Free Kleene algebras with domain

TL;DR

This paper characterizes free algebras of binary relations with domain and Kleene algebra operations, showing their structure, axiomatization, and decidability of their equational theory.

Contribution

It identifies the structure of free algebras with domain and extends to Kleene algebra with domain, providing axiomatizations and decidability results.

Findings

Elements are pointed labelled finite rooted trees

Regular sets of trees form free algebras with union and reflexive transitive closure

Equational theory of Kleene algebras with domain is decidable

Abstract

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.