Finite representability of semigroups with demonic refinement

Author:

Hirsch Robin,Šemrl Jaš

Abstract

AbstractThe motivation for using demonic calculus for binary relations stems from the behaviour of demonic turing machines, when modelled relationally. Relational composition (; ) models sequential runs of two programs and demonic refinement ($$\sqsubseteq $$ ) arises from the partial order given by modeling demonic choice ($$\sqcup $$ ) of programs (see below for the formal relational definitions). We prove that the class $$R(\sqsubseteq , ;)$$ R ( , ; ) of abstract $$(\le , \circ )$$ ( , ) structures isomorphic to a set of binary relations ordered by demonic refinement with composition cannot be axiomatised by any finite set of first-order $$(\le , \circ )$$ ( , ) formulas. We provide a fairly simple, infinite, recursive axiomatisation that defines $$R(\sqsubseteq , ;)$$ R ( , ; ) . We prove that a finite representable $$(\le , \circ )$$ ( , ) structure has a representation over a finite base. This appears to be the first example of a signature for binary relations with composition where the representation class is non-finitely axiomatisable, but where the finite representation property holds for finite structures.

Publisher

Springer Science and Business Media LLC

Subject

Algebra and Number Theory

Cited by 3 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Some results on relation algebra reducts: Residuated and semilattice-ordered semigroups;Journal of Logic and Computation;2022-11-07

2. Reducts of Relation Algebras: The Aspects of Axiomatisability and Finite Representability;Logical Foundations of Computer Science;2021-12-16

3. Domain Range Semigroups and Finite Representations;Relational and Algebraic Methods in Computer Science;2021

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