A new system of algebraic logic is described. it is closely related to cylindric and polyadic algebras, and is axiomatized by a small number of very simple equations.

In the paper some varieties E s n of BCK-algebras with condition defined by W. H. Cornish [1] are considered. A characterization of simple algebras in E s 2 is given and some properties of simple algebras in E s 3 are indicated.

Using techniques developed by Comer on sheaf-theoretic duality for cylindric algebras, we refine a result of Judit Madarász by showing that epimorphisms are not surjective even in the class of simple infinite dimensional cylindric algebras.

We give a simple new construction of representable relation algebras with non-representable completions. Using variations on our construction, we show that the elementary closure of the class of completely representable relation algebras is not finitely axiomatizable.

We give a simple new construction of representable relation algebras with non-representable completions. Using variations on our construction, we show that the elementary closure of the class of completely representable relation algebras is not finitely axiomatizable.

An MV-algebra is an algebra (A, ⊕, ¬, 0), where (A, ⊕, 0) is a commutative monoid and ¬ is an idempotent operation on A satisfying also some additional axioms. Basic algebras are similar algebras that can roughly be characterised as nonassociative (hence, also non-commutative) generalizations of MV-algebras. Basic algebras and commutative basic algebras provide an equivalent algebraic semantics in the sense of Blok and Pigozzi for two recent logical systems. Both are Hilbert-style systems, with (...) implication and negation as the primitive connectives. We present a considerably simpler logic, Lʙ, for basic algebras, where implication and falsum are taken as primitives. We also consider some subvarieties of basic algebras known in the literature, discuss classes of implicational algebras term-equivalent to each of these varieties, and construct axiomatic extensions of Lʙ for which these classes serve as equivalent algebraic semantics. (shrink)

This paper is an attempt to clear some philosophical questions about the nature of logic by setting up a mathematical framework. The notion of congruence in logic is defined. A logical structure in which there is no non-trivial congruence relation, like some paraconsistent logics, is called simple. The relations between simplicity, the replacement theorem and algebraization of logic are studied (including MacLane-Curry’s theorem and a discussion about Curry’s algebras). We also examine how these concepts are related to such (...) notions as semantics, truth-functionality and bivalence. We argue that a logic, which is simple, can deserve the name logic and that the opposite view is connected with a reductionist perspective (reduction of logic to algebra). (shrink)

In this paper we show that the quasivariety generated by an infinite simple MV-algebra only depends on the rationals which it contains. We extend this property to arbitrary families of simple MV-algebras.

A complete classification of simple function germs with respect to Lipschitz equivalence over the field of complex numbers ℂ was given by Nguyen et al. The aim of this article is to implement a classifier in terms of easy computable invariants to compute the type of the Lipschitz simple function germs without computing the normal form in the computer algebra system Singular.

In this paper we show that the quasivariety generated by an infinite simple MV-algebra only depends on the rationals which it contains. We extend this property to arbitrary families of simple MV-algebras.

Famous for the number-theoretic first-order statement known as Goodstein's theorem, author R. L. Goodstein was also well known as a distinguished educator. With this text, he offers an elementary treatment that employs Boolean algebra as a simple medium for introducing important concepts of modern algebra. The text begins with an informal introduction to the algebra of classes, exploring union, intersection, and complementation; the commutative, associative, and distributive laws; difference and symmetric difference; and Venn diagrams. Professor Goodstein proceeds to a (...) detailed examination of three different axiomatizations, and an outline of a fourth system of axioms appears in the examples. The final chapter, on lattices, examines Boolean algebra in the setting of the theory of partial order. Numerous examples appear at the end of each chapter, with full solutions at the end. (shrink)

In this paper we prove that the equational class generated by bounded BCK-algebras is the variety generated by the class of finite simple bounded BCK-algebras. To obtain these results we prove that every simple algebra in the equational class generated by bounded BCK-algebras is also a relatively simple bounded BCK-algebra. Moreover, we show that every simple bounded BCK-algebra can be embedded into a simple integral commutative bounded residuated lattice. We extend our main (...) results to some richer subreducts of the class of integral commutative bounded residuated lattices and to the involutive case. (shrink)

