The Algebra of Partial Maps

Speaker: Marcel Jackson, LaTrobe University

Abstract: The right regular representation essentially identifies semigroup theory with what could reasonably be called the "abstract theory of functions under composition". One can similarly identify semigroup theory with the abstract theory of partial maps under composition and with the abstract theory of binary relations under composition. However there are a number of other natural operations that make sense on partial maps and relations; intersection for example.

Tarski's relation algebras are probably the most structured abstraction of the algebra of binary relations and form a well studied and important part of algebraic logic. The partial map case turns out to have a definite semigroup theoretic feel but is far less understood than the relational case. We will concentrate on the algebra of partial maps, surveying the "classical" literature, as well as recent results obtained by the speaker and T. Stokes.