Partial Maps with Domain and Range: Extending Schein's Representation
Marcel Jackson, Tim Stokes · Communications in Algebra · 2009
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 Schein , B. M. ( 1970a ). Restrictively multiplicative algebras of transformations . Izv. Vysš. Učebn. Zaved. Matematika 4 ( 95 ): 91 – 102 [Russian] . [Google Scholar]] 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 Fountain , J. ( 1979 ). Adequate semigroups . Proc. Edinburgh Math. Soc. 22 : 113 – 125 .[Crossref], [Web of Science ®] , [Google Scholar]]). 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.