A Hilbert algebra with supremum is a Hilbert algebra where the associated order is a join-semilattice. This class of algebras is a variety and was studied in Celani and Montangie . In this paper we shall introduce and study the variety of $${H_{\Diamond}^{\vee}}$$ H ◊ ∨ -algebras, which are Hilbert algebras with supremum endowed with a modal operator $${\Diamond}$$ ◊ . We give a topological representation for these algebras using the topological spectral-like representation for Hilbert (...) class='Hi'>algebras with supremum given in Celani and Montangie . We will consider some particular varieties of $${H_{\Diamond}^{\vee}}$$ H ◊ ∨ -algebras. These varieties are the algebraic counterpart of extensions of the implicative fragment of the intuitionistic modal logic $${\mathbf{IntK}_{\Diamond}}$$ IntK ◊ . We also determine the congruences of $${H_{\Diamond}^{\vee}}$$ H ◊ ∨ -algebras in terms of certain closed subsets of the associated space, and in terms of a particular class of deductive systems. These results enable us to characterize the simple and subdirectly irreducible $${H_{\Diamond}^{\vee }}$$ H ◊ ∨ -algebras. (shrink)

Let (L; f) be a finite simple Ockham algebra and let (X;g) be its dual space. We first prove that every connected component of X is either a singleton or a generalised crown (i.e. an ordered set that is connected, has length 1, and all vertices of which have the same degree). The representation of a generalised crown by a square (0,1)-matrix in which all line sums are equal is used throughout, and a complete description of X, including the (...) number of connected components and the degree of the vertices, is given. We then examine the converse problem of when a generalised crown can be made into a connected component of (X; g). We also determine the number of non-isomorphic finite simple Ockham algebras that belong properly to a given subvariety P 2n,0. Finally, we show that the number of fixed points of (L; f) is 0,1, or 2 according to the nature of X. (shrink)

We provide simple algebraic proofs of two important facts, due to Zakharyaschev and Esakia, about Grzegorczyk algebras.

Two studies examined whether simple algebraic rules that have been shown to be operative in many applied settings may also be found in sport decision-making. The theoretical framework for these studies was the Functional Theory of Cognition (Anderson, Contributions to information integration theory. Hillsdale, NJ: Erlbaum, 1996). The way in which novices but already experienced team sport players (soccer, basketball, and handball players) combine different informational cues (relative importance of the game, numerical status of the team, current score, and (...) time left to play) for deciding a quick restart of play near the end of a match was examined. The basic finding are consistent with the proposition that the knowledge bases at work for judging the appropriateness of this type of sport decisions are structured according to simple algebraic rules. (shrink)

This unique textbook states and proves all the major theorems of many-valued propositional logic and provides the reader with the most recent developments and trends, including applications to adaptive error-correcting binary search. The book is suitable for self-study, making the basic tools of many-valued logic accessible to students and scientists with a basic mathematical knowledge who are interested in the mathematical treatment of uncertain information. Stressing the interplay between algebra and logic, the book contains material never before published, such as (...) a simple proof of the completeness theorem and of the equivalence between Chang's MV algebras and Abelian lattice-ordered groups with unit - a necessary prerequisite for the incorporation of a genuine addition operation into fuzzy logic. Readers interested in fuzzy control are provided with a rich deductive system in which one can define fuzzy partitions, just as Boolean partitions can be defined and computed in classical logic. Detailed bibliographic remarks at the end of each chapter and an extensive bibliography lead the reader on to further specialised topics. (shrink)

The purpose of this paper is to define and investigate the new class of quasi-Stone algebras . Among other things we characterize the class of simple QSA's and the class of subdirectly irreducible QSA's. It follows from this characterization that the subdirectly irreducible QSA's form an elementary class and that the variety of QSA's is locally finite. Furthermore we prove that the lattice of subvarieties of QSA's is an -chain. MSC: 03G25, 06D16, 06E15.

There is a fairly simple algebraic property that distinguishes isols from cosimple isols.

Deontic logic is a branch of symbolic logic interested in notions such as obligatory, permissible, optional, ought, and others similar. There are some equivalent ways to present the Standard Deontic Logic or KD. In this paper, we will mention some of them and highlight one that is of interest. With this presentation we can propose a simple algebraic model for the Standard Deontic Logic.

