Partial Maps with Domain and Range: Extending Schein's Representation

Abstract

The semigroup of all partial maps on a set under the operation of composition admits a number of operations relating to the domain and range of a partial map. Of particular interest are the operations R and L returning the identity on the domain of a map and on the range of a map respectively. Schein [25] gave an axiomatic characterisation of the semigroups with R and L representable as systems of partial maps; the class is a finitely axiomatisable quasivariety closely related to ample semigroups (which were introduced-as type A semigroups-by Fountain, [7]). We provide an account of Schein's result (which until now appears only in Russian) and extend Schein's method to include the binary operations of intersection, of greatest common range restriction, and some unary operations relating to the set of fixed points of a partial map. Unlike the case of semigroups with R and L, a number of the possibilities can be equationally axiomatised.

Domain and Antidomain Semigroups

“…We recall below a fourth instance of such an embedding due to Trokhimenko [Tro73] (cf. [JaS0a]). We present a concise variant of the proof for the domain setting because it uses a general construction that should be of interest for the RelMiCS/AKA community.…”

Domain and range for angelic and demonic compositions

“…Program modelling is just one of several motivations for this work however. Domain and range are already transparently natural features of both relations and functions, and the modelling of these via unary operations can be traced back at least to the work of Menger [30], through the work of Schweizer and Sklar [35,36,37, 38], Trokhimenko [39], Bredikhin [6] and Schein [34] and into the work of the authors and their collaborators [19,22,23,24] as well as in the category-theoretic work of Cockett, Lack, Guo, Hofstra, Manes amongst others [8,9,10,11]. Yet another motivation comes from the structural theory of semigroups, where many authors have enriched the usual associative binary multiplication by the addition of unary operations that map onto idempotent elements; see for example Fountain [16,17], 1 To the best of the authors' knowledge, the earliest reference to demons in this context may be in Broy, Gnatz and Wirsing [7], where Dijkstra [15] is cited as an instance of demonic modelling of nondeterminancy.…”

The finite representation property for composition, intersection, domain and range

“…One of the primary aims is to determine how simply the class of representable algebras can be axiomatised and to find such an axiomatisation. Often, the representation classes have turned out to be axiomatisable by finitely many equations or quasi-equations [8,2,5,3,4,1].…”