On the definition and the representability of quasi-polyadic equality algebras (original) (raw)

Exploring Canonical Axiomatisations of Representable Cylindric Algebras

2011

We show that for finite n ≥ 3 the class of representable cylindric algebras RCA n cannot be axiomatised by canonical first-order formulas. So, although RCA n is known to be canonical, which means that it is closed under canonical extensions, there is no axiomatisation where all the formulas are preserved by canonical extensions. In fact, we show that every axiomatisation contains an infinite number of non-canonical formulas.

Quasi-Algebras versus Regular Algebras - Part I

Scientific Annals of Computer Science, 2015

Starting from quasi-Wajsberg algebras (which are generalizations of Wajsberg algebras), whose regular sets are Wajsberg algebras, we introduce a theory of quasi-algebras versus, in parallel, a theory of regular algebras. We introduce the quasi-RM, quasi-RML, quasi-BCI, (commutative, positive implicative, quasi-implicative, with product) quasi-BCK, quasi-Hilbert and quasi-Boolean algebras as generalizations of RM, RML, BCI, (commutative, positive implicative, implicative, with product) BCK, Hilbert and Boolean algebras respectively. In Part I, the first part of the theory of quasi-algebras-versus the first part of a theory of regular algebras-is presented. We introduce the quasi-RM and the quasi-RML algebras and we present two equivalent definitions of quasi-BCI and of quasi-BCK algebras.

The polyadic generalization of the Boolean axiomatization of fields of sets

Transactions of the American Mathematical Society, 2012

A version of the classical representation theorem for Boolean algebras states that the fields of sets form a variety and that a possible axiomatization is the system of Boolean axioms. An important case for fields of sets occurs when the unit V is a subset of an α-power α U. Beyond the usual set operations union, intersection, and complement, new operations are needed to describe such a field of sets, e.g., the ith cylindrification C i , the constant ijth diagonal D ij , the elementary substitution [i / j] and the transposition [i, j] for all i, j < α restricted to the unit V. Here it is proven that such generalized fields of sets being closed under the above operations form a variety; further, a first order finite scheme axiomatization of this variety is presented. In the proof a crucial role is played by the existence of the operator transposition. The foregoing axiomatization is close to that of finitary polyadic equality algebras (or quasi-polyadic equality algebras).

Categorical Equivalences for Formula quasi-MV Algebras

Journal of Logic and Computation, 2010

p 0 quasi-MV algebras have been mainly viewed as preordered structures w.r.t. the induced preorder relation of their quasi-MV term reducts. In this paper, we shall focus on a di¤erent relation which partially orders cartesian p 0 quasi-MV algebras. We shall prove that: a) every cartesian p 0 quasi-MV algebra is embeddable into an interval in a particular Abelian`-group with operators; b) the category of cartesian p 0 quasi-MV algebras isomorphic with the pair algebras over their own polynomial MV subreducts is equivalent both to the category of such`-groups (with strong order unit), and to the category of MV algebras. As a byproduct of these results we obtain a purely group-theoretical equivalence, namely between the mentioned category of`-groups with operators and the category of Abelian`-groups (both with strong order unit).

Non-commutative EQ-logics and their extensions

Proc. World Congress IFSA-EUSFLAT, 2009

We discuss a formal many-valued logic called EQlogic which is based on a recently introduced special class of algebras called EQ-algebras. The latter have three basic binary operations (meet, multiplication, fuzzy equality) and a top element and, in a certain sense, generalize residuated lattices. The goal of EQ-logics is to present a possible direction in the development of mathematical logics in which axioms are formed as identities. In this paper we propose a basic EQ-logic and three extensions which end up with a logic equivalent to the MTL-logic.

On algebras with a generalized implication

Mathematica Slovaca, 2013

We introduce the notion of gi-algebra as a generalization of dual BCK-algebra, and define the notions of strong, commutative and transitive gi-algebra, and then we show that an interval ↑l = {a ∈ P | l ≤ a} in a strong and commutative gi-algebra P is a lattice. Also, we define a congruence relation ∼D on a transitive gi-algebra P and show that the quotient set P/∼D is a gi-algebra and a dual BCK-algebra.

The Logic of Quasi-MV Algebras

Journal of Logic and Computation, 2010

The algebraic theory of quasi-MV algebras, generalisations of MV algebras arising in quantum computation, is by now rather well-developed. Although it is possible to define several interesting logics from these structures, so far this aspect has not been investigated. The present paper aims at filling this gap.

A preliminary study of MV-algebras with two quantifiers which commute

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 40's, Tasrki 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 is a special cylindric algebra. 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 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 to 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 chains of length n+1 (n<omega). 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.

Relation algebras from cylindric algebras, II

Annals of Pure and Applied Logic, 2001

We prove, for each 4 ≤ n < ω, that SRaCA n+1 cannot be defined, using only finitely many first-order axioms, relative to SRaCA n. The construction also shows that for 5 ≤ n < ω, SRaCA n is not finitely axiomatisable over RA n , and that for 3 ≤ m < n < ω, SNr m CA n+1 is not finitely axiomatisable over SNr m CA n. In consequence, for a certain standard n-variable first-order proof system m,n of m-variable formulas, there is no finite set of m-variable schemata whose m-variable instances, when added to m,n as axioms, yield m,n+1 .