Katsov, Yefim On geometrically equivalent \(S\)-acts. (English) Zbl 1162.20043 Int. J. Algebra Comput. 17, No. 5-6, 1055-1065 (2007). Let \(S\) be a monoid and \(F_X\) a finitely generated free left \(S\)-act. For any binary relation \(T\subseteq F_X\times F_X\) and any \(S\)-act \(G\), define \(T'_G=\{\mu\colon F_X\to G\mid T\subseteq\text{Ker\,}\mu\}\). Denote \(T''=(T'_G)'\). Two \(S\)-acts \(G_1,G_2\) are said to be ‘geometrically equivalent’ if \(T''_{G_1}=T''_{G_2}\) for any binary relation \(T\subseteq F_X\times F_X\) on any finitely generated free \(S\)-act \(F_X\). The author describes all \(S\)-acts geometrically equivalent to a given \(S\)-act in the case when \(S\) is a group or an Abelian group. Reviewer: Peeter Normak (Tallinn) Cited in 1 Document MSC: 20M50 Connections of semigroups with homological algebra and category theory 20M30 Representation of semigroups; actions of semigroups on sets 08C05 Categories of algebras Keywords:geometric equivalence; universal algebraic geometry; free acts over monoids; categories of acts PDF BibTeX XML Cite \textit{Y. Katsov}, Int. J. Algebra Comput. 17, No. 5--6, 1055--1065 (2007; Zbl 1162.20043) Full Text: DOI arXiv References: [1] DOI: 10.1006/jabr.1999.7881 · Zbl 0938.20020 · doi:10.1006/jabr.1999.7881 [2] DOI: 10.1007/BF02673880 · Zbl 0965.08010 · doi:10.1007/BF02673880 [3] DOI: 10.1142/S0218196701000668 · Zbl 1040.08005 · doi:10.1142/S0218196701000668 [4] DOI: 10.1090/S0002-9939-01-06108-1 · Zbl 0990.20018 · doi:10.1090/S0002-9939-01-06108-1 [5] Grätzer G., Universal Algebra (1979) [6] DOI: 10.1080/00927870601115856 · Zbl 1121.16023 · doi:10.1080/00927870601115856 [7] DOI: 10.1080/00927870601169390 · Zbl 1122.20032 · doi:10.1080/00927870601169390 [8] DOI: 10.1515/9783110812909 · doi:10.1515/9783110812909 [9] DOI: 10.1007/978-1-4612-9839-7 · doi:10.1007/978-1-4612-9839-7 [10] Maltsev A. I., Algebraic Systems (1973) [11] Plotkin B., Siberian Adv. Math. 7 pp 64– [12] Plotkin B. I., Algebra i Analiz 9 pp 224– [13] DOI: 10.1080/00927879908826679 · Zbl 1007.20023 · doi:10.1080/00927879908826679 [14] Plotkin B. I., Fundam. Prikl. Mat. 10 pp 181– This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.