We generalize the notion of a monadic algebra to that of a pseudomonadic algebra. In the same way as monadic algebras serve as algebraic models of epistemic modal system S5, pseudomonadic algebras serve as algebraic models of doxastic modal system KD45. The main results of the paper are: Characterization of subdirectly irreducible and simple pseudomonadic algebras, as well as Tokarz's proper filter algebras; Ordertopological representation of pseudomonadic algebras; Complete description of the lattice of subvarieties (...) of the variety of pseudomonadic algebras. (shrink)

In this paper we present a general theory of normal forms, based on a categorial result for the free monoid construction. We shall use the theory mainly for proposictional modal logic, although it seems to have a wider range of applications. We shall formally represent normal forms as combinatorial objects, basically labelled trees and forests. This geometric conceptualization is implicit in and our approach will extend it to other cases and make it more direct: operations of a purely geometric and (...) combinatorial nature will be introduced in order to give a mathematical description of the basic logical/algebraic constructions .We begin by recalling the above-mentioned categorial construction: we need a careful inspection of it because in the various examples considered later we plan to deduce from it in a uniform way the normal forms and the description of finitely generated free algebras. This method always works whenever we can describe the category of algebras corresponding to the logic under consideration as a T-objects category. When this simple description seems not to be available, still the general theory might be of some interest, because a description of the category of algebras as a T-objects category plus equation is possible .The central part of the paper is more advanced and specific: we show how the general approach presented here can provide some insights even in the basic case of the modal system K. Section 4 contains a contribution to the theory of normal forms, namely the description of the uniform substitution. This result will enable us to give a duality theorem for the category of finitely generated free modal algebras and in Section 5 to provide a characterization of the collections of normal forms which happen to be normal forms for a logic, thus giving a description of the lattice of modal logics.Section 6 deals with some applications: we shall show how to use normal forms in order to prove for the modal system K the definability of higher-order propositional quantifiers and of the tense operator F .As to the prerequisites, the paper is almost self-contained. The reader is only assumed to have familiarity with standard techniques in algebraic logic ). Knowledge of the basic facts about adjoint functors is required too, see e.g. McLane or the appendix. (shrink)

This paper is a contribution toward developing a theory of expansions of semi-Heyting algebras. It grew out of an attempt to settle a conjecture we had made in 1987. Firstly, we unify and extend strikingly similar results of [ 48 ] and [ 50 ] to the (new) equational class DHMSH of dually hemimorphic semi-Heyting algebras, or to its subvariety BDQDSH of blended dual quasi-De Morgan semi-Heyting algebras, thus settling the conjecture. Secondly, we give a criterion for (...) a unary expansion of semi-Heyting algebras to be a discriminator variety and give an algorithm to produce discriminator varieties. We then apply the criterion to exhibit an increasing sequence of discriminator subvarieties of BDQDSH . We also use it to prove that the variety DQSSH of dually quasi-Stone semi- Heyting algebras is a discriminator variety. Thirdly, we investigate a binary expansion of semi-Heyting algebras, namely the variety DblSH of double semi-Heyting algebras by characterizing its simples, and use the characterization to present an increasing sequence of discriminator subvarieties of DblSH . Finally, we apply these results to give bases for “small” subvarieties of BDQDSH , DQSSH , and DblSH. (shrink)

We introduce the equational notion of a monadic bounded algebra (MBA), intended to capture algebraic properties of bounded quantification. The variety of all MBA's is shown to be generated by certain algebras of two-valued propositional functions that correspond to models of monadic free logic with an existence predicate. Every MBA is a subdirect product of such functional algebras, a fact that can be seen as an algebraic counterpart to semantic completeness for monadic free logic. The analysis involves the (...) representation of MBA's as powerset algebras of certain directed graphs with a set of "marked" points. It is shown that there are only countably many varieties of MBA's, all are generated by their finite members, and all have finite equational axiomatisations classifying them into fourteen kinds of variety. The universal theory of each variety is decidable. Finitely generated MBA's are shown to be finite, with the free algebra on r generators having exactly $2^{3.2^{r.} } ^{2^{{\rm{2}}^{{\rm{r - 1}}} } } $ elements. An explicit procedure is given for constructing this freely generated algebra as the powerset algebra of a certain marked graph determined by the number r. (shrink)

Using a combinatorial theorem of Herwig on extending partial isomorphisms of relational structures, we give a simple proof that certain classes of algebras, including Crs, polyadic Crs, and WA, have the `finite base property' and have decidable universal theories, and that any finite algebra in each class is representable on a finite set.

