Tarski Algebras Research Articles

On Implicative and Positive Implicative GE Algebras

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.

Amalgamation and Robinson property in universal algebraic logic

Abstract There is a well-established correspondence between interpolation and amalgamation for algebraizable logics that satisfy certain additional assumptions. In this paper, we introduce the Robinson property of a logic and show that a conditionally algebraizable logic without any additional assumptions has the Robinson property if and only if the corresponding class of Lindenbaum–Tarski algebras has the amalgamation property. Moreover, we give the logical characterization of the strong amalgamation property, solving an open problem of Andréka–Németi–Sain. It is also shown that given the mentioned extra assumptions the Robinson property implies the interpolation property. As conditionally algebraizable logics cover algebraizable logics as well as various quantifier logics such as classical first order logic, our results yield a generalization of some of the results concerning interpolation and amalgamation.

Basic modal congruent and monotonic multilattice logics

Abstract In the paper, we introduce multilattice versions of the basic congruent and monotonic modal logics. In the case of congruent and monotonic ones, we also study their extensions by Gödel’s rule. We formulate these logics in the form of sequent calculi and prove syntactic embedding theorems (as a consequence, we obtain cut admissibility and decidability). Then we present them algebraically and semantically: via modal multilattices and via general and descriptive neighbourhood frames. We show the dual equivalency of the categories of modal multilattices and descriptive neighbourhood frames. Using Lindenbaum–Tarski algebras, we prove that the sequent calculi under consideration are sound and complete with respect to modal multilattices.

Subordination Tarski algebras

ABSTRACTIn this work we will study Tarski algebras endowed with a subordination, called subordination Tarski algebras. We will define the notion of round filters, and we will study the class of irreducible round filters and the maximal round filters, called ends. We will prove that the poset of all round filters is a lattice isomorphic to the lattice of the congruences that are compatible with the subordination. We will prove that every end is an irreducible round filter, and that in a topological subordination Tarski algebra A, every irreducible round filter is an end iff A is a monadic subordination Tarski algebra. As corollary of this result we have that the variety of monadic Tarski algebra can be characterised as the topological Tarski algebras where every irreducible open filter is a maximal open filter.

Complete and atomic Tarski algebras

Tarski algebras, also known as implication algebras or semi-boolean algebras, are the $$\left\{ \rightarrow \right\} $$ -subreducts of Boolean algebras. In this paper we shall introduce and study the complete and atomic Tarski algebras. We shall prove a duality between the complete and atomic Tarski algebras and the class of covering Tarski sets, i.e., structures $$\left $$ , where X is a non-empty set and $${\mathcal {K}}$$ is non-empty family of subsets of X such that $$\bigcup {\mathcal {K}}=X$$ . This duality is a generalization of the known duality between sets and complete and atomic Boolean algebras. We shall also analize the case of complete and atomic Tarski algebras endowed with a complete modal operator, and we will prove a duality for these algebras.

Swap structures semantics for Ivlev-like modal logics

In 1988, J. Ivlev proposed some (non-normal) modal systems which are semantically characterized by four-valued non-deterministic matrices in the sense of A. Avron and I. Lev. Swap structures are multialgebras (a.k.a. hyperalgebras) of a special kind, which were introduced in 2016 by W. Carnielli and M. Coniglio in order to give a non-deterministic semantical account for several paraconsistent logics known as logics of formal inconsistency, which are not algebraizable by means of the standard techniques. Each swap structure induces naturally a non-deterministic matrix. The aim of this paper is to obtain a swap structures semantics for some Ivlev-like modal systems proposed in 2015 by M. Coniglio, L. Farinas del Cerro and N. Peron. Completeness results will be stated by means of the notion of Lindenbaum–Tarski swap structures, which constitute a natural generalization to multialgebras of the concept of Lindenbaum–Tarski algebras.

Selfextensional logics with a distributive nearlattice term

We define when a ternary term m of an algebraic language $$\mathcal {L}$$ is called a distributive nearlattice term ( $$\mathrm {DN}$$ -term) of a sentential logic $$\mathcal {S}$$ . Distributive nearlattices are ternary algebras generalising Tarski algebras and distributive lattices. We characterise the selfextensional logics with a $$\mathrm {DN}$$ -term through the interpretation of the DN-term in the algebras of the algebraic counterpart of the logics. We prove that the canonical class of algebras (under the point of view of Abstract Algebraic Logic) associated with a selfextensional logic with a $$\mathrm {DN}$$ -term is a variety, and we obtain that the logic is in fact fully selfextensional.

Coordinatizing some concrete MV algebras and a decomposition theorem

MV algebras are the Lindenbaum–Tarski algebras of Łukasiewicz many-valued logics. From work of D. Mundici, countable such algebras correspond to certain AF C*-algebras. M. Lawson and P. Scott gave a coordinatization theorem for them, representing any countable MV-algebra as the lattice of principal ideals of an AF Boolean inverse monoid. In this note, we give two concrete examples of such a coordinatization, one for $${\mathbb {Q}}\cap [0,1]$$ and another for the so-called Chang algebra. We also discuss Bratteli diagram techniques to further the coordinatization program.

On Tarski algebras with a finite set of free generators

In this paper, we describe a method to determine the structure of the Tarski algebra with a finite set of free generators which is different to that given by Iturrioz and Monteiro in [Les algèbres de Tarski avec un nombre fini de générateurs libres, in Informe Técnico, Vol. 37 (Instituto de Matemática de la Universidad Nacional del Sur, Bahía Blanca, 1994)].

