Override and restricted union for partial functions

Abstract

AbstractThe 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, $$f\sqcup g$$ f ⊔ g is the function with domain $${{\,\textrm{dom}\,}}(f)\cup {{\,\textrm{dom}\,}}(g)$$ dom ( f ) ∪ dom ( g ) that agrees with f on $${{\,\textrm{dom}\,}}(f)$$ dom ( f ) and with g on $${{\,\textrm{dom}\,}}(g) \backslash {{\,\textrm{dom}\,}}(f)$$ dom ( g ) \ 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 $$f\curlyvee g=(f\sqcup g)\cap (g\sqcup f)$$ f ⋎ g = ( f ⊔ g ) ∩ ( g ⊔ f ) for all functions f and g; this is the largest domain restriction of the binary relation $$f\cup g$$ f ∪ g that gives a partial function. Now $$f\cap g=f\backslash (f\backslash g)$$ f ∩ g = f \ ( f \ g ) and $$f\sqcup g=f\curlyvee (f\curlyvee g)$$ f ⊔ g = f ⋎ ( f ⋎ 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.