corresponding to the cut \((\emptyset , Q)\). Denote by \({\textbf{1}}\) the maximal element in \(\text {Pro}(P,Q)\). It sends every \(p \mapsto Q\), corresponding to the cut \((Q,\emptyset )\).
Recall that for a poset P then UP denotes the underlying set, considered as a discrete poset.
$$\begin{aligned} \Gamma f = \{ (q,p)\, | \, q \text { minimal in the complement } f(p)^c\} \subseteq UQ \times UP^\text {op}\,. \end{aligned}$$
(9)
Example 3.3
For a profunctor , see Fig. 3, then \(\Gamma f\) is the red discs, and \(\Lambda f\) is the blue circles.
Note that \(\Lambda f\) and \(\Gamma f\) are disjoint. Also note i) \(\Lambda {\textbf{0}}= \emptyset \) and \(\Gamma {\textbf{0}}= \min Q \times UP\) where \(\min \) denotes the minimal elements, ii) \(\Lambda {\textbf{1}}= UQ \times \min P\) and \(\Gamma {\textbf{1}}= \emptyset \).
Remark 3.4
Their union \(Bf = \Lambda f \cup \Gamma f\), might be called the boundary of the cut (I, F). This union will not play a role here, but it does in [5, Section 2].
4.2 Left and Right Boundaries
We have seen that \(\text {Pro}(P,Q)\), the profunctors from P to Q, identify as \((Q \times P^\text {op}\,)^{\widehat{}}\). So a profunctor corresponds to a cut (I, F) for \(Q \times P^\text {op}\,\) where
\(\Gamma g = (\Lambda f)^\text {op}\,, \quad \Lambda g = (\Gamma f)^\text {op}\,\).
Proof
Part a is by Lemma 2.4. Parts b and c are then immediate from a. \(\square \)
4.3 Extending \(\Lambda \) and \(\Gamma \) to the Next Level
We have looked at cuts (I, F) for \(Q \times P^\text {op}\,\). Proceeding to the next level, we look at cuts \((\mathcal {I},\mathcal {F})\) for \(\text {Pro}(P, Q) = (Q \times P^\text {op}\,)^{\widehat{}}\). Elements in this latter set are cuts (I, F) for \(Q \times P^\text {op}\,\) partially ordered by \((I,F) \le (I^\prime , F^\prime )\) if \(I \subseteq I^\prime \). Thus the down-set \(\mathcal {I}\) consists of cuts (I, F) closed under taking cuts with smaller I’s. Similarly the up-set \(\mathcal {F}\) is closed under taking larger I’s (and so smaller \(F = I^c\)’s).
By (6) the map \(\Lambda \) induces an isotone map of posets:
So the image of the cut \((\mathcal {I},\mathcal {F})\) is the cut in the Boolean lattice \((UQ \times UP^\text {op}\,)^{\widehat{}}\) whose up-set is \(\Lambda (\mathcal {F})^\uparrow \), the up-set generated by all \(\Lambda f\) for \(f \in \mathcal {F}\).
The map \(\Gamma \) induces an isotone map of posets:
So the image of the cut \((\mathcal {I},\mathcal {F})\) is the cut in the Boolean lattice \((UQ \times UP^\text {op}\,)^{\widehat{}}\) whose ideal is \(\Gamma (\mathcal {I})^{\downarrow }\), the ideal generated by the complements (see (12)) of all \(\Gamma f\) for \(f \in \mathcal {I}\),
4.4 The Main Theorem: Preserving the Cut
In order for the left and right boundaries of a cut (I, F) to give enough information we need to make sure that minimal elements of up-sets of P and Q exists. A poset P is well-founded if every subset of P has a minimal element. Equivalently, any descending chain of elements
stabilizes, i.e. for some N we have \(p_n = p_N\) for \(n \ge N\).
The following theorem is a strong generalization of the results in several articles [7, 8, 10, 16], see Sect. 4 for more on this. The most significant tool in the argument is Zorn’s lemma (which is equivalent to the axiom of choice). Note also that \(\text {Pro}(P,Q)\) is a complete distributive lattice and so has all joins (colimits) and meets (limits).
Theorem 3.7
(Preserving the cut) Let P and Q be well-founded posets, and \((\mathcal {I},\mathcal {F})\) a cut for \(\text {Pro}(P, Q)\). Then \((\Gamma (\mathcal {I})^\downarrow ,\Lambda (\mathcal {F})^\uparrow )\) is a cut for the Boolean lattice \((UQ \times UP^\text {op}\,)^{\widehat{}}\). In other words, the maps .
Example 3.8
Consider profunctors \(\text {Hom}([5],\widehat{[3]})\) where \(\widehat{[3]} = [3] \cup \{\infty \}\). Let \(\mathcal {I}\) be the ideal consisting of all f not taking the value \(\infty \). The graph of such an f is shown in Fig.5. Such a path in the rectangle \(U[3] \times U[5]^\text {op}\,\) is a right path. Let \(\mathcal {F}\) be the complement up-set, consisting of all g which take the value \(\infty \) for some argument in [5]. The ascent of such a g is also shown in Fig. 5. Such a path in the rectangle \(U[3] \times U[5]^\text {op}\,\) is an up path. Theorem 3.7 above says that given any subset S of \([5] \times [3]\), exactly one of the following holds: i. S contains an up path, ii. the complement \(S^c\) contains a right path. An earlier observation of this is in [12, Lemma 4.4].
Fig. 5
Profunctors from [5] to [3]. A graph to the left and an (unrelated) ascent to the right
×
Before proving Theorem 3.7 we state two alternative formulations of this theorem.
Theorem 3.9
Let P and Q be well-founded posets, and \((\mathcal {I},\mathcal {F})\) a cut for \(\text {Pro}(P, Q)\). The up-set generated by all \((\Lambda f,-)\) for \(f \in \mathcal {F}\), and the up-set generated by all \(((\Gamma f)^\text {op}\,,-)\) for \(f \in \mathcal {I}\), are Alexander dual up-sets. (This is the version applied in Stanley–Reisner theory, giving Alexander dual monomial ideals, see Sect. 4.)
Proof
By the above Theorem 3.7, \((\Gamma (\mathcal {I})^{\downarrow })^\text {op}\,= \Gamma ^{!}_{U}(\mathcal {I})^\text {op}\,\) is the Alexander dual up-set of the up-set . Then (7) gives \(\Gamma ^{!}_{U}(\mathcal {I},-)^\text {op}\,= (\Gamma ^{\text {op}\,})^!_U(-,\mathcal {I}^{\text {op}\,})\). \(\square \)
In the proof below we write \(\Lambda _{P,Q}\) and \(\Gamma _{Q,P}\) for these horizontal maps.
Theorem 3.10
Let P and Q be well-founded posets. The following diagram commutes:
Proof
By the commutative diagram (13), \(\Gamma _{Q,P}^\text {op}\,= \Lambda _{P,Q}\). Switching P and Q we have \(\Gamma _{P,Q}^\text {op}\,= \Lambda _{Q,P}\). Hence equals . By the above Theorem 3.9 is the dual of and so is the dual of . \(\square \)
However considering the natural maps \(UP \rightarrow P\) and \(UQ \rightarrow Q\) none of the natural functorial ways to get a map as above, gives the map . For instance we can not see a natural factorization of the above map through \(\text {Pro}(UP, Q)^{\widehat{}}\).
Is there a way to understand or formulate the theorem to get a functorial construction, which works for maps \(P^\prime \rightarrow P\) and \(Q^\prime \rightarrow Q\) (or in some other way)?
Proof of Theorem 3.7
We need to show that the down-set \(\Gamma (\mathcal {I})^{\downarrow }\) is the complement of the up-set \(\Lambda (\mathcal {F})^{\uparrow }\) in the Boolean lattice \((UQ \times UP^\text {op}\,)^{\widehat{}}\). \(\square \)
Part I. We first show that for any \((I,F) \in \mathcal {I}\) and \((J,G) \in \mathcal {F}\) that
$$\begin{aligned} \Gamma F \cap \Lambda J \ne \emptyset . \end{aligned}$$
By Lemma 1.5 this shows that \(\Gamma (\mathcal {I})^{\downarrow }\) and \(\Lambda (\mathcal {F})^{\uparrow }\) are disjoint.
