Inspired by the definition of tense operators on distributive lattices presented by Chajda and Paseka in 2015, in this paper, we introduce and study the variety of tense distributive lattices with implication and we prove that these are categorically equivalent to a full subcategory of the category of tense centered Kleene algebras with implication. Moreover, we apply such an equivalence to describe the congruences of the algebras of each variety by means of tense 1-filters (...) and tense centered deductive systems, respectively. (shrink)

The present paper introduces and studies the variety WH of weakly Heyting algebras. It corresponds to the strict implication fragment of the normal modal logic K which is also known as the subintuitionistic local consequence of the class of all Kripke models. The tools developed in the paper can be applied to the study of the subvarieties of WH; among them are the varieties determined by the strict implication fragments of normal modal logics as well as varieties that (...) do not arise in this way as the variety of Basic algebras or the variety of Heyting algebras. Apart from WH itself the paper studies the subvarieties of WH that naturally correspond to subintuitionistic logics, namely the variety of R-weakly Heyting algebras, the variety of T-weakly Heyting algebras and the varieties of Basic algebras and subresiduated lattices. (shrink)

In this paper we introduce and study a variety of algebras that properly includes integral distributive commutative residuated lattices and weak Heyting algebras. Our main goal is to give a characterization of the principal congruences in this variety. We apply this description in order to study compatible functions.

In this paper we introduce and study a variety of algebras that properly includes integral distributive commutative residuated lattices and weak Heyting algebras. Our main goal is to give a characterization of the principal congruences in this variety. We apply this description in order to study compatible functions.

In a previous work we studied, from the perspective ofAlgebraic Logic, the implicationless fragment of a logic introduced by O. Arieli and A. Avron using a class of bilattice-based logical matrices called logical bilattices. Here we complete this study by considering the Arieli-Avron logic in the full language, obtained by adding two implication connectives to the standard bilattice language. We prove that this logic is algebraizable and investigate its algebraic models, which turn out to be distributive bilattices with (...) additional implication operations. We axiomatize and state several results on these new classes of algebras, in particular representation theorems analogue to the well-known one for interlaced bilattices. (shrink)

This paper focuses on order-preserving logics defined from varieties of distributive lattices with negation, and in particular on the problem of whether these can be axiomatized by means Hilbert-style calculi that are finite. On the negative side, we provide a syntactic condition on the equational presentation of a variety that entails failure of finite axiomatizability for the corresponding logic. An application of this result is that the logic of all distributive lattices with negation is not finitely (...) axiomatizable; we likewise establish that the order-preserving logic of the variety of all Ockham algebras is also not finitely axiomatizable. On the positive side, we show that an arbitrary subvariety of semi-De Morgan algebras is axiomatized by a finite number of equations if and only if the corresponding order-preserving logic is axiomatized by a finite Hilbert-style calculus. This equivalence also holds for every subvariety of a Berman variety of Ockham algebras. We obtain, as a corollary, a new proof that the implication-free fragment of intuitionistic logic is finitely axiomatizable, as well as a new corresponding Hilbert-style calculus. Our proofs are constructive in that they allow us to effectively convert an equational presentation of a variety of algebras into a Hilbert-style calculus for the corresponding order-preserving logic, and vice versa. We also consider the assertional logics associated to the above-mentioned varieties, showing in particular that the assertional logics of finitely axiomatizable subvarieties of semi-De Morgan algebras are finitely axiomatizable as well. (shrink)

The logic of (commutative integral bounded) residuated lattices is known under different names in the literature: monoidal logic [26], intuitionistic logic without contraction [1], H BCK [36] (nowadays called by Ono), etc. In this paper we study the -fragment and the -fragment of the logical systems associated with residuated lattices, both from the perspective of Gentzen systems and from that of deductive systems. We stress that our notion of fragment considers the full consequence relation admitting hypotheses. It (...) results that this notion of fragment is axiomatized by the rules of the sequent calculus for the connectives involved. We also prove that these deductive systems are non-protoalgebraic, while the Gentzen systems are algebraizable with equivalent algebraic semantics the varieties of pseudocomplemented (commutative integral bounded) semilatticed and latticed monoids, respectively. All the logical systems considered are decidable. (shrink)

