Override and restricted union for partial functions

Abstract

The override operation \(\sqcup \) is a natural one in computer science, and has connections with other areas of mathematics such as hyperplane arrangements. For arbitrary functions f and g, \(f\sqcup g\) is the function with domain \({{\,\textrm{dom}\,}}(f)\cup {{\,\textrm{dom}\,}}(g)\) that agrees with f on \({{\,\textrm{dom}\,}}(f)\) and with g on \({{\,\textrm{dom}\,}}(g) \backslash {{\,\textrm{dom}\,}}(f)\). Jackson and the author have shown that there is no finite axiomatisation of algebras of functions of signature \((\sqcup )\). But adding operations (such as update) to this minimal signature can lead to finite axiomatisations. For the functional signature \((\sqcup ,\backslash )\) where \(\backslash \) is set-theoretic difference, Cirulis has given a finite equational axiomatisation as subtraction o-semilattices. Define \(f\curlyvee g=(f\sqcup g)\cap (g\sqcup f)\) for all functions f and g; this is the largest domain restriction of the binary relation \(f\cup g\) that gives a partial function. Now \(f\cap g=f\backslash (f\backslash g)\) and \(f\sqcup g=f\curlyvee (f\curlyvee g)\) for all functions f, g, so the signatures \((\curlyvee )\) and \((\sqcup ,\cap )\) are both intermediate between \((\sqcup )\) and \((\sqcup ,\backslash )\) in expressive power. We show that each is finitely axiomatised, with the former giving a proper quasivariety and the latter the variety of associative distributive o-semilattices in the sense of Cirulis.