If \((I,F) \in \mathcal {I}\) and \((J,G) \in \mathcal {F}\) then \((F \cap J) \subseteq Q \times P^\text {op}\,\) is \(\ne \emptyset \) by Lemma 1.5. Let \((q_0,p_0) \in F \cap J\). Define \((q_i,p_i)\) in \(F\cap J\) successively as follows. If there is a \(q < q_i\) such that \((q,p_i)\) is in F, then let \((q_{i+1},p_{i+1})\) be \((q,p_i)\) which is automatically also in J. If there is \(p < p_i\) such that \((q_i,p)\) is also in J, then let \((q_{i+1},p_{i+1})\) be \((q_i,p)\) which is automatically also in F. We get chains \(p_0 \ge p_1 \ge \cdots \) and \(q_0 \ge q_1 \ge \cdots \). These must stabilize and for the stable value pair (q, p) we cannot continue the construction and then this pair will be in \(\Gamma F \cap \Lambda J\).
Part II. Let \(S \subseteq UQ \times UP^\text {op}\,\) be such that \(S \cap \Lambda J \ne \emptyset \) for every\((J,G) \in \mathcal {F}\). This means that \((S^c,S)\) is not in \(\Lambda (\mathcal {F})^{\uparrow }\). We show that \(S \supseteq \Gamma F\) for some\((I,F) \in \mathcal {I}\) and so \((S^c,S)\) is in \(\Gamma (\mathcal {I})^{\downarrow }\). This shows that the union of \(\Gamma (\mathcal {I})^{\downarrow }\) and \(\Lambda (\mathcal {F})^{\uparrow }\) is all of \(UQ \times UP^\text {op}\,\).
Let T consist of \(f \in \text {Pro}(P,Q)\) such that \(\Lambda f \cap S = \emptyset \). Clearly the minimum \({\textbf{0}}\) of \(\text {Pro}(P,Q)\) is in T, so T is non-empty. We use Zorn’s lemma to show that T has a maximal element. So let
be a chain of functions in T, and f be the colimit (join) of these. We claim that \(\Lambda f \cap S = \emptyset \). Let \((q,p) \in \Lambda f\), so \( q \in f(p) \) but \(q \not \in f(p^\prime )\) for every \(p^\prime < p\). Then \(q \in f_i(p)\) for some i and \(q \not \in f_i(p^\prime )\) since \(f_i(p^\prime ) \subseteq f(p^\prime )\). Then \((q,p) \in \Lambda f_i\) and so is not in S. Thus f is an upper bound in T for the chain in (14). By Zorn’s lemma T has a maximal element f, and take note that f must be in \(\mathcal {I}\), by the requirement on S.
Claim 1
\(\Gamma f \subseteq S\).
By the second and third lines of Part II, this finishes the proof.
Proof
Suppose not, so there is \((q_0,p_0) \in \Gamma f \backslash S\). Recall \(q_0\) is minimal in the complement \((f(p_0))^c\). Let
$$\begin{aligned} \Lambda \tilde{f} \subseteq \Lambda f \cup \{ (q_0,p_0)\}.\end{aligned}$$
That \((q,p) \in \Lambda \tilde{f}\) means that \(q \in \tilde{f}(p)\) and \(q \not \in \tilde{f}(p^\prime )\) for \(p^\prime < p\).
1.
If not \(p \ge p_0\) then \(\tilde{f}(p^\prime ) = f(p^\prime )\) for all values \(p^\prime \le p\). Therefore (q, p) is in \(\Lambda f\).
2.
Let \(p \ge p_0\). Then either i. \(q \in f(p)\) or ii. \(q = q_0\). In case i. \((q,p) \in \Lambda f\). In case ii, where \(q = q_0\), if \(p > p_0\) then \((q_0,p)\) would not be in \(\Lambda \tilde{f}\) since \(q \in \tilde{f}(p)\) and \(q \in \tilde{f}(p_0)\). We are thus left with \((q,p) = (q_0,p_0)\).
As a consequence we also have \(\Lambda \tilde{f} \cap S = \emptyset \). But by definition of S then \(\tilde{f} \in T\). This contradicts f being maximal in T. Therefore no such \((q_0,p_0)\) could exist, and \(\Gamma f \subseteq S\). \(\square \)
5 Applications to Finite Posets and Stanley–Reisner Ideals
In [7] they show that letterplace ideals L([n], P) and co-letterplace ideals L(P, [n]) are Alexander dual ideals, and this was taken considerably further in [10].
Here we define such square free ideals in the full general setting for finite posets P and Q and cuts in \(\text {Pro}(P,Q)\). We show how the result of [7] above is a consequence. In the following for ease of notation we denote a polynomial ring \(k[x_{UQ \times UP^{op}}]\) as \(k[x_{Q \times P^\text {op}\,}]\).
Definition 4.1
Let P and Q be finite posets. From the cut \((\mathcal {I},\mathcal {F})\) for \(\text {Pro}(P,{Q})\), we get the cut \((\Gamma (\mathcal {I})^\downarrow , \Lambda (\mathcal {F})^\uparrow )\) for \((UQ \times UP^\text {op}\,)^{\widehat{}}\).
The \(\Lambda \)-ideal. Since \(UQ \times UP^\text {op}\,\) is simply a set, this gives a Stanley–Reisner ideal in \(k[x_{Q\times P^\text {op}\,}]\). We denote it \(L_\Lambda (\mathcal {F};P,Q)\), or simply \(L_\Lambda (\mathcal {F})\). As f varies in \(\mathcal {F}\), it is generated by the \(\Lambda f\) (or rather the squarefree monomials \( \prod _{(q,p)\in \Lambda f}x_{q,p}\)).
The \(\Gamma \)-ideal. The dual cut \((\Lambda (\mathcal {F})^{\uparrow \text {op}\,}, \Gamma (\mathcal {I})^{\downarrow \text {op}\,})\) for \((UP \times UQ^\text {op}\,)^{\widehat{}}\), also corresponds to a Stanley–Reisner ideal in \(k[x_{P \times Q^\text {op}\,}]\). We denote this ideal as \(L_\Gamma (\mathcal {I};P,Q)\), or simply \(L_\Gamma (\mathcal {I})\). As f varies in \(\mathcal {I}\), it is generated by the \((\Gamma f)^{\text {op}\,}\) (or rather the squarefree monomials \(\prod _{(p,q) \in (\Gamma f)^{\text {op}\,}} x_{p,q}\)).
By Theorem 3.9 above, the \(\Lambda \)-ideal and \(\Gamma \)-ideal are Alexander dual ideals.
Remark 4.2
Let \(Q = [n]\), the chain on n elements, and \(\mathcal {I}\subseteq \text {Pro}^{< \infty }(P,{[n]})\). The \(\Lambda \)-ideals \(L_\Lambda (\mathcal {F};P,[n])\) are the letterplace ideals of [10] and are shown to be Cohen-Macaulay ideals. The Alexander dual \(\Gamma \)-ideals \(L_\Gamma (\mathcal {I};P,[n])\) are the co-letterplace ideals in loc.cit. and thus have linear resolutions, [19, Thm.5.56]. The above definition and Theorem 3.9 may thus be seen as a full generalization of the setting of [10].
Remark 4.3
With the same setting as in the above remark, the \(\Lambda \)-ideals \(L_\Lambda (\mathcal {F};P,[n])\) define simplicial balls by the Stanley–Reisner correspondence, [5]. Furthermore there is a very simple description of the Stanley–Reisner ideal of their boundaries, which are simplicial spheres. This gives the construction of an enormous amount of simplicial spheres, due to the freedom in choosing P, [n] and \(\mathcal {I}\).
Corollary 2.8 has the following consequence, see also Example 3.8.
Corollary 4.4
If P or Q has a maximal element, the ideals \(L_\Gamma (\text {Pro}^{< \infty }(P, {Q}))\) and \(L_\Gamma (\text {Pro}^{< \infty }(Q,{P}))\) are Alexander dual ideals.