Algebraic functions in Łukasiewicz implication algebras

In this article we study algebraic functions in [Formula: see text]-subreducts of MV-algebras, also known as Łukasiewicz implication algebras. A function is algebraic on an algebra [Formula: see text] if it is definable by a conjunction of equations on [Formula: see text]. We fully characterize algebraic functions on every Łukasiewicz implication algebra belonging to a finitely generated variety. The main tool to accomplish this is a factorization result describing algebraic functions in a subproduct in terms of the algebraic functions of the factors. We prove a global representation theorem for finite Łukasiewicz implication algebras which extends a similar one already known for Tarski algebras. This result together with the knowledge of algebraic functions allowed us to give a partial description of the lattice of classes axiomatized by sentences of the form [Formula: see text] within the variety generated by the 3-element chain.

Endomorphisms and subalgebras of Tarski algebras

Endomorphisms and subalgebras of Tarski algebras

Free Monadic Tarski and MMI3-Algebras

Abstract MMI3-algebras are a generalization of the monadic Tarski algebras as defined by A. Monteiro and L. Iturrioz, and a particular case of the MMIn+1-algebras defined by A. Figallo. They can also be seen as monadic three-valued Łukasiewicz algebras without a first element. By using this point of view, and the free monadic extensions, we construct the free MMI3-algebras on a finite number of generators, and indicate the coordinates of the generators. As a byproduct, we also obtain a construction of the free monadic Tarski algebras.

On Cantor identities

In this paper, some special varieties which generalize Jónsson–Tarski algebras are considered. We prove that every nontrivial algebra from such a variety is term infinite and contains infinitely many distinct proper diagonal term operations of every arity.

Finite Tarski algebras are determined by their endomorphisms

We prove that if two finite Tarski algebras have isomorphic endomorphism monoids then they are isomorphic.

Fuzzifications of generalized Tarski filters in Tarski algebras

The fuzzification of generalized Tarski filters of generalized Tarski algebras is considered, and related properties are investigated. Characterizations of a fuzzy generalized Tarski filter are established. Based on the theory of a falling shadow, the notion of a falling fuzzy generalized Tarski filter of a generalized Tarski algebra is introduced. Relations between fuzzy generalized Tarski filters and falling fuzzy generalized Tarski filters are provided. A characterization of a falling fuzzy generalized Tarski filter is established.

Canonical extensions for congruential logics with the deduction theorem

We introduce a new and general notion of canonical extension for algebras in the algebraic counterpart Alg S of any finitary and congruential logic S . This definition is logic-based rather than purely order-theoretic and is in general different from the definition of canonical extensions for monotone poset expansions, but the two definitions agree whenever the algebras in Alg S are based on lattices. As a case study on logics purely based on implication, we prove that the varieties of Hilbert and Tarski algebras are canonical in this new sense.

Representation and duality for Hilbert algebras

Abstract In this paper we introduce a special kind of ordered topological spaces, called Hilbert spaces. We prove that the category of Hilbert algebras with semi-homomorphisms is dually equivalent to the category of Hilbert spaces with certain relations. We restrict this result to give a duality for the category of Hilbert algebras with homomorphisms. We apply these results to prove that the lattice of the deductive systems of a Hilbert algebra and the lattice of open subsets of its dual Hilbert space, are isomorphic. We explore how this duality is related to the duality given in [6] for finite Hilbert algebras, and with the topological duality developed in [7] for Tarski algebras.

Join-semilattices whose sections are residuated PO-monoids

We generalize the concept of an integral residuated lattice to join-semilattices with an upper bound where every principal order-filter (section) is a residuated semilattice; such a structure is called a sectionally residuated semilattice. Natural examples come from propositional logic. For instance, implication algebras (also known as Tarski algebras), which are the algebraic models of the implication fragment of the classical logic, are sectionally residuated semilattices such that every section is even a Boolean algebra. A similar situation rises in case of the Lukasiewicz multiple-valued logic where sections are bounded commutative BCK-algebras, hence MV-algebras. Likewise, every integral residuated (semi)lattice is sectionally residuated in a natural way. We show that sectionally residuated semilattices can be axiomatized as algebras (A, r, →, ⇝, 1) of type 〈3, 2, 2, 0〉 where (A, →, ⇝, 1) is a {→, ⇝, 1}-subreduct of an integral residuated lattice. We prove that every sectionally residuated lattice can be isomorphically embedded into a residuated lattice in which the ternary operation r is given by r(x, y, z) = (x · y) ∨ z. Finally, we describe mutual connections between involutive sectionally residuated semilattices and certain biresiduation algebras.

Topological duality for Tarski algebras

In this paper we will generalize the representation theory developed for finite Tarski algebras given in [7]. We will introduce the notion of Tarski space as a generalization of the notion of dense Tarski set, and we will prove that the category of Tarski algebras with semi-homomorphisms is dually equivalent to the category of Tarski spaces with certain closed relations, called T-relations. By these results we will obtain that the algebraic category of Tarski algebras is dually equivalent to the category of Tarski spaces with certain partial functions. We will apply these results to give a topological characterization of the subalgebras.

Cyclic Tarski algebras

The variety of cyclic Boolean algebras is a particular subvariety of the variety of tense algebras. The objective of this paper is to study the variety  of {→,g, h}-subreducts of cyclic Boolean algebras, which we call cyclic Tarski algebras. We prove that  is generated by its finite members and we characterise the locally finite subvarieties of . We prove that there are no splitting varieties in the lattice Λ() of subvarieties of . Finally, we prove that the subquasivarieties and the subvarieties of a locally finite subvariety of  coincide.