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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.
Type
Journal Article
Type of thesis
Series
Citation
Jackson, M. & Stokes, T. (2009). Partial maps with domain and range: extending Schein's representation. Communications in Algebra, 37(8), 2845-2870.
Date
2009
Publisher
Taylor & Francis
Degree
Supervisors
Rights
This is an author's accepted version of an article published in the journal: Communications in Algebra. Copyright © Taylor & Francis Group, LLC.