Representation of De Morgan and (Semi-)Kleene Lattices

For the related class of Sugihara monoids, a twist-structure representation has been given in Galatos and Raftery (2015). An interesting question for further research is how the twist construction of Galatos and Raftery (2015) is related to (in particular, whether it can be viewed as a special case of) the ones introduced in Sects. 4 and 5.

As observed in Rivieccio and Spinks (2019), these conditions imply \(n(a_+) \wedge _- a_- = 0_-\) for all \(\langle a_+, a_- \rangle \in A\) and \(\pi _2[A] = L_-\). (I shall use these facts, sometimes without warning, in proofs.) The first holds because, since \(a_+ \wedge _+ p(a_-) = 0_+\), we can apply n to both sides of the equation and, using its properties, we obtain \(n(a_+ \wedge _+ p(a_-)) = n(a_+) \wedge _- n(p(a_-)) = n(a_+) \wedge _- a_- = 0_-= n(0_+) \) as required. Likewise, \(\pi _2[A] = L_-\) follows from \(\pi _1[A] = L_+\). In fact, for all \(a_- \in L_-\), we know that \(a_- = n(p(a_-))\), where \(p(a_-) \in L_+\). Then, \(\pi _1[A] = L_+\) guarantees that there is \(b_- \in L_- \) such that \(\langle p(a_-), b_- \rangle \in A\), which means that \(\lnot \langle p(a_-), b_- \rangle = \langle p(b_-), n(p(a_-) \rangle = \langle p(b_-), a_- \rangle \in A\). Thus, \(a_- \in \pi _2[A] \) as required.

I do not know how one could further relax Definition 5.1 so as to include \(\mathbf {A} _4\) as well. On the one hand, this would be desirable, because \(\mathbf {A} _4\)can indeed by represented by a twist-structure, as is easy to check. However, if we delete (SK 5) from Definition 5.1, then it is not clear how to make the twist-structure construction work in general. In this sense, \(\mathbf {A} _4\) is perhaps a fortunate case because, being a chain, it satisfies the “Ockham identity” \(\lnot (x \wedge y ) \approx \lnot x \vee \lnot y \) (see Proposition 5.3). This ensures that \(\mathbf {A} _4\) can be represented as a twist-structure, but of course replacing (SK 5) in Definition 5.1 with the Ockham identity would confine us to a quite specific subclass of semi-De Morgan lattices (i.e., Ockham lattices).

Implicative meet-semilattices are the \(\langle \wedge , \rightarrow \rangle \)-subreducts of Heyting algebras, corresponding to the conjunction–implication fragment of intuitionistic logic. Thus, in particular, the \(\langle \wedge , \rightarrow \rangle \)-reduct of every Heyting algebra is an implicative meet-semilattice.

This is probably the best occasion to point out an unfortunate terminological clash with the previous literature, of which M. Spinks and I were not aware while writing (Rivieccio and Spinks 2019). Cignoli (1986) and Odintsov (2010) in their wake, call a Kleene algebra that satisfies item (i) of Theorem 6.2 a quasi-Nelson algebra. I shall prove in Theorem 6.3 that every quasi-Nelson algebra in the sense of Rivieccio and Spinks (2019) indeed satisfies both items of Theorem 6.2; obviously, however, it need not be involutive. Thus, neither the quasi-Nelson algebras of Cignoli–Odintsov are a special case of ours, nor the other way around.

Indeed, the property of being \(({\mathbf {0}}, {\mathbf {1}})\)-congruence orderable does not seem unrelated to that of being representable by some kind of twist-structure construction, for both essentially rely on a filter separation property. This is quite clear at least in the context of residuated lattices, where filters are in one–one correspondence with congruences; see (Spinks et al. 2018, Section 7). In the quasi-Kleene context, one may observe, for instance, that the quasi-identity (iv) from Proposition 6.14 is essentially saying that two elements must be equal if they generate the same consistent filter and the same total filter...