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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| Published in | Journal of logical and algebraic methods in programming Vol. 117; p. 100606 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
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Elsevier Inc
01.12.2020
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| ISSN | 2352-2208 |
| DOI | 10.1016/j.jlamp.2020.100606 |
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| 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. |
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| AbstractList | 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. |
| ArticleNumber | 100606 |
| Author | McLean, Brett |
| Author_xml | – sequence: 1 givenname: Brett surname: McLean fullname: McLean, Brett email: brett.mclean@unice.fr organization: Laboratoire J. A. Dieudonné UMR CNRS 7351, Université Nice Sophia Antipolis, 06108 Nice Cedex 02, France |
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| Cites_doi | 10.1016/j.jlap.2010.07.019 10.1142/S0218196709005214 10.1112/plms/s3-29.3.385 10.1112/jlms/s1-29.4.396 10.1145/1183278.1183285 10.1145/256167.256195 10.2307/2268577 10.1016/j.scico.2010.05.007 10.1023/A:1008271805106 10.1007/BF01190768 10.1016/0022-0000(79)90046-1 10.1145/1456650.1456653 10.1016/0022-0000(80)90061-6 10.1006/inco.1994.1037 10.1017/S0017089500008168 10.1016/j.jpaa.2015.12.007 |
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| Keywords | Kleene algebra Domain Equational theory Decidable |
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