Definition 4.5
Given an isotone map \(f: P \rightarrow Q\), let the graph
$$\begin{aligned} \Gamma _i f = \{ (f(p),p^\text {op}\,)\, | \, p \in P \} \subseteq UQ \times UP^\text {op}\,. \end{aligned}$$
The isotonian ideal L(P, Q) is the ideal in \(k[x_{P \times Q^\text {op}\,}]\) generated by the monomials \(\prod _{(p,q) \in (\Gamma _i f)^\text {op}\,} x_{(p,q)}\) as f varies in \(\text {Hom}(P,Q)\). If P is the chain [n], it is called a letterplace ideal, and if \(Q = [n]\), it is a co-letterplace ideal, [10].
The canonical map \(Q \rightarrow \widehat{Q}\) induces an isotone map
Note that the isotonian ideal L(P, Q) is the squarefree monomial ideal associated to the image by of the cut \((\emptyset , \text {Hom}(P,Q)^\text {op}\,)\) for \(\text {Hom}(P,Q \widehat{)^{\text {op}\,}}\).
The poset P is a forest if for every two incomparable \(p_1\) and \(p_2\) in P, there is no \(p \in P\) such that \(p \le p_1\) and \(p \le p_2\). The connected components of the Hasse diagram of P are then trees with the roots on top.
Lemma 4.6
If P is a forest, then:
a.
For the map in (16), \(\alpha ^{!}(\text {Hom}(P,Q))\) is the down-set \(\text {Pro}^{< \infty }(P,Q)\).
By the right diagram of (17), part a above implies part b. So we do part a. We show that if f is a profunctor in \(\text {Pro}^{< \infty }(P,Q)\), then there exists a \(g: P \rightarrow Q\) such that \(\Gamma f \supseteq \Gamma _i g\). This implies that \(\alpha (g) \ge f\) giving part a. It also implies part b.
1.
If p is maximal in P then let q be a minimal element in the non-empty set \(f(p)^c\). Define then \(g(p) = q\).
2.
Suppose p is not maximal. Suppose \(g(p^\prime )\) is defined for \(p^\prime > p\), such that \((g(p^\prime ), p^{\prime \, \text {op}\,}) \in \Gamma f\). Let \(p^\prime \) be the unique cover of p (since P is a forest). Then \(f(p) \subseteq f(p^\prime )\) and so \(f(p)^c \supseteq f(p^\prime )^c\). Given \(g(p^\prime ) \in f(p^\prime )^c\). Then there will exist a minimal q in \(f(p)^c\) with \(q \le g(p^\prime )\). Then define \(g(p) = q\). This gives the map g.
\(\square \)
We specialize to \(Q = [n]\) and have the consequence:
Corollary 4.7
The letterplace ideal L([n], P) and the co-letterplace ideal L(P, [n]) are Alexander dual ideals.
Proof
The ideal L(P, [n]) is \(L_\Gamma (\text {Pro}^{< \infty }(P,[n]))\) since \(\widehat{[n]} = [n] \cup \{ \infty \}\). Its Alexander dual is the ideal \(L_\Gamma (\text {Pro}^{< \infty }([n],P)\) by Corollary 4.4. By the above lemma this identifies as L([n], P). \(\square \)
In [15] they fully characterize for which P and Q the isotonian ideals L(P, Q) and L(Q, P) are Alexander dual.
6 Topology on \(\text {Pro}(P,Q)\)
We want to get a setting where P and Q may be infinite but the ascents \(\Lambda f\) and graphs \(\Gamma f\) are finite. Then we get \(\Lambda \)- and \(\Gamma \)-ideals in infinite-dimensional polynomial rings. This can be achieved if we put a suitable topology on \(\text {Hom}(P,Q)\).
6.1 Defining the Topology
We define a topology on \(\text {Pro}(P,Q)\) by defining a basis of open subsets.
Definition 5.1
Given a large profunctor \(\overline{f}\) in \(\text {Pro}^L(P,Q)\) and a small profunctor \(\underline{f}\) in \(\text {Pro}_S(P,Q)\). The set of profunctors f such that \(\underline{f} \le f \le \overline{f}\) is denoted \(U(\underline{f},\overline{f})\).
The following is immediate.
Lemma 5.2
Given two such sets \(U_1 = U(\underline{f}_1, \overline{f}_1)\) and \(U_2 = U(\underline{f}_2, \overline{f}_2)\), let \(\underline{f}\) and \(\overline{f}\) be the profunctors defined by the join and meet
As a consequence the sets \(U(\underline{f}, \overline{f})\) form the basis of a topology on \(\text {Pro}(P,Q)\). Note that if P and Q are finite we get the discrete topology on \(\text {Pro}(P,Q)\).
Observation 5.3
a. There is a one-one correspondence between open down-sets in \(\text {Pro}(P,Q)\) and down-sets in \(\text {Pro}^L(P,Q)\).
b. There is a one-one correspondence between open up-sets in \(\text {Pro}(P,Q)\) and up-sets in \(\text {Pro}_S(P,Q)\).
Note also by Lemma 2.3 that the open set \(U(\underline{f}, \overline{f})\) is mapped by the dual D to the open set \(U(D\overline{f}, D\underline{f})\). Hence we get:
Lemma 5.4
The duality map D of Lemma 2.3 is a homeomorphism of topological spaces.
6.2 Generalities on Topology
In any topological space X there is distinguished type of open subsets: those that are the interiors of closed subsets, called regular open sets. In fact, let \( \text {op}\,X\) be the poset of open subsets of X, and \(\text {cl}X\) the poset of closed subsets of X. There is a Galois correspondence:
$$\begin{aligned}{} & {} \text {op}\,X \overset{\overline{()}}{\underset{()^\circ }{\rightleftarrows }} \text {cl}X, \\{} & {} U \mapsto \overline{U} = \text { closure of } U, \quad C \mapsto C^{\circ } = \text {interior of } C \end{aligned}$$
The fix points in \(\text {op}\,X\) for the Galois connection, i.e. the fix points in \(\text {op}\,X\) for the composition of closure and interior \({}^\circ \), are the regular open sets. We denote these as \(\text {reg}\,X\), the open subsets which are interiors of closed subsets.
Lemma 5.5
The map on open subsets \(U \mapsto \overline{U}^c\) where c denotes complement, gives an involution \( \text {reg}\,X {\mathop {\longrightarrow }\limits ^{i}} \text {reg}\,X\). Furthermore \(\overline{U}^c = (U^c)^\circ \).
Proof
Let \(V = \overline{U}^c\). We will show that \(\overline{V}^c = U\) by proving inclusion either way.
1.
We get \(V \subseteq U^c\) and since the latter is closed, \(\overline{V} \subseteq U^c\) and so \(\overline{V}^c \supseteq U\).
2.
We also have \(\overline{U}^c \subseteq \overline{V}\) and so \(\overline{U} \supseteq \overline{V}^c\). Since U is the interior of \(\overline{U}\) we get \(U \supseteq \overline{V}^c\).
3.
We now show that V is regular. Let W be an open subset of \(\overline{V}\). Then \(W^c \supseteq \overline{V}^c = U\), so \(W^c \supseteq \overline{U} = V^c\). Thus \(W \subseteq V\). Hence V is the largest open subset of \(\overline{V}\), and so is the interior.
4.
Since \(U = \overline{V}^c\) we get \(U^c = \overline{V}\), and so
$$\begin{aligned} (U^c)^\circ = \overline{V}^\circ = V = \overline{U}^c. \end{aligned}$$
\(\square \)
6.3 Open and Closed Down-Sets
Lemma 5.6
Let \(\mathcal {I}\) be a down-set in \(\text {Pro}(P,Q)\). The closure \(\overline{\mathcal {I}}\) and the interior \((\mathcal {I})^\circ \) are also down-sets. By duality the analog also holds for up-sets.
Proof
Let \(F \in \overline{\mathcal {I}}\) and let \(G \le F\). We will show G is also in \(\overline{\mathcal {I}}\). If not there is an open U(b, c) disjoint from \(\mathcal {I}\) with \(G \in U(b,c)\), so \(b \le G \le c\).
Then \(F \ge b\) with b small and not in \(\mathcal {I}\). Then \(F \in U(b,{\textbf{1}})\) with this basis open set disjoint from \(\mathcal {I}\) since b is not in \(\mathcal {I}\) and \(\mathcal {I}\) is a down-set. But then F is not in the closure of \(\mathcal {I}\), contrary to assumption.