The well-known logic first degree entailment logic (FDE), introduced by Belnap and Dunn, is defined with |$\wedge $|⁠, |$\vee $| and |$\sim $| as the sole primitive connectives. The aim of this paper is to establish the lattice formed by the class of all 4-valued C-extending implicative expansions of FDE verifying the axioms and rules of Routley and Meyer’s basic logic B and its useful disjunctive extension B|$^{\textrm {d}}$|⁠. It is to be noted that Boolean negation (so, classical propositional (...) logic) is definable in the strongest element in the said class. (shrink)

Lattice BCK logic is the expansion of the well known Meredith implicational logic BCK expanded with lattice conjunction and disjunction. Although its natural axiomatization has three rules named modus ponens, ∨‐rule and ∧‐rule, we show that we can give an equivalent presentation with just modus ponens and ∧‐rule, however it is impossible to obtain an equivalent presentation with modus ponens as unique rule. In this paper we study and characterize all axiomatic extensions of lattice BCK logic (...) class='Hi'>with modus ponens as unique rule. We obtain an infinite chain of proper axiomatic extensions with this property. Moreover, we prove that there is no weakest axiomatic extension of Lattice BCK‐logic admitting modus ponens as unique rule. (shrink)

A topological duality is presented for a wide class of lattice-ordered structures including lattice-ordered groups. In this new approach, which simplifies considerably previous results of the author, the dual space is obtained by endowing the Priestley space of the underlying lattice with two binary functions, linked by set-theoretical complement and acting as symmetrical partners. In the particular case of l-groups, one of these functions is the usual product of sets and the axiomatization of the dual space is given by (...) very simple first-order sentences, saying essentially that both functions are associative and that the space is a residuated semigroup with respect to each of them. (shrink)

The best known algebraizable logics with a conjunction and an implication have the property that the conjunction defines a meet semi-lattice in the algebras of their algebraic counterpart. This property makes it possible to associate with them a semi-lattice based deductive system as a companion. Moreover, the order of the semi-lattice is also definable using the implication. This makes that the connection between the properties of the logic and the properties of its semi-lattice based companion is (...) strong. We introduce a class of algebraizable deductive systems that includes those systems, and study some of their properties and of their semi-lattice based companions. We also study conditions which, when satisfied by a deductive system in the class, imply that it is strongly algebraizable. This brings some information on the open area of research of Abstract Algebraic Logic which consists in finding interesting characterizations of classes of algebraizable logics that are strongly algebraizable. (shrink)

In this paper we study the variety \(\textsf{WL}\) of bounded distributive lattices endowed with an implication, called weak Lewis distributive lattices. This variety corresponds to the algebraic semantics of the \(\{\vee,\wedge,\Rightarrow,\bot,\top \}\) -fragment of the arithmetical base preservativity logic \(\mathsf {iP^{-}}\). The variety \(\textsf{WL}\) properly contains the variety of bounded distributive lattices with strict implication, also known as weak Heyting algebras. We introduce the notion of WL-frame and we prove a representation theorem for (...) WL-lattices by means of WL-frames. We extended this representation to a topological duality by means of Priestley spaces endowed with a special neighbourhood relation between points and closed upsets of the space. These results are applied in order to give a representation and a topological duality for the variety of weak Heyting–Lewis algebras, i.e., for the algebraic semantics of the arithmetical base preservativity logic \(\textsf{iP}^{-}\). (shrink)

