A Course in BE-algebras

Residuation is one of the most important concepts of the theory of ordered algebraic structures which naturally arises in many other fields of mathematics. The study of abstract residuated structures has originated from the investigation of ideal lattices of commutative rings with 1. In general, a partially ordered monoid is residuated if for all a, b in its universe there exist \(a\rightarrow b = \max \{c: ca\le b\}\) and \(a\rightsquigarrow b = \max \{c:ac\le b\}\), and in other words, if for every a the translations \(x\rightarrow xa\) and \(x\rightsquigarrow ax\) are residuated mappings. If the multiplicative identity is the greatest element in the underlying order, then the monoid is integral. Residuation structures include lattice order groups and their negative cones as well as algebraic models of various propositional logics. In the logical context, the monoid operation \(\cdot \) can be interpreted as conjunction and the residuals \(\rightarrow \) and \(\rightsquigarrow \) as two implications (they coincide if and only if the conjuncture is commutative).

Preliminary pages include everything up to the main body of the text or introduction. In this chapter, a few important definitions and results are collected from various sources for the use in the forthcoming chapters. Some basic and important properties of BE-algebras, BCK-algebras, some special classes of BE-algebras like self-distributive, commutative, transitive, implicative, and ordered BE-algebras are observed. Some preliminary properties of ordered relations, partially ordered sets and congruences, are quoted for further reference to the reader.

BE-algebras are important tools for certain investigations in algebraic logic since they can be considered as fragments of any propositional logic containing a logical connective implication and the constant 1 which is considered as the logical value “true.” The notion of BE-algebras was introduced and extensively studied by H.S. Kim and Y.H. Kim in (Sci Math Jpn, Online e-2006, 1299–1302, [153]). These classes of BE-algebras were introduced as a generalization of the class of BCK-algebras of K. I\(\acute{s}\)eki and S. Tanaka (Math Jpn 23(1):1–26, 1979, [124]). I. Chajda and J. Kuhr (Miskolc Math Notes 8(1):11–21, 2007, [38]) studied the algebraic structure derived from a BCK-algebra. They viewed commutative BCK-algebras as semilattices whose sections have antitone involutions, and it is known that bounded commutative BCK-algebras are equivalent to MV-algebras. In (Rezaei and Saeid, Afrika Matematika 22(2):115–127, 2011, [202]), A. Rezaei and A.B. Saeid introduced the concept of fuzzy subalgebras of BE-algebras and studied its nature. They stated and proved the Foster’s results on homomorphic images and inverse images in fuzzy topological BE-algebras.

In a BE-algebra, filters are important substructures which play an important role in the characterization of many special classes of BE-algebras. Also, as it is well known that filters are exactly the kernels of congruences, many authors tried to define various filters of BE-algebras in order to construct quotient BE-algebras and investigate some of their properties.

Inspired by the considerations of Zadeh (Synthesis, 30:407–428, 1975, [252]), Hajek in (Fuzzy Sets and Systems, 124:329–333, 2001, [105]) formalized the fuzzy truth-value very true. He enriched the language of the basic fuzzy logic BL by adding a new unary connective vt and introduced the propositional logic \(BL_{vt}\). The completeness \(BL_{vt}\) was proved in Liu and Wang (On v-filters of commutative residuated lattices with weak vt-operators, 2009, [168]) by using the so-called \(BL_{vt}\)-algebra, an algebraic counterpart of \(BL_{vt}\). In 2006, Vychodil (Fuzzy sets and systems, 157:2074–2090, 2006, [237]) proposed an axiomatization of unary connectives like slightly true and more or less true and introduced \(BL_{vt, st}\)-logic which extends \(BL_{vt}\)-logic by adding a new unary connective “slightly true” denoted by “st.” Noting that bounded commutative \(R\ell \)-monoids are algebraic structures which generalize, e.g., both BL-algebras and Heyting algebras (an algebraic counterpart of the intuitionistic propositional logic), Rachunek and Salounova taken bounded commutative \(R\ell \)-monoids with a vt-operator as an algebraic semantics of a more general logic than Hajeks fuzzy logic and studied algebraic properties of \(R\ell _{vt}\)-monoids in Rachunek (Soft Comput, 15:327–334, 2011, [196]).