As for the interior, if \(\mathcal {F}\) is the complement up-set of \(\mathcal {I}\), then by a similar argument \(\overline{\mathcal {F}}\) is a up-set, and so the complement \((\overline{\mathcal {F}})^c = \mathcal {I}^\circ \) is a down-set. \(\square \)
Lemma 5.7
Let \(\mathcal {I}\) be an open down-set \(\mathcal {I}\). Then \(\mathcal {I}\) is regular if and only if \(\overline{\mathcal {I}}\backslash \mathcal {I}\) does not contain large maps.
Similarly an open up-set \(\mathcal {F}\) is regular if and only if \(\overline{\mathcal {F}}\backslash \mathcal {F}\) does not contain small maps.
Proof
We only show the first statement. Suppose \(\mathcal {I}\) is regular. Let f in \(\overline{\mathcal {I}}\backslash \mathcal {I}\) be large. The closure \(\overline{\mathcal {I}}\) is a down-set. It contains f and so the open set \(U({\textbf{0}},f)\), which must then be in the interior of \(\overline{\mathcal {I}}\) which is \(\mathcal {I}\). This contradicts f being in the gap. Similarly one can argue that an f in the gap cannot be bounded, by considering the up-set \(\mathcal {F}\).
If \(\mathcal {I}\) is not regular, there is some open \(U = U(\underline{f}, \overline{f}) \subseteq \overline{\mathcal {I}}\) with \(U \not \subseteq \mathcal {I}\). But then \(\overline{f} \in \overline{\mathcal {I}} \backslash \mathcal {I}\), and so \(\overline{\mathcal {I}} \backslash \mathcal {I}\) contains \(\overline{f}\) which is large. \(\square \)
Definition 5.8
A Dedekind cut in the poset \(\text {Pro}(P,Q)\) is a pair \([\mathcal {I},\mathcal {F}]\) where \(\mathcal {I}\) is an regular open down-set of \(\text {Pro}(P,Q)\) and \(\mathcal {F}\) is its image by the involution i (so \(\mathcal {F}\) is an regular open up-set). The gap of the Dedekind cut is
Let \(\mathcal {I}\) be an open down-set and \(\mathcal {F}\) a disjoint open up-set. Then \(\mathcal {I}\) and \(\mathcal {F}\) form a Dedekind cut if and only if the gap \(\text {Pro}(P, Q) \backslash (\mathcal {I}\cup \mathcal {F})\) between them, does not contain large or small maps.
Remark 5.10
For the rational numbers \({\mathbb Q}\) the distributive lattice \(\widehat{{\mathbb Q}}\) consists of \(\pm \infty \), the irrational numbers, and the rational numbers doubled, since a rational number q gives two cuts:
However if we have the natural topology on \({\mathbb Q}\), Dedekind cuts give the real numbers together with \(\pm \infty \).
7 Profunctors Between Natural Posets
We introduce the class of natural posets as a suitable generalization of the poset of natural numbers. We then have good criteria for when open down-sets of \(\text {Pro}(P,Q)\) are closed or regular. We show, Theorem 6.10, that an open down-set is clopen (closed and open) if and only if it has a finite number of maximal elements. The purpose of this section is to get in Sect. 7 a version of Theorem 3.7 which applies to construct Alexander dual ideals in infinite-dimensional polynomial rings. Our main application is to profunctors , see [9], and briefly indicated in the last Sect. 8.
7.1 Well Partially Ordered Sets
A poset P is well partially ordered if the following two conditions holds:
i.
Any descending chain in P stabilizes.
ii.
Any antichain in P is finite.
The following characterization of well partially ordered sets is classical, see for instance [18].
Proposition 6.1
The following are equivalent for a poset P.
(1)
P is well partially ordered.
(2)
Any up-set of P is finitely generated.
(3)
The ascending chain condition holds for up-sets in P, equivalently the descending chain condition holds for \(\widehat{P}\).
Lemma 6.2
Let P and Q be well partially ordered sets, and an isotone map.
a.
If f is large, then \(\Gamma f\) is finite.
b.
If f is small, then \(\Lambda f\) is finite.
Proof
a. Let f have profile (I, F). The projection of \(\Gamma f\) onto P will be contained in I since the complement \(f(p)^c\) is empty for \(p \in F\). Furthermore for \(p \in I\) the up-set \(f(p)^c\) has only a finite set of minimal elements by the above proposition.
b. Let f have co-profile (J, G). The projection of \(\Lambda f\) onto Q is contained in J. If \(q \in J\), the set of \(p \in P\) such that \(q \in f(p)\) is a up-set in P and so has a finite set of minimal elements. Hence \(\Lambda f\) is finite. \(\square \)
7.2 Natural Posets
The class of well partially ordered sets is very large, it includes all well-ordered sets. For our purposes we need an extra condition so that our posets are more like natural numbers.
Definition 6.3
A poset P is down finite if for each p in P, the principal down-set \(\downarrow p\) is finite. A natural poset (in analogy with natural numbers) is a poset which is well partially ordered and down finite.
Lemma 6.4
Suppose every antichain in P is finite. Then P is natural iff there exists a chain of finite down-sets in P
The if statement is clear. Suppose then P is natural. For \(p \in P\) the principal ideal \(\downarrow p\) is finite. Let the height of p, denoted \(\ell (p)\), be the length of the longest chain in \(\downarrow p\). Then let \(I^r\) be the set of elements of P of height \(\le r\). Then \(I^r\backslash I^{r-1}\) is the set of elements of height exactly r. These form an anti-chain in P and so a finite set. Hence each \(I^r\) is finite. Clearly the union of the \(I^r\) is all of P. \(\square \)
Lemma 6.5
Let Q be down finite. Then for any large f in \(\text {Pro}(P,Q)\) the open set \(U({\textbf{0}},f)\) is also closed.
Proof
Let f have profile (I, F). Consider the set X of all pairs (p, x) where \(p \in I\) and x is minimal in \(f(p)^c\). For each pair \((p,x) \in X\), let \(g_{px}\) be the smallest isotone map \(g: P \rightarrow \widehat{Q}\) such that g(p) is the principal down-set \(\downarrow x\). Then \(g_{px}\) is a small map since \(\downarrow x\) is finite. Furthermore if h is an isotone map such that h is not \(\le f\), then for at least one pair \((p,x) \in X\) we must have \(x \in h(p)\) and so \(h \ge g_{px}\). Then the complement
be a weakly increasing set of maps in \(\text {Pro}(P,Q)\), write \(\text {colim}f_r\) for their join. If the \(f_r\)’s are a decreasing sequence we write \(\lim f_r\) for their meet.
Posets in our application will typically be natural posets, like finite posets and the natural numbers \({\mathbb N}\). We assume in this subsection that our posets are natural. The following seems to provide the best way to check whether an open down-set of isotone maps is regular.
Proposition 6.6
Let P and Q be natural posets and \(\mathcal {I}\) an open down-set in \(\text {Pro}(P,Q)\).
a.
\(\mathcal {I}\) is also closed if and only if it contains the colimit of any increasing sequence of isotone maps in \(\mathcal {I}\).
b.
\(\mathcal {I}\) is regular if and only if it contains any large colimit of an increasing sequence of isotone maps in \(\mathcal {I}\).
There are analogous statements for open up-sets and decreasing sequences of maps in \(\mathcal {F}\).
Proof
a1. Suppose \(\mathcal {I}\) is closed. Let \(\{f_i\}\) be a weakly increasing sequence of isotone maps in \(\mathcal {I}\), and \(f = \text {colim}f_i\). If f is not in \(\mathcal {I}\), there is an open subset U(b, c) in the complement \(\mathcal {I}^c\) containing f.
Since b(p) is finite and \(\lim f_i(p) = f(p)\) which contains b(p), there is a number N(p) such that \(f_i(p)\) contains b(p) for \(i \ge N(p)\).
Let T be the projection of \(\Lambda b\) on P. Since b is small, \(\Lambda b\) is finite and so is T. Let N be the maximum of N(p) for \(p \in T\). We show below that \(f_i \ge b\) for \(i \ge N\). But then since \(f_i \in \mathcal {I}\) and \(\mathcal {I}\) is a down-set, this gives \(b \in \mathcal {I}\), a contradiction since U(b, c) and \(\mathcal {I}\) are disjoint. Thus f must be in \(\mathcal {I}\).
