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We show that the Conway identities and the identities associated with the members of a subclass $\\mathcal{Q}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of finite automata is complete for iteration categories iff for every finite simple group G<\/jats:italic> there is an automaton Q<\/jats:bold> \u2208 $\\mathcal{Q}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> such that G<\/jats:italic> is a quotient of a group in the monoid M<\/jats:italic>(Q<\/jats:bold>) of the automaton Q<\/jats:bold>. We also prove a stronger result that concerns identities associated with finite automata with a distinguished initial state.<\/jats:p>","DOI":"10.1017\/s0960129515000031","type":"journal-article","created":{"date-parts":[[2015,4,8]],"date-time":"2015-04-08T09:02:22Z","timestamp":1428483742000},"page":"54-69","source":"Crossref","is-referenced-by-count":1,"title":["Equational axioms associated with finite automata for fixed point operations in cartesian categories"],"prefix":"10.1017","volume":"27","author":[{"given":"ZOLT\u00c1N","family":"\u00c9SIK","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2015,4,8]]},"reference":[{"key":"S0960129515000031_ref12","doi-asserted-by":"publisher","DOI":"10.1006\/inco.1998.2746"},{"key":"S0960129515000031_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-78034-9"},{"key":"S0960129515000031_ref16","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.50.5.869"},{"key":"S0960129515000031_ref13","first-page":"65","article-title":"Axiomatizing iteration categories","volume":"14","author":"\u00c9sik","year":"1999","journal-title":"Acta Cybernetica"},{"key":"S0960129515000031_ref8","volume-title":"Automata, Languages, and Machines","author":"Eilenberg","year":"1976"},{"key":"S0960129515000031_ref7","unstructured":"DeBakker J. 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