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On geometrically equivalent $$S$$-acts. (English) Zbl 1162.20043
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.

##### MSC:
 20M50 Connections of semigroups with homological algebra and category theory 20M30 Representation of semigroups; actions of semigroups on sets 08C05 Categories of algebras
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##### 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–
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