To show that \(f_i \ge b\), if the contrary were true, let \(q \in b(p)\backslash f_i(p)\) for some p and let \(p^\prime \le p\) be minimal in P with \(q \in b(p^\prime )\). Then:
i.
\((q,p^\prime ) \in \Lambda b\) and so \(f_i(p^\prime ) \supseteq b(p^\prime )\) since \(i \ge N \ge N(p^\prime )\).
ii.
Since \(p^\prime \le p\) then \(f_i(p^\prime ) \subseteq f_i(p)\) and so \(q \not \in f_i(p^\prime )\).
But these give a contradiction, and so we must have \(f_i \ge b\).
a2. Suppose any co-limit of an increasing sequence in \(\mathcal {I}\), is in \(\mathcal {I}\). Suppose \(\mathcal {I}\) is not closed. Then there is \(f \in \mathcal {I}^c\) such that any open subset of f intersects \(\mathcal {I}\). Now there exists the following:
$$\begin{aligned} f_k(p) = \cup _{p^\prime \le p} J^k_{p^\prime } \text { when } p \in I^k, \quad f_k(p) = \cup _{p^\prime \in I^k,p^\prime \le p} f_k(p^\prime ) \text { when } p \not \in I^k. \end{aligned}$$
Then
\(f_k\) is isotone and small,
\(i \le j\) implies \(f_i \le f_j\),
The open subset \(U(f_i,{\textbf{1}})\) contains f and so intersects \(\mathcal {I}\). Then \(f_i \in \mathcal {I}\) since \(\mathcal {I}\) is a down-set,
\(\text {colim}f_i = f\) and so \(f \in \mathcal {I}\).
But the latter contradicts \(f \not \in \mathcal {I}\).
b1. Let \(\mathcal {I}\) be regular and \(\{f_i\}\) an increasing sequence in \(\mathcal {I}\), whose colimit f is large. By part a the colimit of \(f_i\) is in the closure \(\overline{\mathcal {I}}\). Since \(\mathcal {I}\) is regular \(\overline{\mathcal {I}} \backslash \mathcal {I}\) does not contain large maps by Lemma 5.7. Hence \(f \in \mathcal {I}\).
b2. Suppose \(\mathcal {I}\) contains large limits of increasing sequences in \(\mathcal {I}\). Let f be a large map in the closure \(\overline{\mathcal {I}}\). By the construction in a2 there is a sequence \(\{ f_i \}\) of small maps such that \(f = \text {colim}f_i\). Each open subset \(U(f_i,f)\) intersects \(\mathcal {I}\) and so \(f_i\) is in \(\mathcal {I}\). But then f is in \(\mathcal {I}\). Then by Lemma 5.7, \(\mathcal {I}\) is regular. \(\square \)
Example 6.7
Let the open down-set \(\mathcal {I}\) of \(\text {Pro}({\mathbb N}, {\mathbb N})\) be generated by the large maps \(f_1, f_2, f_3, \cdots \) given by
$$\begin{aligned} f_r(n) = {\left\{ \begin{array}{ll} r &{} n \le r \\ \infty &{} n > r \end{array}\right. } \end{aligned}$$
Then \(\text {colim}_r f_r\) is the maximal isotone map \({\textbf{1}}\), which is not in \(\mathcal {I}\). Hence \(\mathcal {I}\) is not regular. Also, its closure is all of \(\text {Pro}({\mathbb N}, {\mathbb N})\).
Example 6.8
Let the open up-set \(\mathcal {F}\) in \(\text {Pro}({\mathbb N}, {\mathbb N})\) be generated by the small maps \(g_1, g_2, \cdots \) given by
$$\begin{aligned} g_r(n) = {\left\{ \begin{array}{ll} 1 &{} n < r \\ 2 &{} n \ge r \end{array}\right. }. \end{aligned}$$
The limit \(\lim _r g_r\) is the minimal isotone map \({\textbf{0}}\), which is not in \(\mathcal {F}\). Hence \(\mathcal {F}\) is not regular. Also, its closure is all of \(\text {Pro}({\mathbb N}, {\mathbb N})\).
Example 6.9
Let the profunctor \(h: {\mathbb N}\rightarrow {\mathbb N}\) be given by \(h(i) = i\). Let \(\mathcal {I}\) be the down-set generated by the large maps \(h^r\) for \(r \ge 1\), and \(\mathcal {F}\) be the up-set generated by the small maps \(h_r\) for \(r \ge 0\), where
$$\begin{aligned} h^r(i) = {\left\{ \begin{array}{ll} i &{} i \le r \\ r &{} i = r+1 \\ \infty &{} i \ge r+2 \end{array}\right. }, \quad h_r(i) = {\left\{ \begin{array}{ll} i &{} i \le r \\ r+2 &{} i \ge r+1 \end{array}\right. }. \end{aligned}$$
Then \([\mathcal {I}, \mathcal {F}]\) is a Dedekind cut and the gap consists of the function h.
In the argument that follows now we use the following construction: Let \(I \subseteq J \subseteq P\) be down-sets and \(f: I \rightarrow \widehat{Q}\) an isotone map. Then there exists a unique minimal extension \(f^J: J \rightarrow \widehat{Q}\) such that the restriction\((f^J)_{|I} = f\). For each \(p \in J\) we let
Note that if f is small and I is finite, the extension \(f^J\) is also small.
The following characterizes precisely when the gap is empty.
Theorem 6.10
Let P and Q be natural posets. A down-set \(\mathcal {I}\) of \(\text {Pro}(P,Q)\) is clopen (closed and open) if and only if \(\mathcal {I}\) is a finite union of basis open subsets \(U({\textbf{0}},f)\). Alternatively formulated, an open down-set \(\mathcal {I}\) is clopen if and only if it is finitely generated.
Proof
The open subsets \(U({\textbf{0}},f)\) are closed, by Proposition 6.6. Hence any finite union of them is clopen.\(\square \)
Suppose now \(\mathcal {I}\) is clopen. We will show it has only a finite number of minimal generators. Assume the opposite, that there are infinitely many generators of \(\mathcal {I}\), none of which are comparable.
such that the union of the \(I_r\)’s is P. Let \(T_1\) be the set of isotone maps \(F_1: I_1 \rightarrow \widehat{Q}\) such that:
For every \(f: I_1 \rightarrow \widehat{Q}\) with \(f \le F_1\) and the f(p) finite down-sets in Q for \(p \in I_1\), there are infinitely many of the minimal generators g of \(\mathcal {I}\) such that the restriction \(g_{|I_1} \ge f\). Clearly \(T_1\) is non-empty: we may take \(F_1\) to be the constant map with value the empty set.
2. Let us show that we can use Zorn’s lemma on \(T_1\). If
is a chain in \(T_1\), let \(F_1 = \text {colim}F_1^r\). We claim \(F_1\) is in \(T_1\). Let \(f \le F_1\) with f having finite values. For each p, for some integer N(p) we have, since f(p) is finite, that \(f(p) \subseteq F^{N(p)}_1(p)\). Let N be the maximum of the N(p) for \(p \in I_1\). Then \(f \le F_1^N\) and so there are infinitely many minimal generators g of \(\mathcal {I}\) with \(g_{|I_1} \ge f\). This shows that Zorn’s lemma applies. Furthermore, there is an increasing sequence \(\{ f\}\) of such functions whose limit is \(F_1\). Extending these f’s to P we get an increasing sequence \(\{ f^P \}\) whose limit is the extension \(F_1^P\). Each \(f^P\) is in \(\mathcal {I}\) since there are minimal generators g of \(\mathcal {I}\) with \(g_{|I_1} \ge f\). Since \(\mathcal {I}\) is closed, by Proposition 6.6 we have \(F_1^P \in \mathcal {I}\), a fact we use in 5. below.
3. Let \(F_1\) be maximal in \(T_1\). We now show the following refined statement:
Claim 2
For each finite \(f \le F_1\) there are infinitely many minimal generators g such that \(f \le g_{|I_1} \le F_1\).
