{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,1,3]],"date-time":"2025-01-03T05:09:35Z","timestamp":1735880975678,"version":"3.32.0"},"reference-count":28,"publisher":"World Scientific Pub Co Pte Ltd","issue":"08","funder":[{"name":"the Austrian Science Fund","award":["FWF P33878"],"award-info":[{"award-number":["FWF P33878"]}]},{"name":"the Charles University Prague","award":["PRIMUS\/24\/SCI\/008"],"award-info":[{"award-number":["PRIMUS\/24\/SCI\/008"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2024,12]]},"abstract":"<jats:p> In this paper, we investigate properties of varieties of algebras described by a novel concept of equation that we call commutator equation. A commutator equation is a relaxation of the standard term equality obtained substituting the equality relation with the commutator relation. Namely, an algebra [Formula: see text] satisfies the commutator equation [Formula: see text] if for each congruence [Formula: see text] in [Formula: see text] and for each substitution [Formula: see text] of elements in the same [Formula: see text]-class, we have [Formula: see text]. This notion of equation draws inspiration from the definition of a weak difference term and allows for further generalization of it. Furthermore, we present an algorithm that establishes a connection between congruence equations valid in the variety generated by the abelian algebras of the idempotent reduct of a given variety and congruence equations that hold in the entire variety. Additionally, we provide a proof that if the variety generated by the abelian algebras of the idempotent reduct of a variety satisfies a nontrivial idempotent Mal\u2019cev condition, then also the entire variety satisfies a nontrivial idempotent Mal\u2019cev condition, a statement that follows also from [ 12 , Theorem 3.13]. <\/jats:p>","DOI":"10.1142\/s0218196724500541","type":"journal-article","created":{"date-parts":[[2024,10,25]],"date-time":"2024-10-25T01:01:33Z","timestamp":1729818093000},"page":"1273-1291","source":"Crossref","is-referenced-by-count":0,"title":["Commutator equations"],"prefix":"10.1142","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6918-1805","authenticated-orcid":false,"given":"Stefano","family":"Fioravanti","sequence":"first","affiliation":[{"name":"Department of Algebra, Stefano Fioravanti, Faculty of Mathematics and Physics, Charles University, Sokolovsk\u00e1 83 18675 Praha 8, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2024,12,28]]},"reference":[{"key":"S0218196724500541BIB001","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2017.37"},{"key":"S0218196724500541BIB002","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8130-3"},{"key":"S0218196724500541BIB003","first-page":"3","volume":"43","author":"Cz\u2019edli G.","year":"1981","journal-title":"Acta Sci. 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