Axiomatization of identity-free equations valid in relation algebras (original) (raw)

The equational theory of union-free algebras of relations

Algebra Universalis, 1995

We solve a problem of J6nsson by showing that the class Y/of (isomorphs of) algebras of binary relations, under the operations of relative product, conversion, and intersection, and with the identity element as a distinguished constant, is not axiomatizable by a set of equations. We also show that the set of equations valid in ~ is decidable, and in fact the set of equations true in the class of all positive algebras of relations is decidable.

Axiomatizability of positive algebras of binary relations

Algebra universalis, 2011

We consider all positive fragments of Tarski's representable relation algebras and determine whether the equational and quasiequational theories of these fragments are finitely axiomatizable in first-order logic. We also look at extending the signature with reflexive, transitive closure and the residuals of composition.

On the equational theory of representable polyadic equality algebras (extended abstract)

Logic Journal of IGPL, 1998

Among others we will see that the equational theory of ω dimensional representable polyadic equality algebras (RP EAω's) is not schema axiomatizable. This result is in interesting contrast with the Daigneault-Monk representation theorem, which states that the class of representable polyadic algebras is finite schema-axiomatizable (and hence the equational theory of this class is finite schemaaxiomatizable, as well). We will also see that the complexity of the equational theory of RP EAω is also extremely high in the recursion theoretic sense. Finally, comparing the present negative results with the positive results of Ildikó Sain and Viktor Gyuris [10], the following methodological conclusions will be drawn: the negative properties of polyadic (equality) algebras can be removed by switching from what we call the "polyadic algebraic paradigm" to the "cylindric algebraic paradigm". 1

Reducts of Relation Algebras: The Aspects of Axiomatisability and Finite Representability

Logical Foundations of Computer Science, 2021

In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is isomorphic to some algebra over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson [11]. We also show that the class of representable join semilattice-ordered semigroups has a recursively enumerable axiomatisation using back-andforth games.

Gentzen-style axiomatizations in equational logic

Algebra Universalis, 1995

The notion of a Gentzen-style axiomatization of equational theories is presented. In the standard deductive systems for equational logic axioms take the form of equations and the inference rules can be viewed as quasi-equations. In the deductive systems for quasi-equational logic the axioms, which are quasi-equations, can be viewed as sequents and the inference rules as Gentzen-style rules. It is conjectured that every finite algebra of finite type has a finite Gentzen-style axiomatization for its quasi-identities. We verify this conjecture for a class of algebras that includes all finite algebras without nontrivial proper subalgebras, and all finite simple algebras that are embeddable into the free algebra of their variety.

Expressibility of properties of relations

We investigate in an algebraic setting the question in which logical languages the properties integral, permutational, and rigid of algebras of relations can be expressed.