In this paper, we define the notion of PC-lattice, as a generalization of finite positive implicative BCK-algebras with condition and bounded commutative BCK-algebras. We investiate some results for Pc-lattices being a new class of BCK-lattices. Specially, we prove that any Boolean lattice is a PC-lattice and we show that if X is a PC-lattice with condition S, then X is an involutory BCK-algebra if and only if X is a commutative BCK-algebra. Finally, we prove that any (...) PC-lattice with condition is a distributive BCK-algebra. (shrink)

A hemi-implicative lattice is an algebra \((A,\wedge,\vee,\rightarrow,1)\) of type (2, 2, 2, 0) such that \((A,\wedge,\vee,1)\) is a lattice with top and for every \(a,b\in A\), \(a\rightarrow a = 1\) and \(a\wedge (a\rightarrow b) \le b\). A new variety of hemi-implicative lattices, here named sub-Hilbert lattices, containing both the variety generated by the \(\{\wedge,\vee,\rightarrow,1\}\) -reducts of subresiduated lattices and that of Hilbert lattices as proper subvarieties is defined. It is shown that any sub-Hilbert lattice is (...) determined (up to isomorphism) by a triple (_L_, _D_, _S_) which satisfies the following conditions: _L_ is a bounded distributive lattice, _D_ is a sublattice of _L_ containing 0, 1 such that for each \(a, b \in L\) there is an element \(c \in D\) with the property that for all \(d \in D\), \(a \wedge d \le b\) if and only if \(d \le c\) (we write \(a \rightarrow _D b\) for the element _c_), and _S_ is a non void subset of _L_ such that _S_ is closed under \(\rightarrow _D\) and _S_, with its inherited order, is itself a lattice. Finally, the congruences of sub-Hilbert lattices are studied. (shrink)

If $\mathbf{S}$ is a semilattice with operators, then there is an implicational theory $\mathscr{Q}$ such that the congruence lattice $\operatorname{Con}$ is isomorphic to the lattice of all implicational theories containing $\mathscr{Q}$.

Automated reasoning issues are addressed for a finite lattice-valued propositional logic LnP(X) with truth-values in a finite lattice-valued logical algebraic structure—lattice implication algebra. We investigate extended strategies and rules from classical logic to LnP(X) to simplify the procedure in the semantic level for testing the satisfiability of formulas in LnP(X) at a certain truth-value level α (α-satisfiability) while keeping the role of truth constant formula played in LnP(X). We propose a lock resolution method at a certain truth-value level (...) α (α-lock resolution) in LnP(X) and have proved its theorems of soundness and weak completeness, respectively. We provide more efficient resolution based automated reasoning in LnP(X) and key supports for α-resolution-based automated reasoning approaches and algorithms in lattice based linguistic truth-valued logic. (shrink)

This is the continuation of the paper :417–446, 2010). We continue the abstract study of non-classical logics based on the kind of generalized implication connectives they possess and we focus on semilinear logics, i.e. those that are complete with respect to the class of models where the implication defines a linear order. We obtain general characterizations of semilinearity in terms of the intersection-prime extension property, the syntactical semilinearity metarule and the class of finitely subdirectly irreducible models. Moreover, (...) we consider extensions of the language with lattice connectives and generalized disjunctions, study their interplay with implication and obtain axiomatizations and further descriptions of semilinear logics in terms of disjunctions and the proof by cases property. (shrink)

This paper discusses Crawley completions of residuated lattices. While MacNeille completions have been studied recently in relation to logic, Crawley completions (i.e. complete ideal completions), which are another kind of regular completions, have not been discussed much in this relation while many important algebraic works on Crawley completions had been done until the end of the 70’s. In this paper, basic algebraic properties of ideal completions and Crawley completions of residuated lattices are studied first in their conncetion (...) class='Hi'>with the join infinite distributivity and Heyting implication. Then some results on algebraic completeness and conservativity of Heyting implication in substructural predicate logics are obtained as their consequences. (shrink)

