![Sample our Mathematics & Statistics journals, sign in here to start your FREE access for 14 days]({"type":"image" , "src" : "/sda/5943/TOC-Subject-Sample-2021-Mathematics-Statistics.jpg"})
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 [Citation25] 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, [Citation7]). 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.
ACKNOWLEDGEMENT
The first author was supported by ARC Discovery Project Grant DP0342459.
Notes
1In this article, partial maps act on the left of a set.
2We mention here that the class of {L, R}-semigroups satisfying right twistedness of R, left twistedness of L (the left dual of law (R6) for L), law (L6) and its right dual for R have received substantial attention in the literature: they are precisely the ample semigroups of Fountain [Citation7] (there they are called type A semigroups, but later renamed as ample semigroups).
3Schein's representation for stacks is in terms of partial maps acting on the right, while we (and Schweizer and Sklar) are representing partial maps acting on the left of a set. Obvious modifications of axioms and definitions allow one to translate between these two possibilities.
Communicated by V. Gould.