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The {\em override} operation $\sqcup$ is a natural one in computer science,and has connections with other areas of mathematics such as hyperplanearrangements. For arbitrary functions $f$ and $g$, $f\sqcup g$ is the functionwith domain $dom(f)\cup dom(g)$ that agrees with $f$ on $dom(f)$ and with $g$on $dom(g) \backslash dom(f)$. Jackson and the author have shown that there isno finite axiomatisation of algebras of functions of signature $(\sqcup)$. Butadding operations (such as {\em update}) to this minimal signature can lead tofinite axiomatisations. For the functional signature $(\sqcup,\backslash)$where $\backslash$ is set-theoretic difference, Cirulis has given a finiteequational axiomatisation as subtraction o-semilattices. Define $f\curlyveeg=(f\sqcup g)\cap (g\sqcup f)$ for all functions $f$ and $g$; this is thelargest domain restriction of the binary relation $f\cup g$ that gives apartial function. Now $f\cap g=f\backslash(f\backslash g)$ and $f\sqcupg=f\curlyvee(f\curlyvee g)$ 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 finitelyaxiomatised, with the former giving a proper quasivariety and the latter thevariety of associative distributive o-semilattices in the sense of Cirulis.

Override and restricted union for partial functions