We study effectively inseparable prelattices $\wedge, \vee$ are binary computable operations; ${ \le _L}$ is a computably enumerable preordering relation, with $0{ \le _L}x{ \le _L}1$ for every x; the equivalence relation ${ \equiv _L}$ originated by ${ \le _L}$ is a congruence on L such that the corresponding quotient structure is a nontrivial bounded lattice; the ${ \equiv _L}$ -equivalence classes of 0 and 1 form an effectively inseparable pair of sets). Solving a problem in we show, that (...) if L is an e.i. prelattice then ${ \le _L}$ is universal with respect to all c.e. preordering relations, i.e., for every c.e. preordering relation R there exists a computable function f reducing R to ${ \le _L}$, i.e., $xRy$ if and only if $f\left{ \le _L}f\left$, for all $x,y$. In fact ${ \le _L}$ is locally universal, i.e., for every pair $a{ < _L}b$ and every c.e. preordering relation R one can find a reducing function f from R to ${ \le _L}$ such that the range of f is contained in the interval $\left\{ {x:a{ \le _L}x{ \le _L}b} \right\}$. Also ${ \le _L}$ is uniformly dense, i.e., there exists a computable function f such that for every $a,b$ if $a{ < _L}b$ then $a{ < _L}f\left{ < _L}b$, and if $a{ \equiv _L}a\prime$ and $b{ \equiv _L}b\prime$ then $f\left{ \equiv _L}f\left$. Some consequences and applications of these results are discussed: in particular for $n \ge 1$ the c.e. preordering relation on ${{\rm{\Sigma }}_n}$ sentences yielded by the relation of provable implication of any c.e. consistent extension of Robinson’s system R or Q is locally universal and uniformly dense; and the c.e. preordering relation yielded by provable implication of any c.e. consistent extension of Heyting Arithmetic is locally universal and uniformly dense. (shrink)

N4-lattices provide algebraic semantics for the logic N4, the paraconsistent variant of Nelson's logic with strong negation. We obtain the representation of N4-lattices showing that the structure of an arbitrary N4-lattice is completely determined by a suitable implicative lattice with distinguished filter and ideal. We introduce also special filters on N4-lattices and prove that special filters are exactly kernels of homomorphisms. Criteria of embeddability and to be a homomorphic image are obtained for N4-lattices in (...) terms of the above mentioned representation. Finally, subdirectly irreducible N4-lattices are described. (shrink)

With a certain graphic interpretation in mind, we say that a function whose value at every point in its domain is a nonempty set of real numbers is an Abacus. It is shown that to every collection C of abaci there corresponds a logic, called an abacus logic, i.e., a certain set of propositions partially ordered by generalized implication. It is also shown that to every collection C of abaci there corresponds a theory JC in a classical propositional (...) calculus such that the abacus logic determined by C is isomorphic to the poset of JC. Two examples are given. In both examples abacus logic is a lattice in which there happens to be an operation of orthocomplementation. In the first example abacus logic turns out to be the Lindenbaum algebra of JC. In the second example abacus logic is a lattice isomorphic to the ortholattice of subspaces of a Hilbert space. Thus quantum logic can be regarded as an abacus logic. Without suggesting "hidden variables" it is finally shown that the Lindenbaum algebra of the theory in the second example is a subalgebra of the abacus logic B of the kind studied in example 1. It turns out that the "classical observables" associated with B and the "quantum observables" associated with quantum logic are not unrelated. The value of a classical observable contains, in coded form, information about the "uncertainty" of a quantum observable. This information is retrieved by decoding the value of the corresponding classical observable. (shrink)

