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Zbl 1101.20033
Grillet, Pierre Antoine
Cancellative actions.
(English)
[J] Commentat. Math. Univ. Carol. 44, No. 3, 399-411 (2003). ISSN 0010-2628

Let $S$ be a semigroup, $X$ be a set. A mapping $\cdot$ from $S\times X$ into $X$ is called a left semigroup action. We say that $S$ acts simply on $X$ if $s\cdot x=t\cdot x$ implies $s=t$ and $S$ acts on $X$ transitively if for every $x,y\in X$ there exists $s\in S$ with $s\cdot x=y$. An act morphism from $(S,X,\cdot)$ into $(T,Y,\cdot)$ is a pair $(\alpha,\iota)$ where $\alpha\colon S\to T$ is a semigroup homomorphism and $\iota\colon X\to Y$ is a mapping with $\iota(s\cdot x)=\alpha(s)\cdot\iota(x)$.\par It is proved that for a left semigroup act $(S,X,\cdot)$ and a semigroup homomorphism $\alpha\colon S\to T$ there exists a universal act morphism $(\alpha,\iota)\colon(S,X,\cdot)\to(T,Y,\cdot)$ (i.e., for every act morphism $(\alpha,\beta)\colon(S,X,\cdot)\to(T,Z,\cdot)$ there exists a unique act morphism $(id_T,\gamma)\colon(T,Y,\cdot)\to(T,Z,\cdot)$ with $\beta=\gamma\circ\iota$). Hence if $(S,X,\cdot)$ is a semigroup act then for a reflexion $\gamma\colon S\to G(S)$ of $S$ into groups there exists a universal act morphism $(\gamma,\iota)\colon(S,X,\cdot)\to(G(S),Y,\cdot)$ (i.e., for every act morphism $(\alpha,\beta)\colon(S,X,\cdot)\to(G,Z,\cdot)$ where $G$ is a group there exists a unique act morphism $(\varphi,\psi)\colon(G(S),Y,\cdot)\to(G,Z,\cdot)$ with $\alpha=\varphi\circ\gamma$ and $\beta=\psi\circ\iota$). The properties of group universal acts are studied. As a consequence a necessary and sufficient condition is given such that for a left semigroup act $(S,X,\cdot)$ with a cancellative semigroup $S$ there exist a simple transitive left group act $(G(S),Y,\cdot)$ and an act morphism $(\gamma,\iota)\colon(S,X,\cdot)\to(G(S),Y,\cdot)$ such that $\iota$ is injective.
[Václav Koubek (Praha)]
MSC 2000:
*20M50 Connections of semigroups with homological algebra
18B40 Generalizations of groups viewed as categories
20M30 Representation of semigroups

Keywords: semigroup acts; universal acts; group actions; cancellative semigroups; transitive representations of groups; act morphisms; semigroup homomorphisms; universal group acts

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