In mathematics, particularly in order theory, a pseudo-complement is one generalization of the notion of complement. In a lattice L with bottom element 0, an element \(x\in L\) is said to have a pseudo-complement if there exists a greatest element \(x^{*}\in L\), disjoint from x, with the property that \(x\wedge x^{*} = 0\). More formally, \(x^{*} = \max \{y\in L~|~x\wedge y = 0\}\). The lattice L itself is called a pseudo-complemented lattice if every element of L is pseudo-complemented. Every pseudo-complemented lattice is necessarily bounded; i.e., it has a 1 as well. Since the pseudo-complement is unique by definition (if it exists), a pseudo-complemented lattice can be endowed with a unary operation * mapping every element to its pseudo-complement. The theory of pseudo-complements in lattices, and particularly in distributive lattices, was developed by M.H. Stone (Trans Am Math Soc 40:37–111, 1936), [228], O. Frink (Duke Math J 29:505–514, 1962), [97], and G. Gratzer (General Lattice Theory, Academic Press, New York 1978), [103]. Later many authors like R. Balbes (Distributive Lattices, University of Missouri Press, Columbia, 1974), [12], O. Frink (Duke Math J 29:505–514, 1962), [97] extended the study of pseudo-complements to characterize Stone lattices. In 2013, Cilo\(\breve{g}\)lu and Ceven (Algebra, 1–5, 2013), [53] studied the properties of the elements \(x*0\) in a commutative and bounded BE-algebras. Recently in 2014, R. Borzooei et. al (J Math Appl 37:13–26, 2014), [23] studied some structural properties of bounded and involutory BE-algebras and investigate the relationship between them.

A stabilizer is a part of an algebra acting on a set. Specifically, let X be any algebra operating on a set X and let A be a subset of X. The stabilizer of A, sometimes denoted St(A), is the set of elements a of A for which \(a(S)\subseteq S\). The strict stabilizer is the set of \(a\in A\) for which \(a(A) = A\). In the other words, the stabilizer of A is the transporter of A to itself. The concept of stabilizers is introduced in Hilbert algebras by I. Chajda and R. Hala\(\check{s}\) (Mult. Valued Logic 8:139–148, 2002), [37]. In this paper, the authors studied the properties of stabilizers and relative stabilizers of a given subset of a Hilbert algebra. They proved that every stabilizer of a deductive system C of \(\mathcal {H}\) is also a deductive system which is a pseudo-complement of C in the lattice of all deductive systems of \(\mathcal {H}\). In (Borumand et al., in Sci. Bull. Ser. A 74(2):65–74, 2012), [15], A. Borumand Saeid and N. Mohtashamnia constructed quotient of residuated lattices via stabilizer and studied its properties. L. Torkzadeh (Math Sci 3(2):111–132, 2009), [232] introduced dual right and dual left stabilizers in bounded BCK-algebras and investigated the relationship between of them.

Many new fields of science require a probability theory based on non-classical logics. We know multiple-valued logics are non-classical logics and became popular in computer science since it was understood that they play a fundamental role in fuzzy logics. In analogous to probability measure, the states on multiple-valued algebras proved to be the most suitable models for averaging the truth-value in their corresponding logics. Mundici introduced states (an analog of probability measures) on MV-algebras in 1995, as averaging of the truth-value in Mundici (Studia Logica, 55:113–127, 1995) [184]. Since middle 1990s, mainly after Mundici’s paper (Mundici in Studia Logica, 55:113–127, 1995, [184]), on probability theory on MV-algebras, there has been an increasing amount of study on generalizations of probability theoretical concepts, most notable states, on various logic origin algebraic structures. In (Borzooei et al., in Kochi Math, 9:27–42, 2014, [24]), R. Borzooei et al. studied the states on BE-algebras. Bosbach state was introduced by R. Bosbach in (Axiomatic und Arithmetic, Fundamenta Mathe maticae 64:257–287, 1969), [25] and (Kongruenzen and Quotiente, Fundamenta Math ematicae 69:1-14, 1970, [26]). The notion of a Bosbach state has been studied for other algebras of fuzzy structures such as pseudo BL-algebras (Georgescu in Boshbatch states on fuzzy structures, Soft Comput, 8:217–230, 2004, [99]), bounded non-commutative \(R\mathfrak {l}\)-monoids (Dvurecenskij and Rachunek in Discrete Math 306:1317–1326, 2006, [90]), (Dvurecenskij and Rachunek in Semigroup Forum 56:487–500, 2006, [91]), residuated lattices (Ciungu in Appl Funct Anal 2:175–188, 2002 [55]), pseudo BCK-semilattices, and pseudo BCK-algebras (Kuhr in Pseudo-BCK-algebras and related structures, Univerzite Palackeho Olomouci, 2007, [163]). In (Busneag in Math Comp Sci Ser, 37:58–64, 2010, [31]), C. Busneag developed the theory of state-morphisms on Hilbert algebras and got some results relative to the theory of Bosbach states on bounded and non-bounded Hilbert algebras.