Proof
Let \(p_1, p_2, \ldots , p_m\) be the elements of \(I_1\) in an order such that \(p_i < p_j\) implies \(i < j\). (This is usually called a linear extension of \(I_1\).) Suppose we have verified that for each \(f \le F_1\) there are infinitely many g’s such \(f \le g_{|I_1}\) and with \(f(p_i) \le g(p_i) \le F_1(p_i)\) for \(i < s\). Let \(X_s\) be the (finite) set of minimal elements of the complement \(F_1(p_s)^c\). If not \(g(p_s) \subseteq F_1(p_s)\) then \(g(p_s)\) contains some \(x \in X_s\). Define the isotone \(F_{1x}: I_1 \rightarrow \widehat{Q}\) by
$$\begin{aligned} F_{1x}(p) = {\left\{ \begin{array}{ll} F_1(p), &{} \text { not } p \ge p_s \\ F_1(p) \cup \{x \} &{} p \ge p_s \end{array}\right. } \end{aligned}$$
Since \(F_1\) is maximal, there is some \(f_x \le F_{1x}\) where \(f_x\) has finite values, such that there are not infinitely many g’s with \(f_x \le g_{|I_1}\). Let \(f_x^\prime \) be the meet \(f_x \wedge F_1\). Then let \(f^\prime \) be the join \(f \vee (\vee _{x \in X_s} f_x^\prime )\), an isotone with finite values. Note that \(f^\prime \le F_1\). Then there are infinitely many g’s such that: i) \(f^\prime \le g\) and ii) \(f^\prime (p_i) \le g(p_i) \le F_1(p_i)\) for \(i < s\). But only a finite number of g’s are larger than \(f_x\) for each x in the finite set \(X_s\). So taking away a finite number of the g’s, none of the infinitely many rest will have \(g(p_s)\) containing any \(x \in X_s\). Hence for these \(g(p_s) \subseteq F_1(p_s)\). \(\square \)
4. Suppose we now have constructed \(F_j: I_j \rightarrow \widehat{Q}\) for \(j = 1, \cdots , r\) such that \(F_{j|I_i} = F_i\) for each \(i \le j\) and the \(F_j\) are such that for each finite \(f: I_j \rightarrow \widehat{Q}\) with \(f \le F_j\) there are infinitely many generators g such that \(f \le g_{|I_j} \le F_j\), and \(F_j\) is maximal such. We will extend \(F_r\) to a function \(F_{r+1}: I_{r+1} \rightarrow \widehat{Q}\). Let \(T_{r+1}\) be the set of isotone \(F: I_{r+1} \rightarrow \widehat{Q}\) such that \(F_{|I_r} = F_r\) and for each finite \(f \le F\) there are infinitely many g’s with \(g_{|I_r} \ge f\) and \(F_r \ge g_{|I_r} \ge f\). The set \(T_{r+1}\) is non-empty since it is easy to see that the extension \(F_r^{I_{r+1}}\) fulfills the condition of being in \(T_{r+1}\) since \(F_r\) does so for \(T_r\). As above \(T_{r+1}\) fulfills the condition for Zorn’s lemma and so \(T_{r+1}\) has a maximal element \(F_{r+1}\). As above we may show that for each finite \(f \le F_{r+1}\) there are infinitely many minimal generators g of \(\mathcal {I}\) such that \(F_{r+1} \ge g_{|I_{r+1}} \ge f\)
Let \(F = \text {colim}F_r^P\). Since each \(F_r^P \in \mathcal {I}\) (by the same reason as at the end of part 2) and \(\mathcal {I}\) is closed we get \(F \in \mathcal {I}\). Since \(\mathcal {I}\) is open there exists then a large \(G \ge F\) in \(\mathcal {I}\). Let I be the finite profile down-set of G. Due to I being finite we must have \(I_r \supseteq I\) for sufficiently large r. But there are infinitely many minimal generators g such that \(g_{|I_r} \le F_{r}\) which again is \(\le G_{|I_r}\). But since \(G(p) = \infty \) for p in the complement \(I_r^c\), we would have \(G \ge g\). This is a contradiction since the g’s are (infinitely many) minimal generators of \(\mathcal {I}\).
The upshot is that there cannot be infinitely many generators of \(\mathcal {I}\). \(\square \)
Problem 6.11
What subsets of \(\text {Pro}^u({\mathbb N},{\mathbb N})\) can be gaps?
8 Natural Posets and Finite Type Cuts
Here we give the variant of Theorem 3.7 such that the \(\Lambda \)- and \(\Gamma \)-ideals are finitely generated ideals in a polynomial ring. This works when P and Q are natural posets, which we assume in this section.
By Lemma 6.2 the map \(\Lambda \) of (11) restricts to a map:
For a cut \((\mathcal {I}, \mathcal {F})\) for \(\text {Pro}(P,Q)^{\widehat{}}\) let \(\mathcal {F}_S\) denote the small maps in \(\mathcal {F}\), and \(\mathcal {I}^L\) the large maps in \(\mathcal {I}\).
By (6) the map \(\Lambda \) induces an isotone map:
Let P and Q be natural posets and \([\mathcal {I},\mathcal {F}]\) a Dedekind cut for \(\text {Pro}(P, Q)\). Then \((\Gamma (\mathcal {I}^L)^{\downarrow },\Lambda (\mathcal {F}_S)^{\uparrow })\) is a finite type cut for the Boolean lattice \((UQ \times UP^\text {op}\,)^{\widehat{}}\).
Proof
We must show 1 and 2 in Definition 1.7. We do 1 as 2 is analogous. \(\square \)
Part I. We show the implication \(\Rightarrow \) of 1 in Definition 1.7. This amounts to show for any \((I,F) \in \mathcal {I}^L \) and \((J,G) \in \mathcal {F}_S\) that
$$\begin{aligned} \Gamma F \cap \Lambda J \ne \emptyset . \end{aligned}$$
The argument for this is the same as in Part I for Theorem 3.7.
Part II. We show the implication \(\Leftarrow \) of 1 in Definition 1.7. Let \(S \subseteq UQ \times UP^\text {op}\,\) be a finite set such that \(S \cap \Lambda J \ne \emptyset \) for every\((J,G) \in \mathcal {F}_S\). We show that \(S \supseteq \Gamma F\) for some\((I,F) \in \mathcal {I}^L\).
Let \(I_0 \subseteq P\) and \(J_0 \subseteq Q\) be finite down-sets such that the projections of S are contained respectively in \(I_0\) and in \(J_0\). (We know they are finite due to P and Q being down-finite.) For a large f in \(\text {Pro}(P,Q)\) denote by \(\Lambda _0 f\) the subset of \(\Lambda f\) lying above \(I_0\). Let T consist of the large f such that:
Clearly the minimum element of those f fulfilling the first point, also fulfills the second, so T is non-empty. We use Zorn’s lemma to show that T has a maximal element. So let
be a chain of maps in T and let \(f = \text {colim}f_r\). We claim that \(\Lambda _0 f \cap S = \emptyset \). So suppose \((q,p) \in \Lambda f\), so \( q \in f(p) \) but \(q \not \in f(p^\prime )\) for every \(p^\prime < p\). Then \(q \in f_i(p)\) for some i and \(q \not \in f_i(p^\prime )\) since \(f_i(p^\prime ) \subseteq f(p^\prime )\). Then \((q,p) \in \Lambda f_i\) and so is not in S. Thus f is an upper bound in T for the chain. By Zorn’s lemma T has a maximal element, which we denote f.
Claim 3
f is in \(\mathcal {I}\).
Proof
The map f is not in the gap between \(\mathcal {I}\) and \(\mathcal {F}\) since it is large. So if f is not in \(\mathcal {I}\) it must be in \(\mathcal {F}\). Since \(\mathcal {F}\) is open, for \(B \supseteq J_0\) large enough \(f_B\) given by
$$\begin{aligned} f_B(p) = {\left\{ \begin{array}{ll} f(p), &{} p \in I_0 \\ B, &{} p \not \in I_0 \end{array}\right. }, \end{aligned}$$
is in \(\mathcal {F}\). But \(\Lambda _0 f_B \subseteq \Lambda _0 f\), and so \(\Lambda f_B \cap S = \Lambda _0 f_B \cap S\) is the empty set, which is not so by the requirement on S. Hence f is not in \(\mathcal {F}\) and not in the gap, so it must be in \(\mathcal {I}\). \(\square \)
Claim 4
\( \Gamma f \subseteq S\). This finishes the proof by the second line of Part II.
