SAMS Subject Classification Number: 34

A relation algebra is an algebraic structure consisting of a non-empty set, the Boolean operations and constants, an associative binary operation that has a unit, and an involutive unary operation, such that these operations and constants satisfy certain equations forming an axiomatization of a calculus of binary relations. The example of a relation algebra that motivated this definition is the algebra of all binary relations on a set, with the associative binary operation interpreted as the usual composition of binary relations, its unit as the identity relation, and the involutive unary operation as the converse relation.

A generalization of the motivating example mentioned above is the power set algebra of an equivalence relation on a set, with the operations defined in the same way. This relation algebra plays an important role in the definition of a representable relation algebra as any relation algebra isomorphic to a subalgebra of the power set algebra of some equivalence relation on some set.

Representable relation algebras have been studied for many decades. In 1950 Lyndon showed that there is a non-representable relation algebra [2]. In 1955 Tarski showed that representable relation algebras is a variety [4]. In 1964, Monk showed that representable relation algbras is not finitely axiomatizable [3], unlike relation algebras, which is finitely axiomatizable by definition.

In the 1940s, Alfred Tarski proved that relation algebras have an undecidable equational theory [5]. To remedy this, Galatos and Jipsen identified a larger variety, called quasi relation algebras, similar to relation algebras that has a decidable equational theory [1].

In this talk we will introduce representable distributive quasi relation algebras. These are defined via a construction involving posets with certain symmetry requirements. This gives rise to many interesting questions about the representability of distributive quasi relation algebras and the properties of representable quasi relation algebras.

[1] N. Galatos, P. Jipsen, Relation algebras as expanded FL-algebras, Algebra Universalis 69 (2013), 1–21.

[2] R.C. Lyndon, The representation of relation algebras, Ann. Math. 51 (1950), 707–729.

[3] J.D. Monk, On representable relation algebras, Michigan Math. J. 11 (1964), 207–210.

[4] A. Tarski, Contributions to the theory of models. III, Nederl. Akad. Wetensch. Proc. Ser. A. 58 (1955), 56–64. s [5] A. Tarski, S. Givant, A Formalization of Set Theory Without Variables. American Mathematical Society, Providence (1987).