A simple model for the description of atomic and ionic species with spectra exhibiting a quantum defect is solved using the Lie algebra su(1, 1). The quantum defect of bound states is related to the phase shift of scattering states. The resonances are discussed in terms of the nonunitary representations of this algebra.

Ewald considered tense operators G, H, F and P on intuitionistic propositional calculus and constructed an intuitionistic tense logic system called IKt. In 2014, Figallo and Pelaitay introduced the variety IKt of IKt-algebras and proved that the IKt system has IKt-algebras as algebraic counterpart. In this paper, we introduce and study the variety of tense Nelson algebras. First, we give some examples and we prove some properties. Next, we associate an IKt-algebra to each tense Nelson algebras. (...) This result allowed us to determine the congruence of the tense Nelson algebras and also to characterize the subdirectly irreducible tense Nelson algebras and particularly the simple tense Nelson algebras. Finally, we prove that there exists an equivalence between the category of IKt-algebras and the category of tense centered Nelson algebras. (shrink)

In the study of categories whose morphisms display a behaviour similar to that of partial functions, the concept of morphism domain is, obviously, central. In this paper an operation defined on morphisms describes those properties which are related to morphisms being regarded as abstractions of partial functions. This operation allows us to characterise the morphism domains directly, and gives rise to an algebra defined by a simple set of identities. No product-like categorical structures are needed therefore. We also develop (...) the construction of topologies together with the notion of continuous morphism, in order to test the effectiveness of this approach. It is interesting to see how much of the computational character of the morphisms is translated into continuity. (shrink)

In this paper we make an algebraic study of the variety of MV*-algebras introduced by C. C. Chang as an algebraic counterpart for a logic with positive and negative truth values.We build the algebraic theory of MV*-algebras within its own limits using a concept of ideal and of prime ideal that are very naturally related to the corresponding concepts in l-groups. The main results are a subdirect representation theorem, a completeness theorem, a study of simple and semisimple (...) algebras, and a characterization of the ideal of infinitesimals as an l-group. In the last section we develop a detailed proof a the one-dimensional theorem of McNaughton, that is, the free MV*-algebra in one generator is the algebra of McNaughton functions over [−1,1]. In contrast with the rest of the paper, this last result is based on work done for MV-algebras. (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)

The paper has two parts preceded by quite comprehensive preliminaries.In the first part it is shown that a subvariety of the variety ${\cal T}$ of all tense algebras is discriminator if and only if it is semisimple. The variety ${\cal T}$ turns out to be the join of an increasing chain of varieties ${\cal D}_n$, which are discriminator varieties. The argument carries over to all finite type varieties of boolean algebras with operators satisfying some term conditions. In the (...) case of tense algebras, the varieties ${\cal D}_n$ can be further characterised by certain natural conditions on Kripke frames.In the second part it is shown that the lattice of subvarieties of ${\cal D}_0$ has two atoms, the lattice of subvarieties of ${\cal D}_1$ has countably many atoms, and for $n>1$, the lattice of subvarieties of ${\cal D}_n$ has continuum atoms. The proof of the second of the above statements involves a rather detailed description of zero-generated simple algebras in ${\cal D}_1$.Almost all the arguments are cast in algebraic form, but both parts begin with an outline describing their contents from the dual point of view of tense logics. (shrink)

Using a combinatorial theorem of Herwig on extending partial isomorphisms of relational structures, we give a simple proof that certain classes of algebras, including Crs, polyadic Crs, and WA, have the `finite base property' and have decidable universal theories, and that any finite algebra in each class is representable on a finite set.