Proof
Suppose not, so there is \((q_0,p_0) \in \Gamma f \backslash S\). Recall \(q_0\) is minimal in the complement \(f(p_0)^c\). Let
That \((q,p) \in \Lambda _0 \tilde{f}\) means that \(p \in I_0\) and \(q \in \tilde{f}(p)\), and \(q \not \in \tilde{f}(p^\prime )\) for \(p^\prime < p\).
1.
If not \(p \ge p_0\) then \(\tilde{f}(p^\prime ) = f(p^\prime )\) for all values \(p^\prime \le p\). Therefore (q, p) is in \(\Lambda _0 f\).
2.
Let \(p \ge p_0\). Then either i. \(q \in f(p)\) or ii. \(q = q_0\). In case i. \((q,p) \in \Lambda f\). In case ii. where \(q = q_0\), if \(p > p_0\) then \((q_0,p)\) would not be in \(\Lambda \tilde{f}\) since \(q_0 \in \tilde{f}(p)\) and \(q_0 \in \tilde{f}(p_0)\). We are thus left with \((q,p) = (q_0,p_0)\).
The upshot is that \(\tilde{f}\) fulfills the requirements to be in T. But this contradicts f being maximal in T. Hence no such \((q_0,p_0)\) can exist, and so \(\Gamma f \subseteq S\). \(\square \)
We may now define the \(\Lambda \)- and \(\Gamma \)-ideals for (infinite) natural posets P and Q. These ideals then live in infinite-dimensional polynomial rings. These ideals will be square-free monomial ideals.
Definition 7.2
Let \([\mathcal {I},\mathcal {F}]\) be a Dedekind cut for \(\text {Pro}(P,Q)\). Let \(L_\Lambda (\mathcal {F})\) the ideal in \(k[x_{Q \times P^\text {op}\,}]\) generated by the monomials \(\prod _{(q,p) \in I} x_{q,p}\) for \((I,-) \in \Lambda (\mathcal {F}_S)\). Let \(L_\Gamma (\mathcal {I})\) be the ideal in \(k[x_{P \times Q^\text {op}\,}]\) generated by the monomials \(\prod _{(p,q) \in F^\text {op}\,}x_{p,q}\) for \((F^\text {op}\,,-) \in \Gamma (\mathcal {I}^L)^{\text {op}\,}\).
By the theorem above, these ideals are Alexander dual ideals, meaning that the monomials in \(L_\Lambda (\mathcal {F})\) are precisely those monomials with non-trivial common divisor with every monomial in \(L_\Gamma (\mathcal {I})\), and vice versa.
9 Monomial Ideals
For the case \(Q = {\mathbb N}\) we show that open down-sets in \(\text {Pro}(P,{\mathbb N})\) induce monomial ideals in the polynomial ring \(k[x_P]\). In particular when \(P = {\mathbb N}\), these monomial ideals are precisely the class of strongly stable ideals in the infinite-dimensional polynomial ring \(k[x_{\mathbb N}]\). The results in Sect. 5 then enables defining a duality on a distinguished class of strongly stable ideals, see [9].
Given a map of sets \(A {\mathop {\longrightarrow }\limits ^{\alpha }} B\). Let \(\text {Hom}_{\text {fin}}(A,{\mathbb N}_0)\) be the maps \(A {\mathop {\longrightarrow }\limits ^{u}} {\mathbb N}_0\) such that the support \(\{ a \, | \, u(a) > 0 \}\) is a finite set. Then \(\alpha \) induces a map
Given a map \(\tau : UP \times UQ^\text {op}\,\rightarrow \textbf{2}\) corresponding to a finite subset \(T \subseteq UP \times UQ^\text {op}\,\) (the arguments for which \(\tau \) is non-zero), then \(\tilde{\pi }_1 \circ {\tilde{i}}(\tau )\) is the map sending p to the cardinality of the set of pairs \((p,q) \in T\) with first coordinate p.
Note that the set \(\text {Hom}_{\text {fin}}(UP, {\mathbb N}_0)\) identifies as the set of monomials \(\text {Mon}(x_P)\) in the variables \(x_p\) for \(p \in P\).
Lemma 8.1
Let \(Q = {\mathbb N}\). The composite \(\lambda = \tilde{\pi _1} \circ \tilde{i} \circ \Lambda \):
The up-sets in \(\text {Hom}_{\text {fin}}(UP,{\mathbb N}_0)\) identifies as the monomial ideals in \(k[x_P]\). Recall that open up-sets \(\mathcal {F}\) in \(\text {Pro}(P,{\mathbb N})\) are in bijection with up-sets in \(\text {Pro}_S(P,{\mathbb N})\). So we get an upset \(\mathcal {F}_S\) in the latter. By we get a monomial ideal in \(k[x_P]\).
Application. When \(P = {\mathbb N}\) the monomial ideals one gets are precisely the strongly stable ideals in \(k[x_{\mathbb N}]\). There is a one-one correspondence between open up-sets in \(\text {Pro}({\mathbb N},{\mathbb N})\) and strongly stable ideals in \(k[x_{\mathbb N}]\), see [9].
Definition 8.3
Given a Dedekind cut \([\mathcal {I},\mathcal {F}]\) for \(\text {Pro}({\mathbb N}, {\mathbb N})\). We get an up-set in \(\text {Hom}_{\text {fin}}({\mathbb N},{\mathbb N}_0)\) which corresponds to a monomial ideal \(I_\lambda \) of \(k[x_{\mathbb N}]\). Similarly we get a up-set in \(\text {Hom}_{\text {fin}}({\mathbb N},{\mathbb N}_0)\) which corresponds to a monomial ideal \(I_\gamma \) of \(k[x_{\mathbb N}]\). The ideals \(I_\lambda \) and \(I_\gamma \) are dual strongly stable ideals.
Note that this definition is reasonable. Given \(I_\lambda \) we can reconstruct \(\mathcal {F}_S\) and thereby \(\mathcal {F}\). Since \([\mathcal {I},\mathcal {F}]\) is a Dedekind cut, we can construct \(\mathcal {I}\) from \(\mathcal {F}\) and thereby determine \(I_\gamma \). And of course we may go the opposite way, from \(I_\gamma \) we may determine \(I_\lambda \). Note also that not every strongly stable ideal I has a dual. Only the strongly stable ideals such that the corresponding open down-set \(\mathcal {I}\) is regular, have duals. Versions of this duality for finite-dimensional polynomial rings were first given in [8, Section 7] and independently by [23].
Acknowledgements
We are grateful to an anonymous referee for suggestions concerning notation and pointers to the literature.
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Conflict of interest
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as more effective than the notion of isotone (order preserving) maps \(P \rightarrow Q\) between posets, especially for applications in algebra. When Q is totally ordered, these notions are practically the same, but when Q is not, profunctors seem to have a clear advantage for developing natural theory.
is simply an isotone map \(P \times Q^\text {op}\,\rightarrow \textbf{2}\). If P and Q are sets (discrete posets), then this is simply a relation between P and Q. The notion of profunctor may generally be defined between categories or between categories enriched in a symmetric monoidal closed category (like \(\textbf{2}\)), see [2, 3], or, for a recent gentle introduction focusing on applications, [13, Section 4].
. This is again a a partially ordered set and the opposite of this poset is \(\text {Pro}(Q,P)\). The basic notions we introduce associated to a profunctor
between posets are the notions of its graph \(\Gamma f\) and its ascent \(\Lambda f\). These are dual notions in the sense that if
is the dual profunctor, the graph of f equals the ascent of g. Let UP and UQ denote the underlying sets of P and Q. In \(\text {Pro}(P,Q)\) let \(\mathcal {I}\) be a down-set and \(\mathcal {F}\) its complement up-set, so \((\mathcal {I},\mathcal {F})\) is a cut for \(\text {Pro}(P,Q)\). Let \(\mathcal {F}_\Lambda \) be the up-set in the Boolean lattice of all subsets of \(UQ \times UP^\text {op}\,\) generated by the ascents \(\Lambda f\) for \(f \in \mathcal {F}\). Let \(\mathcal {I}_\Gamma \) be the down-set in this Boolean lattice generated by the complements of the graphs \(\Gamma f\) for \(f \in \mathcal {I}\).