In a preceding paper [1] it was shown that quantum logic, given by the tableaux-calculus Teff, is complete and consistent with respect to the dialogic foundation of logics. Since in formal dialogs the special property of the 'value-definiteness' of propositions is not postulated, the calculus $T_{eff}$ represents a calculus of effective (intuitionistic) quantum logic. Beginning with the tableaux-calculus the equivalence of $T_{eff}$ to calculi which use more familiar figures such as sequents and implications can be investigated. In this (...) paper we present a sequents-calculus of Gentzen-type and a propositional calculus of Brouwer-type which are shown to be equivalent to $T_{eff}$ . The effective propositional calculus provides an interpretation for a lattice structure, called quasi-implicative lattice. If, in addition, the value-definiteness of quantum mechanical propositions is postulated, a propositional calculus is obtained which provides an interpretation for a quasi-modular orthocomplemented lattice which, as is well-known, has as a model the lattice of subspaces of a Hilbert space. (shrink)

This paper is devoted to the investigation of term-definable connexive implications in substructural logics with exchange and, on the semantical perspective, in sub-varieties of commutative residuated lattices (FL ${}_{\scriptsize\mbox{e}}$ -algebras). In particular, we inquire into sufficient and necessary conditions under which generalizations of the connexive implication-like operation defined in [6] for Heyting algebras still satisfy connexive theses. It will turn out that, in most cases, connexive principles are equivalent to the equational Glivenko property with respect to (...) Boolean algebras. Furthermore, we provide some philosophical upshots like, e.g., a discussion on the relevance of the above operation in relationship with G. Polya’s logic of plausible inference, and some characterization results on weak and strong connexivity. (shrink)

We present dualities for implicative and residuated lattices. In combination with our recent article on a discrete duality for lattices with unary modal operators, the present article contributes in filling in a gap in the development of Orłowska and Rewitzky’s research program of discrete dualities, which seemed to have stumbled on the case of non-distributive lattices with operators. We discuss dualities via truth, which are essential in relating the non-distributive logic of two-sorted frames (...) class='Hi'>with their sorted, residuated modal logic, as well as full Stone duality for residuated lattices. Our results have immediate applications to the semantics of related substructural logical calculi. (shrink)

A $\Pi^0_1$ class can be defined as the set of infinite paths through a computable tree. For classes $P$ and $Q$, say that $P$ is Medvedev reducible to $Q$, $P \leq_M Q$, if there is a computably continuous functional mapping $Q$ into $P$. Let $\mathcal{L}_M$ be the lattice of degrees formed by $\Pi^0_1$ subclasses of $2^\omega$ under the Medvedev reducibility. In "Non-branching degrees in the Medvedev lattice of $\Pi \sp{0}\sb{1} classes," I provided a characterization of nonbranching/branching and a classification of (...) the nonbranching degrees. In this paper, I present a similar classification of the branching degrees. In particular, $P$ is separable if there is a clopen set $C$ such that $P \cap C \neq \emptyset \neq P \cap C^c$ and $P \cap C \perp_M P \cap C^c$. By the results in the first paper, separability is an invariant of a Medvedev degree and a degree is branching if and only if it contains a separable member. Further define $P$ to be hyperseparable if, for all such $C$, $P \cap C \perp_M P \cap C^c$ and totally separable if, for all $X,Y \in P$, $X \perp_T Y$. I will show that totally separable implies hyperseparable implies separable and that the reverse implications do not hold, that is, that these are three distinct types of branching degrees. Along the way I will show some related results and present a combinatorial framework for constructing $\Pi^0_1$ classes with priority arguments. (shrink)

We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the given (...) signature; if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation; in an appendix, we further extend this representation to lattices with implications. Finally, we discuss adding to our logic a conditional obeying only introduction and elimination rules, interpreted as a modality using a family of accessibility relations. (shrink)

A non-classical subsystem of orthomodular quantum logic is proposed. This system employs two basic operations: the Sasaki hook as implication and the _and-then_ operation as conjunction. These operations successfully satisfy modus ponens and the deduction theorem. In other words, they form an adjunction in terms of category theory. Two types of semantics are presented for this logic: one algebraic and one physical. The algebraic semantics deals with orthomodular lattices, as in traditional quantum logic. The physical semantics is (...) given as a procedure for deriving a final segment of a series of yes-no experiments. (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)