In this paper we investigate the class of MV-algebras equipped with two quantifiers which commute as a natural generalization of diagonal-free two-dimensional cylindric algebras. In the 40s, Tarski first introduced cylindric algebras in order to provide an algebraic apparatus for the study of classical predicate calculus. The diagonal–free two-dimensional cylindric algebras are special cylindric algebras. The treatment here of MV-algebras is done in terms of implication and negation. This allows us to simplify some results (...) due to Di Nola and Grigolia :125–139, 2004) related to the characterization of a quantifier in terms of some special sub-algebra associated to it. On the other hand, we present a topological duality for this class of algebras and we apply it to characterize the congruences of one algebra via certain closed sets. Finally, we study the subvariety of this class generated by a chain of length n + 1. We prove that the subvariety is semisimple and we characterize their simple algebras. Using a special functional algebra, we determine all the simple finite algebras of this subvariety. (shrink)

We give a complete axiomatization of the identities of the basic game algebra valid with respect to the abstract game board semantics. We also show that the additional conditions of termination and determinacy of game boards do not introduce new valid identities.En route we introduce a simple translation of game terms into plain modal logic and thus translate, while preserving validity both ways, game identities into modal formulae.The completeness proof is based on reduction of game terms to a certain (...) ‘minimal canonical form’, by using only the axiomatic identities, and on showing that the equivalence of two minimal canonical terms can be established from these identities. (shrink)

In this paper, we study the field of algebraic numbers with a set of elements of small height treated as a predicate. We prove that such structures are not simple and have the independence property. A real algebraic integer is called a Salem number if α and are Galois conjugate and all other Galois conjugates of α lie on the unit circle. It is not known whether 1 is a limit point of Salem numbers. We relate the simplicity of (...) a certain pair with Lehmer's conjecture and obtain a model‐theoretic characterization of Lehmer's conjecture for Salem numbers. (shrink)

We consider a classM of Boolean algebras with strictly positive, finitely additive measures. It is shown thatM is closed under iterations with finite support and that the forcing via such an algebra does not destroy the Lebesgue measure structure from the ground model. Also, we deduce a simple characterization of Martin's Axiom reduced to the classM.

In this paper we prove that the equational class generated by bounded BCK-algebras is the variety generated by the class of finite simple bounded BCK-algebras. To obtain these results we prove that every simple algebra in the equational class generated by bounded BCK-algebras is also a relatively simple bounded BCK-algebra. Moreover, we show that every simple bounded BCK-algebra can be embedded into a simple integral commutative bounded residuated lattice. We extend our main (...) results to some richer subreducts of the class of integral commutative bounded residuated lattices and to the involutive case. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

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)

A characterization of the subdirectly irreducible separable dynamic algebras is presented. The notions develo- ped for this study were also suitable to describe the previously found class of simple separable dynamic algebras.

A Boolean product construction is used to give examples of existentially closed algebras in the universal Horn class ISP generated by a universal classKof finitely subdirectly irreducible algebras such that Γa has the Fraser-Horn property. If ⟦a≠b⟧ ∩ ⟦c≠d⟧ = ∅ is definable inKandKhas a model companion ofK-simple algebras, then it is shown that ISP has a model companion. Conversely, a sufficient condition is given for ISP to have no model companion.

The relation between least and diagonal fixed points is a well known and completely studied question for a large class of partially ordered models of the lambda calculus and combinatory logic. Here we consider this question in the context of algebraic recursion theory, whose close connection with combinatory logic recently become apparent. We find a comparatively simple and rather weak general condition which suffices to prove the equality of least fixed points with canonical (corresponding to those produced by the (...) Curry combinator in lambda calculus) diagonal fixed points in a class of partially ordered algebras which covers both combinatory spaces of Skordev and operative spaces of Ivanov. Especially, this yields an essential improvement of the axiomatization of recursion theory via combinatory spaces. (shrink)

ABSTRACT We propose a combination of a fragment of four-valued logic and process algebra. This fragment is geared to a simple relation with process algebra via the conditional guard construct, and can easily be extended to a truth-functionally complete logic. We present an operational semantics in SOS-style, and a completeness result for ACP with conditionals and four- valued logic. Completeness is preserved under the restriction to some other non-classical logics.