Boolean and classical restriction categories

Abstract

A restriction category is an abstract category of partial maps. A Boolean restriction category is a restriction category that supports classical (Boolean) reasoning. Such categories are models of loop-free dynamic logic that is deterministic in the sense that < α > Q ⊂ [α]Q. Classical restriction categories are restriction categories with a locally Boolean structure: it is shown that they are precisely full subcategories of Boolean restriction categories. In particular, a Boolean restriction category may be characterised as a classical restriction category with finite coproducts in which all restriction idempotents split.Every restriction category admits a restriction embedding into a Boolean restriction category. Thus, every abstract category of partial maps admits a conservative extension that supports classical reasoning. An explicit construction of the classical completion of a restriction category is given.