. The blue circles give the ascent of this profunctor. The graphs and the ascents are then subsets of \(U[4] \times U[5]^{\text {op}\,}\). The up-set \(\mathcal {F}_\Lambda \) is the up-set of the Boolean lattice of subsets of \(U[4] \times U[5]^\text {op}\,\) generated by the ascents of \(f \in \mathcal {F}\). The down-set \(\mathcal {I}_\Gamma \) is the down-set of this Boolean lattice generated by the complements of graphs of \(f \in \mathcal {I}\).
are the topic of many our examples. These benefit the reader with a quick access to many of our notions and results: 1.2, 1.4, 2.5, 2.12, 2.13, 3.3, 3.8, 5.10, 6.7, 6.8, 6.9. See also the end of Sect. 8.
(i.e. order preserving maps \(f: {\mathbb N}\rightarrow {\mathbb N}\cup \{ \infty \}\)) to the duality theory of strongly stable ideals in the the infinite polynomial ring \(k[x_i]_{i \in {\mathbb N}}\).
between posets. Such a profunctor has a dual
and we investigate this correspondence. For an introduction to profunctors, see [13, Chap.4]. See also [3, Section 7] and [2] where they are called distributors.
is simply a poset homomorphism \(P \rightarrow \widehat{Q}\). By the adjunctionThus a profunctor is equivalently an isotone map \(P \times Q^\text {op}\,\rightarrow \textbf{2}\) and this is often taken as the definition. It is also equivalently an element of the distributive lattice \((Q \times P^\text {op}\,)^{\widehat{}}\), and so corresponds to a cut (I, F) for \(Q \times P^\text {op}\,\). (Our convention differs somewhat from [13], since there a profunctor
corresponds to an isotone map \(P^\text {op}\,\times Q \rightarrow \textbf{2}\).)
as \(\text {Pro}(P,Q) (= \text {Hom}(P,\widehat{Q}))\). It is itself a partially ordered set, in fact a distributive lattice.
. 
and its dual
. In particular \(f(p) = \infty \) if and only if \(g(q) \le \widehat{p}\) for every \(q \in Q\).
(see Example 1.2) which is an isotone map \(f: {\mathbb N}\rightarrow {\mathbb N}\cup \{\infty \}\). Let its values beIn Fig. 3 the graph of f are marked with red discs 
(which identifies as an isotone \(f: {\mathbb N}\rightarrow \widehat{{\mathbb N}}\)), then \(f(1) \ge 2\) iff the dual map \(g = Df\) has \(g(1) = 1\). Hence there are no self-dual maps f.
is an isotone \(f: P \rightarrow \widehat{Q}\).
be a profunctor. If \(g = Df\) is the dual, the co-profile of f equals the profile of g (by Lemma 2.4a).
. By Example 1.2 recall \(\widehat{{\mathbb N}} = {\mathbb N}\cup \{ \infty \}\). Such a map is large if \(f(n) = \infty \) for some n. It is small if f is eventually constant, so \(f(n) = c\) for all \(n \ge n_0\). The maps in \(\text {Pro}^u({\mathbb N}, {\mathbb N})\) are the maps \(f: {\mathbb N}\rightarrow {\mathbb N}\) which are not bounded, so \(\lim _{n \rightarrow \infty } f(n) = \infty \). We see the naturalness of considering \(\text {Pro}({\mathbb N}, {\mathbb N})\) instead of \(\text {Hom}({\mathbb N}, {\mathbb N})\): The latter is not self-dual while the former is. \(\text {Pro}({\mathbb N}, {\mathbb N})\) has two countable dual “shores” \(\text {Pro}^L({\mathbb N}, {\mathbb N})\) and \(\text {Pro}_S({\mathbb N},{\mathbb N})\) and between them an uncountable self-dual “ocean” \(\text {Pro}^u({\mathbb N},{\mathbb N})\).
where \(f(F)^\uparrow \) is the smallest up-set in Q containing f(F). All these maps are functorial in P and Q.
Here
sends a cut (I, F) of P to the cut (J, G) of Q where \(G = g(F)^\uparrow \) is the up-set of Q generated by the g(p) for \(p \in F\). Note that the above composite
has a left adjoint \(i^! \circ g^*\) (although it will not play a role for us).
, or equivalently the right and left boundaries of the cut (I, F) for \(Q \times P^\text {op}\,\) corresponding to this profunctor. Then we state several versions of the main theorem of this article, Theorem 3.7, on preserving cuts.
be a profunctor. Its ascent isIts graph is
, see Fig. 3, then \(\Gamma f\) is the red discs, and \(\Lambda f\) is the blue circles.
corresponds to a cut (I, F) for \(Q \times P^\text {op}\,\) where
Let (I, F) be the cut for \(Q \times P^\text {op}\,\) associated to f, and (J, G) the cut for \(P \times Q^\text {op}\,\) associated to g. 
So the image of the cut \((\mathcal {I},\mathcal {F})\) is the cut in the Boolean lattice \((UQ \times UP^\text {op}\,)^{\widehat{}}\) whose up-set is \(\Lambda (\mathcal {F})^\uparrow \), the up-set generated by all \(\Lambda f\) for \(f \in \mathcal {F}\).
So the image of the cut \((\mathcal {I},\mathcal {F})\) is the cut in the Boolean lattice \((UQ \times UP^\text {op}\,)^{\widehat{}}\) whose ideal is \(\Gamma (\mathcal {I})^{\downarrow }\), the ideal generated by the complements (see (12)) of all \(\Gamma f\) for \(f \in \mathcal {I}\),
.
generated by all \((\Lambda f,-)\) for \(f \in \mathcal {F}\), and the up-set
generated by all \(((\Gamma f)^\text {op}\,,-)\) for \(f \in \mathcal {I}\), are Alexander dual up-sets. (This is the version applied in Stanley–Reisner theory, giving Alexander dual monomial ideals, see Sect. 4.)
. Then (7) gives \(\Gamma ^{!}_{U}(\mathcal {I},-)^\text {op}\,= (\Gamma ^{\text {op}\,})^!_U(-,\mathcal {I}^{\text {op}\,})\). \(\square \)
In the proof below we write \(\Lambda _{P,Q}\) and \(\Gamma _{Q,P}\) for these horizontal maps.
. For instance we can not see a natural factorization of the above map through \(\text {Pro}(UP, Q)^{\widehat{}}\).
Note that the isotonian ideal L(P, Q) is the squarefree monomial ideal associated to the image by
of the cut \((\emptyset , \text {Hom}(P,Q)^\text {op}\,)\) for \(\text {Hom}(P,Q \widehat{)^{\text {op}\,}}\).
and interior \({}^\circ \), are the regular open sets. We denote these as \(\text {reg}\,X\), the open subsets which are interiors of closed subsets.
, see [9], and briefly indicated in the last Sect. 8.
an isotone map. 
The map \(\Gamma \) induces an isotone map:
We have the following variation of Theorem 3.7.
Note that the vertical map sends a cut (I, F) to a cut \((F^\text {op}\,, I^\text {op}\,)\).
The up-sets in \(\text {Hom}_{\text {fin}}(UP,{\mathbb N}_0)\) identifies as the monomial ideals in \(k[x_P]\). Recall that open up-sets \(\mathcal {F}\) in \(\text {Pro}(P,{\mathbb N})\) are in bijection with up-sets in \(\text {Pro}_S(P,{\mathbb N})\). So we get an upset \(\mathcal {F}_S\) in the latter. By
we get a monomial ideal in \(k[x_P]\).
in \(\text {Hom}_{\text {fin}}({\mathbb N},{\mathbb N}_0)\) which corresponds to a monomial ideal \(I_\lambda \) of \(k[x_{\mathbb N}]\). Similarly we get a up-set
in \(\text {Hom}_{\text {fin}}({\mathbb N},{\mathbb N}_0)\) which corresponds to a monomial ideal \(I_\gamma \) of \(k[x_{\mathbb N}]\). The ideals \(I_\lambda \) and \(I_\gamma \) are dual strongly stable ideals.