Free Kleene algebras with domain

First we identify the free algebras of the class of algebras of binary relations equipped with the composition and domain operations. Elements of the free algebras are pointed labelled finite rooted trees. Then we extend to the analogous case when the signature includes all the Kleene algebra with d...

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Bibliographic Details
Published inJournal of logical and algebraic methods in programming Vol. 117; p. 100606
Main Author McLean, Brett
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.12.2020
Subjects
Decidable
Domain
Equational theory
Kleene algebra
Kleene algebra
Domain
Equational theory
Decidable
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Summary:First we identify the free algebras of the class of algebras of binary relations equipped with the composition and domain operations. Elements of the free algebras are pointed labelled finite rooted trees. Then we extend to the analogous case when the signature includes all the Kleene algebra with domain operations; that is, we add union and reflexive transitive closure to the signature. In this second case, elements of the free algebras are ‘regular’ sets of the trees of the first case. As a corollary, the axioms of domain semirings provide a finite quasiequational axiomatisation of the equational theory of algebras of binary relations for the intermediate signature of composition, union, and domain. Next we note that our regular sets of trees are not closed under complement, but prove that they are closed under intersection. Finally, we prove that under relational semantics the equational validities of Kleene algebras with domain form a decidable set.
ISSN:2352-2208
DOI:10.1016/j.jlamp.2020.100606