How to generalise demonic composition

Abstract

Demonic composition is defined on the set of binary relations over the non-empty set X, 𝑅𝑒𝑙𝑋, and is a variant of standard or “angelic” composition. It arises naturally in the setting of the theory of non-deterministic computer programs, and shares many of the nice features of ordinary composition (it is associative, and generalises composition of functions). When equipped with the operations of demonic composition and domain, 𝑅𝑒𝑙𝑋 is a left restriction semigroup (like 𝑃𝑇𝑋, the semigroup of partial functions on X), whereas usual composition and domain give a unary semigroup satisfying weaker laws. By viewing 𝑅𝑒𝑙𝑋 under a restricted version of its usual composition and domain as a constellation (a kind of “one-sided” category), we show how this demonic left restriction semigroup structure arises on 𝑅𝑒𝑙𝑋, placing it in a more general context. The construction applies to any unary semigroup with a “domain-like” operation satisfying certain minimal conditions which we identify. In particular it is shown that using the construction, any Baer ∗-semigroup S can be given a left restriction semigroup structure which is even an inverse semigroup if S is ∗-regular. It follows that the semigroup of 𝑛×𝑛 matrices over the real or complex numbers is an inverse semigroup with respect to a modified notion of product that almost always agrees with the usual matrix product, and in which inverse is pseudoinverse (Moore–Penrose inverse).