Flaminio and Montagna were the first to present a unified approach to states and probabilistic many-valued logic in a logical and algebraic setting (Flaminio and Montagna, Inter J Approx Reason, 50:138–152, 2009, [95]). They added a unary operation, called internal state or state operator to the language of MV-algebras which preserves the usual properties of states. A more powerful type of logic can be given by algebraic structures with internal states, and they are also very interesting varieties of universal algebras. Di Nola and Dvurecenskij introduced the notion of a state-morphism MV-algebra which is a stronger variation of a state MV-algebra (Di Nola and Dvurecenskij, Ann Pure Appl Logic 161:161–173, 2009, [65]). The notion of a state operator was extended by Rachunek and Salounova in (Soft Comput 15:327–334, 2011, [196]) for the case of GMV-algebras (pseudo MV-algebras). State operators and state-morphism operators on BL-algebras were introduced and investigated in Ciungu et al. (Soft Comput 15:619–634, 2011, [57]), and subdirectly irreducible state-morphism BL-algebras were studied in Dvurecenskij (Arch Math Logic 50:145–160, 2011, [89]). Recently, the state BCK-algebras and state-morphism BCK-algebras were defined and studied in Borzooei et al. (State BCK-algebras and state-morphism BCK-algebras, 2013, [22]).

Self-mappings of special kinds are the subjects of many important theories: see, for instance, Lie group, mapping class group, permutation group. In category theory, “map” is often used as a synonym for morphism or arrow, thus for something more general than a function. In formal logic, the term map is sometimes used for a functional predicate, whereas a function is a model of such a predicate in set theory.

The term endomorphism is derived from the Greek adverb endon (“inside”) and morphosis (“to form” or “to shape”). In an algebra, an endomorphism of a group, module, ring, vector space, etc., is a homomorphism from the algebra to itself (with surjectivity not required). In 2001, Sergio Celani (Int J Math Math Sci, 29(1):55–61, 2002) [34] gave a representation theorem for Hilbert algebras by means of ordered sets and characterized the homomorphisms of Hilbert algebras in terms of applications defined between the sets of all irreducible deductive systems of the associated algebras. In [11], Chul Kon Bae (J Korea Soc Math Edu, 24(1):7–10, 1985) investigated some properties on homomorphisms in BCK-algebras. In his paper, he mainly studied the properties of the compositions of homomorphisms of BCK-algebras. In [46], Z. Chen, Y. Huang and E.H. Roh (Comm Korean Math Soc, 10(3):499–518, 1995) considered the centralizer C(S) of a given set with respect to the semigroup End(X) of all endomorphisms of an implicative BCK-algebras X with the condition (S). They obtained a series of interesting results those indicated the embedding of X into the centralizer C(S).

It is well-known that an important task of the artificial intelligence is to make computer simulate human being in dealing with certainty and uncertainty in information. Logic gives a technique for laying the foundations of this task. Information processing dealing with certain information is based on the classical logic. Non-classical logic includes many valued logic and fuzzy logic which takes the advantage of the classical logic to handle information with various facets of uncertainty (Zadeh in Inform Sci, 172:1–40, 2005, [253]), such as fuzziness and randomness. Therefore, non-classical logic has become a formal and useful tool for computer science to deal with fuzzy information and uncertain information.

In 2001, Turunen [233] proposed the notions of implicative filters and Boolean filters (Boolean deductive systems) and proved that implicative filters are equivalent to Boolean filters in BL-algebras. Boolean filters are important filters, because the quotient algebras induced by Boolean filters are Boolean algebras. Jun and Ahn [132] provided several degrees in defining a fuzzy implicative filter. Meng [171] introduced the concept of implicative ideals in BCK-algebras and investigated the relationship of it with the concepts of positive implicative ideals and commutative ideals. Meng et al. [179] and Mostafa [182] fuzzified the concept of implicative ideals in BCK-algebras independently. Also, Xu and Qin [245] proposed the notions of implicative filters and fuzzy implicative filters of lattice implication algebras.

It is a general observation in a BE-algebra that the BE-ordering \(\le \) is reflexive but neither antisymmetric nor transitive. If the BE-algebra X is commutative, then the pair \((X, \le )\) is a partially ordered set. If X is transitive, then \(\le \) satisfies only the transitive property. This is the exact reason to concentrate on the transitive property of the filters and to introduce the filters called transitive filters. It is also observed that every filter of a BE-algebra satisfies the transitive property whenever the BE-algebra is transitive. Along with the transitive property of filters, the distributive property is also studied in BE-algebras and introduced the notion of distributive and strong distributive filters in BE-algebras. It is also observed that every filter of a BE-algebra satisfies the distributive property whenever the BE-algebra is self-distributive.