  The {\em 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 $dom(f)\cup dom(g)$ that agrees with $f$ on $dom(f)$ and with $g$
on $dom(g) \backslash dom(f)$. Jackson and the author have shown that there is
no finite axiomatisation of algebras of functions of signature $(\sqcup)$. But
adding operations (such as {\em 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.