Uncertainty natural language processing has always been a research focus in the artificial intelligence field. In this paper, we continue to study the linguistic truth-valued concept lattice and apply it to the disease intelligent diagnosis by building an intelligent model to directly handle natural language. The theoretical bases of this model are the classical concept lattice and the lattice implication algebra with natural language. The model includes the case library formed by patients, attributes matching, and the matching degree (...) calculation about the new patient. According to the characteristics of the patients, the disease attributes are firstly divided into intrinsic invariant attributes and extrinsic variable attributes. The calculation algorithm of the linguistic truth-valued formal concepts and the constructing algorithm of the linguistic truth-valued concept lattice based on the extrinsic attributes are proposed. And the disease bases of the different treatments for different patients with the same disease are established. Secondly, the matching algorithms of intrinsic attributes and extrinsic attributes are given, and all the linguistic truth-valued formal concepts that match the new patient’s extrinsic attributes are found. Lastly, by comparing the similarity between the new patients and the matching formal concepts, we calculate the best treatment options to realize the intelligent diagnosis of the disease. (shrink)

Codes of conduct have proliferated not only at company level, but also at supra-and suborganisational levels. However, the latter have remained an under-researched area within the CSR literature. Hence, this article examined what range of organisational and sub-organisational codes large companies - here the FTSE100 constituent companies -have developed. The article isolated seven different types of organisational and sub-organisational codes, which together with six supraorganisational ones form a lattice-work of intermeshing documents. Such a division of labour between types of (...) codes has two significant implications for CSR practice and research. In terms of corporate practice, an analysis of the content of the organisational and sub-organisational codes indicates that companies seem to enter into generalised commitments in the more visible documents, whereas other aspects, in particular more coercive aspects of CSR, get buried in the lower levels of the code hierarchy. In terms of research methodology, the differentiation between codes highlights that an analysis of codes of conduct alone is insufficient to establish what the CSR approach of a company is. Rather, it is the entire range of codes at organisational and sub-organisational levels that scholars of codes of conduct should be concerned with. (shrink)

A latarre is a lattice with an arrow. Its axiomatization looks natural. Latarres have a nontrivial theory which permits many constructions of latarres. Latarres appear as an end result of a series of generalizations of better known structures. These include Boolean algebras and Heyting algebras. Latarres need not have a distributive lattice.

We consider a simple modal logic whose nonmodal part has conjunction and disjunction as connectives and whose modalities come in adjoint pairs, but are not in general closure operators. Despite absence of negation and implication, and of axioms corresponding to the characteristic axioms of _T_, _S4_, and _S5_, such logics are useful, as shown in previous work by Baltag, Coecke, and the first author, for encoding and reasoning about information and misinformation in multiagent systems. For the propositional-only fragment of (...) such a dynamic epistemic logic, we present an algebraic semantics, using lattices with agent-indexed families of adjoint pairs of operators, and a cut-free sequent calculus. The calculus exploits operators on sequents, in the style of “nested” or “tree-sequent” calculi; cut-admissibility is shown by constructive syntactic methods. The applicability of the logic is illustrated by reasoning about the muddy children puzzle, for which the calculus is augmented with extra rules to express the facts of the muddy children scenario. (shrink)

In this note we introduce and study algebras of type such that is a bounded distributive lattice and ⌝ is an operator that satisfies the condition ⌝ = a ⌝ b and ⌝ 0 = 1. We develop the topological duality between these algebras and Priestley spaces with a relation. In addition, we characterize the congruences and the subalgebras of such an algebra. As an application, we will determine the Priestley spaces of quasi-Stone algebras.

The lattices of the title generalize the concept of a De Morgan lattice. A representation in terms of ordered topological spaces is described. This topological duality is applied to describe homomorphisms, congruences, and subdirectly irreducible and free lattices in the category. In addition, certain equational subclasses are described in detail.

