In this work we provide a new topological representation for implication algebras in such a way that its one-point compactification is the topological space given in [1]. Some applications are given thereof.

Łukasiewicz implication algebras are the {→,1}-subreducts of MV- algebras. They are the algebraic counterpart of Super-Łukasiewicz Implicational Logics investigated in Komori (Nogoya Math J 72:127–133, 1978). In this paper we give a description of free Łukasiewicz implication algebras in the context of McNaughton functions. More precisely, we show that the |X|-free Łukasiewicz implication algebra is isomorphic to ${\bigcup_{x\in X} [x_\theta)}$ for a certain congruence θ over the |X|-free MV-algebra. As corollary we describe the free (...) algebras in all subvarieties of Łukasiewicz implication algebras. (shrink)

Łukasiewicz implication algebras are {→,1}-subreducts of Wajsberg algebras (MV-algebras). They are the algebraic counterpart of Super-Łukasiewicz Implicational logics investigated in Komori, Nogoya Math J 72:127–133, 1978. The aim of this paper is to study the direct decomposability of free Łukasiewicz implication algebras. We show that freely generated algebras are directly indecomposable. We also study the direct decomposability in free algebras of all its proper subvarieties and show that infinitely freely generated algebras (...) are indecomposable, while finitely free generated algebras can be only decomposed into a direct product of two factors, one of which is the two-element implication algebra. (shrink)

AbstractŁukasiewicz implication algebras are {→,1}-subreducts of Wajsberg algebras (MV-algebras). They are the algebraic counterpart of Super-Łukasiewicz Implicational logics investigated in Komori, Nogoya Math J 72:127–133, 1978. The aim of this paper is to study the direct decomposability of free Łukasiewicz implication algebras. We show that freely generated algebras are directly indecomposable. We also study the direct decomposability in free algebras of all its proper subvarieties and show that infinitely freely generated algebras (...) are indecomposable, while finitely free generated algebras can be only decomposed into a direct product of two factors, one of which is the two-element implication algebra. (shrink)

AbstractŁukasiewicz implication algebras are the {→,1}-subreducts of MV- algebras. They are the algebraic counterpart of Super-Łukasiewicz Implicational Logics investigated in Komori (Nogoya Math J 72:127–133, 1978). In this paper we give a description of free Łukasiewicz implication algebras in the context of McNaughton functions. More precisely, we show that the |X|-free Łukasiewicz implication algebra is isomorphic to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bigcup_{x\in X} [x_\theta)}$$\end{document} for a certain congruence θ over (...) the |X|-free MV-algebra. As corollary we describe the free algebras in all subvarieties of Łukasiewicz implication algebras. (shrink)

The fact that it is possible to define three different material conditionals in orthomodular lattices suggests that there exist three different orthomodular logics whose conditionals are material conditionals and whose models are orthomodular lattices. The purpose of this paper is to provide equationally definable implication algebras for each of these material conditionals.

In this paper we study some questions concerning Łukasiewicz implication algebras. In particular, we show that every subquasivariety of Łukasiewicz implication algebras is, in fact, a variety. We also derive some characterizations of congruence permutable algebras. The starting point for these results is a representation of finite Łukasiewicz implication algebras as upwardly-closed subsets in direct products of MV-chains.

In this paper we see that the answer of this question is affirmative. We prove this for Dco-algebras and as special case we obtain the result for Positive Implication algebras. First we give, without proof, the properties of Dco-algebras and S-algebras and their connection with Positive Implication algebras and Implication algebras. These results can be found in [T] and [IT].

In this paper we study some questions concerning Łukasiewicz implication algebras. In particular, we show that every subquasivariety of Łukasiewicz implication algebras is, in fact, a variety. We also derive some characterizations of congruence permutable algebras. The starting point for these results is a representation of finite Łukasiewicz implication algebras as upwardly-closed subsets in direct products of MV-chains.

