Restriction in Program Algebra

Author:

Jackson Marcel1,Stokes Tim2

Affiliation:

1. Department of Mathematical and Physical Sciences, La Trobe University , Victoria 3086, Australia

2. Department of Mathematics, University of Waikato , Hamilton 3240, New Zealand

Abstract

Abstract We provide complete classifications of algebras of partial maps for a significant swathe of combinations of operations not previously classified. Our focus is the many subsidiary operations that arise in recent considerations of the ‘override’ and ‘update’ operations arising in specification languages. These other operations turn out to have an older pedigree: domain restriction, set subtraction and intersection. All signatures considered include domain restriction, at least as a term. Combinations of the operations are classified and given complete axiomatizations with and without the presence of functional composition. Each classification is achieved by way of providing a concrete representation of the corresponding abstract algebras as partial maps acting on special kinds of filters determined with respect to various induced orders. In contrast to many negative results in the broader area, all of the considered combinations lead to finite axiomatizations.

Publisher

Oxford University Press (OUP)

Subject

Logic

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2. The axiomatization of override and update;Berendsen;Journal of Applied Logic,2010

3. Skew Boolean algebras and discriminator varieties;Bignall;Algebra Universalis,1995

4. Representations of ordered semigroups and lattices by binary relations;Bredikhin;Colloquium Mathematicum,1978

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