Equational unification is the problem of solving an equation modulo equational axioms. In this paper, we provide a relationship between equational unification and homological algebra for equational theories. We will construct a functor from the category of sets of equational axioms to the category of abelian groups. Then, our main theorem gives a necessary condition of equational unifiability that is described in terms of abelian groups associated with equational axioms and homomorphisms between them. To construct our functor, we use a ringoid (a category enriched over the category of abelian groups) obtained from the equational axioms and a free resolution of a "good" module over the ringoid, which was developed by Malbos and Mimram.
@InProceedings{ikebuchi:LIPIcs.MFCS.2021.61, author = {Ikebuchi, Mirai}, title = {{A Homological Condition on Equational Unifiability}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {61:1--61:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.61}, URN = {urn:nbn:de:0030-drops-145010}, doi = {10.4230/LIPIcs.MFCS.2021.61}, annote = {Keywords: Equational unification, Homological algebra, equational theories} }
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