Axiomatizability of positive algebras of binary relations

Abstract

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.

Undecidability of representability for lattice-ordered semigroups and ordered complemented semigroups

“…We note that non-finite axiomatisability of representability of algebras with signature τ where {·, ∧, ∨} ⊆ τ ⊆ {·, ∧, ∨,˘, e, 0, T} was shown by Andréka [1]; see also Andréka and Mikulás [2].…”

“…We have already observed that both versions of (3) are known to be undecidable, and so too are the items in (1) and (2). The goal of this paper is to introduce equivalent statements regarding decidability of representability of signatures weaker than that of a Boolean monoid.…”

On Varieties of Groupoids of Relations with Operation of Binary Cylindrification

“…Usually, these operations can be expressed in terms of operations of Tarski's algebras of relations. Algebras with such operations are called reducts of Tarski's algebras of relations [23,1,2,8].…”

Positive Fragments of Relevance Logic and Algebras of Binary Relations

“…It is known that RRA is a variety [Ta55], but it cannot be defined by any finite set of first-order formulas [Mo64], hence the equational theory is not finitely based. There has been some interest in identifying subsignatures τ of the full relation algebra signature such that the corresponding representation class Q(τ ) (defined formally below) is defined by a finite set of axioms, or where the set of equations valid over Q(τ ) follows from a finite set of equations, see [Sc91,Mi04,AM10] for surveys. It is known that Q(τ ) is a finitely axiomatizable quasivariety if τ ⊆ {0, 1, +, • , −, 1 , }.…”