Undecidable theories of Lyndon algebras

Abstract

With each projective geometry we can associate a Lyndon algebra. Such an algebra always satisfies Tarski's axioms for relation algebras and Lyndon algebras thus form an interesting connection between the fields of projective geometry and algebraic logic. In this paper we prove that if G is a class of projective geometries which contains an infinite projective geometry of dimension at least three, then the class L(G) of Lyndon algebras associated with projective geometries in G has an undecidable equational theory. In our proof we develop and use a connection between projective geometries and diagonal-free cylindric algebras.

Inequivalent representations of geometric relation algebras

“…Jonsson [10] used them to show that RRA cannot be axiomatized by any set of equations that involves only finitely many variables. More recently, classes of these algebras have been the objects of study in investigations regarding decision problems (see Andreka-Givant-Nemeti [1] and Stebletsova-Venema [19]), axiomatizability (see Givant [6] and Stebletsova [18]), and inequivalent (non-baseisomorphic) representations (see Hirsch-Hodkinson [8]). 'The assumption that the representation is complete is needed when the projective geometry P is infinite, and is therefore inserted parenthetically.…”