GE algebras (generalized exchange algebras), transitive GE algebras (tGE algebras, for short) and aGE algebras (that is, GE algebrasverifying the antisymmetry) are a generalization of Hilbert algebras. Here some properties and characterizations of these algebras are investigated. Connections between GE algebras and other classes of algebras of logic are studied. The implicative and positive implicative properties are discussed. It is shown that the class of positive implicative GE algebras (resp. the (...) class of implicative aGE algebras) coincides with the class of generalized Tarski algebras (resp. the class of Tarski algebras). It is proved that for any aGE algebra the property of implicativity is equivalent to the commutative property. Moreover, several examples to illustrate the results are given. Finally, the interrelationships between some classes of implicative and positive implicative algebras are presented. (shrink)

The logic DAI of demodalised analytic implication has been introduced by J.M. Dunn as a variation on a time-honoured logical system by C.I. Lewis’ student W.T. Parry. The main tenet underlying this logic is that no implication can be valid unless its consequent is “analytically contained” in its antecedent. DAI has been investigated both proof-theoretically and model-theoretically, but no study so far has focussed on DAI from the viewpoint of abstract algebraic logic. We provide several different algebraic semantics (...) for DAI, showing their equivalence with the known semantics by Dunn and Epstein. We also show that DAI is algebraisable and we identify its equivalent quasivariety semantics. This class turns out to be a linguistic and axiomatic expansion of involutive bisemilattices, a subquasivariety of which forms the algebraic counterpart of Paraconsistent Weak Kleene logic. This fact sheds further light on the relationship between containment logics and logics of nonsense. (shrink)

This paper combines two classes of generalized logics, one of which is the class of weakly implicative logics introduced by Cintula and the other of which is the class of gaggle logics introduced by Dunn. For this purpose we introduce implicational tonoid logics. More precisely, we first define implicational tonoid logics in general and examine their relation to weakly implicative logics. We then provide algebraic semantics for implicational tonoid logics. Finally, we consider relational semantics, called Routley–Meyer–style semantics, for finitary those (...) logics. (shrink)

We review some of the essential novel ideas introduced by Bohm through the implicate order and indicate how they can be given mathematical expression in terms of an algebra. We also show how some of the features that are needed in the implicate order were anticipated in the work of Grassmann, Hamilton, and Clifford. By developing these ideas further we are able to show how the spinor itself, when viewed as a geometric object within a geometric algebra, can be given (...) a meaning which transcends the notion of the usual metric geometry in the sense that it must be regarded as an element of a broader and more general pregeometry. (shrink)

The notion of positive implicative soju ideal in BCK-algebra is introduced, and several properties are investigated. Relations between soju ideal and positive implicative soju ideal are considered, and characterizations of positive implicative soju ideal are established. Finally, extension property for positive implicative soju ideal is constructed.

It is well-known that intuitionistic logics can be formalized by means of Heyting algebras, i.e. relatively pseudocomplemented semilattices. Within such algebras the logical connectives implication and conjunction are formalized as the relative pseudocomplement and the semilattice operation meet, respectively. If the Heyting algebra has a bottom element |$0$|⁠, then the relative pseudocomplement with respect to |$0$| is called the pseudocomplement and it is considered as the connective negation in this logic. Our idea is to consider an arbitrary (...) meet-semilattice with |$0$| satisfying only the Ascending Chain Condition (these assumptions are trivially satisfied in finite meet-semilattices) and introduce the operators formalizing the connectives negation |$x^{0}$| and implication |$x\rightarrow y$| as the set of all maximal elements |$z$| satisfying |$x\wedge z=0$| and as the set of all maximal elements |$z$| satisfying |$x\wedge z\leq y$|⁠, respectively. Such a negation and implication is ‘unsharp’ since it assigns to one entry |$x$| or to two entries |$x$| and |$y$| belonging to the semilattice, respectively, a subset instead of an element of the semilattice. Surprisingly, this kind of negation and implication still shares a number of properties of these connectives in intuitionistic logic, in particular the derivation rule Modus Ponens. Moreover, unsharp negation and unsharp implication can be characterized by means of five, respectively seven simple axioms. We present several examples. The concepts of a deductive system and of a filter are introduced as well as the congruence determined by such a filter. We finally describe certain relationships between these concepts. (shrink)

It has been proposed that the implicate order can be given mathematical expression in terms of an algebra and that this algebra is similar to that used in quantum theory. In this paper we bring out in a simple way those aspects of the algebraic formulation of quantum theory that are most relevant to the implicate order. By using the properties of the standard ket introduced by Dirac we describe in detail how the Heisenberg algebra can be generalized to produce (...) an algebraic structure in which it is possible to describe space translations in a way that is analogous to the description of rotations in a Clifford algebra. This approach opens up the possibility of going beyond the limits of the present quantum formalism and we discuss briefly some of the new implications. (shrink)

Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well (...) as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets. (shrink)

This monograph presents a general theory of weakly implicative logics, a family covering a vast number of non-classical logics studied in the literature, concentrating mainly on the abstract study of the relationship between logics and their algebraic semantics. It can also serve as an introduction to algebraic logic, both propositional and first-order, with special attention paid to the role of implication, lattice and residuated connectives, and generalized disjunctions. Based on their recent work, the authors develop a powerful uniform framework (...) for the study of non-classical logics. In a self-contained and didactic style, starting from very elementary notions, they build a general theory with a substantial number of abstract results. The theory is then applied to obtain numerous results for prominent families of logics and their algebraic counterparts, in particular for superintuitionistic, modal, substructural, fuzzy, and relevant logics. The book may be of interest to a wide audience, especially students and scholars in the fields of mathematics, philosophy, computer science, or related areas, looking for an introduction to a general theory of non-classical logics and their algebraic semantics. (shrink)

An algebraic semantics, based on factor algebras, for one-way and two-way implicative verbs is proposed. Implicative verbs denote elements of filters or of ideals generated by identity functions in factor algebras. This semantics explains in particular the problem of implicational equivalence raised by two-way implicative verbs, and shows that the negation necessary to establish the implicativity of these verbs is the negation which preserves the presuppositions of sentences with implicative verbs. In addition, it follows from the proposed semantics (...) that any two implicative verbs denoting in the same algebra but belonging to different categories, are semantically related. (shrink)

Orthologic is defined by weakening the axioms and rules of inference of the classical propositional calculus. The resulting Lindenbaum-Tarski quotient algebra is an orthoimplication algebra which generalizes the author's implication algebra. The associated order structure is a semi-orthomodular lattice. The theory of orthomodular lattices is obtained by adjoining a falsity symbol to the underlying orthologic or a least element to the orthoimplication algebra.

It can be said that Formal Logic begun by studying an idealization of the statements ’if p, then q’, something coming from long ago in both Greek and Scholastic Philosophy. Nevertheless, only in the XX Century it arrived at a stage of formalization once in 1910 Russell introduced and identified the ’material conditional’ with the expresion ”not p or q”. In 1918, and from paradoxical conditionals like ”If the Moon is a cheese, it is a Lyon’s face”, Lewis critiziced the (...) material conditional and introduced the so-called ’strict conditional’ as the modal necessity of the material one. In 1934, Huntington proved that both material and strict implications are coincidental in the setting of Boolean algebras and, hence, that they only can be actually different in algebraic structures weaker than Boolean algebras. Boolean algebras have too much laws for supporting the difference of both conditionals. This paper is nothing else than an algebraic trial to find simple binary Boolean operations able to express Lewis’ strict implication, and contains a new and simpler proof than that of Huntington, made by identifying ’possible’ with ’non self-contradictory’. By the way, it is shown that this proof is only valid in Boolean algebras, but neither in proper De Morgan algebras, nor in proper Ortholattices, with which it still remains open where the two conditional relations are actually different. (shrink)

It is shown that in a linearly ordered MV-algebra A , the implication is unique if and only if the identity function is the unique De Morgan automorphism on A . Modulo categorical equivalence, our uniqueness criterion recalls Ohkuma's rigidness condition for totally ordered abelian groups. We also show that, if A is an Archimedean totally ordered MV-algebra, then each non-trivial De Morgan automorphism of the underlying involutive lattice of A yields a new implication on A , which (...) is not isomorphic to the original implication. (shrink)

In the paper, pre-Hilbert algebras are defined as a generalization of Hilbert algebras (namely, a Hilbert algebra is just a pre-Hilbert algebra satisfying the property of antisymmetry). Pre-Hilbert algebras have been inspired by Henkin's Positive Implicative Logic. Their properties and characterizations are investigated. Some important results and examples are given. Moreover, positive implicative pre-Hilbert algebras are introduced and studied, their connections with some algebras of logic are presented. The hierarchies existing between the classes of (...) class='Hi'>algebras considered here are shown. (shrink)