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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.

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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