It was shown in [3] (see also [5]) that there is a duality between the category of bounded distributive lattices endowed with a join-homomorphism and the category of Priestley spaces endowed with a Priestley relation. In this paper, bounded distributive lattices endowed with a join-homomorphism, are considered as algebras and we characterize the congruences of these algebras in terms of the mentioned duality and certain closed subsets of Priestley spaces. This enable us to characterize the (...) simple and subdirectly irreducible algebras. In particular, Priestley relations enable us to characterize the congruence lattice of the Q-distributive lattices considered in [4]. Moreover, these results give us an effective method to characterize the simple and subdirectly irreducible monadic De Morgan algebras [7].The duality considered in [4], was obtained in terms of the range of the quantifiers, and such a duality was enough to obtain the simple and subdirectly irreducible algebras, but not to characterize the congruences. (shrink)

An Ockham lattice is defined to be a distributive lattice with 0 and 1 which is equipped with a dual homomorphic operation. In this paper we prove: (1) The lattice of all equational classes of Ockham lattices is isomorphic to a lattice of easily described first-order theories and is uncountable, (2) every such equational class is generated by its finite members. In the proof of (2) a characterization of orderings of with respect to which the successor (...) function is decreasing is given. (shrink)

There is substantial evidence that methylphenidate (MPH; Ritalin), is being used by healthy university students for non-medical motives such as the improvement of concentration, alertness, and academic performance. The scope and potential consequences of the non-medical use of MPH upon healthcare and society bring about many points of view.

In this paper we prove the equivalence between the Gentzen system G LJ*\c , obtained by deleting the contraction rule from the sequent calculus LJ* (which is a redundant version of LJ), the deductive system IPC*\c and the equational system associated with the variety RL of residuated lattices. This means that the variety RL is the equivalent algebraic semantics for both systems G LJ*\c in the sense of [18] and [4], respectively. The equivalence between G LJ*\c and IPC*\c (...) is a strengthening of a result obtained by H. Ono and Y. Komori [14, Corollary 2.8.1] and the equivalence between G LJ*\c and the equational system associated with the variety RL of residuated lattices is a strengthening of a result obtained by P.M. Idziak [13, Theorem 1].An axiomatization of the restriction of IPC*\c to the formulas whose main connective is the implication connective is obtained by using an interpretation of G LJ*\c in IPC*\c. (shrink)

A symmetric residuated lattice is an algebra such that is a commutative integral bounded residuated lattice and the equations x=x and =xy are satisfied. The aim of the paper is to investigate the properties of the unary operation ε defined by the prescription εx=x→0. We give necessary and sufficient conditions for ε being an interior operator. Since these conditions are rather restrictive →0)=1 is satisfied) we consider when an iteration of ε is an interior operator. In particular we consider the (...) chain of varieties of symmetric residuated lattices such that the n iteration of ε is a boolean interior operator. For instance, we show that these varieties are semisimple. When n=1, we obtain the variety of symmetric stonean residuated lattices. We also characterize the subvarieties admitting representations as subdirect products of chains. These results generalize and in many cases also simplify, results existing in the literature. (shrink)

In this paper we study 12 four-valued logics arisen from Belnap's truth and/or knowledge four-valued lattices, with or without constants, by adding one or both or none of two new non-regular operations—classical negation and natural implication. We prove that the secondary connectives of the bilattice four-valued logic with bilattice constants are exactly the regular four-valued operations. Moreover, we prove that its expansion by any non-regular connective (such as, e.g., classical negation or natural implication) is strictly (...) functionally complete. Further, finding axiomatizations of the quasi varieties generated by the 12 logics involved (that prove to be varieties), we find naturell equational axiomatizations of these logics. Finally, applying Pynko's general theory of algebraizable sequential consequence operations, we also find equivalent natural sequentiell axiomatizations of the logics under consideration that expand either of two Pynko's sequential calculi for the constant-free truth-lattice four-valued logic. (shrink)