Divisible and exponent actions of semigroups
Suppose that \(S\) is a monoid (i.e., semigroup with identity) and \(X\) a set. An \textit{action of \(S\) on} \(X\) is a function \((s,x)\mapsto sx\colon S\times X\to X\) such that \(1x=x\) and \((st)x=s(tx)\) for all \(s,t\in S\), \(x\in X\). For each monoid and each set we always have a \textit{trivial action} given by \(sx=x\). If \(X^ X\) denotes the monoid of all self-maps of \(X\) under composition, then there is a bijection between the set of all actions of \(S\) on \(X\) and all monoid morphisms (\textit{representations}) \(S\to X^ X\) sending \((s,x)\mapsto sx\) to \(s\mapsto (x\mapsto sx)\). (The reviewer writes actions on the left, the author on the right.) If \(X\) is itself a monoid then for an action of \(S\) on \(X\) every map \(x\mapsto sx\) is a monoid endomorphism of \(X\) iff the associated representation maps \(S\) into \(\Hom(X,X)\subseteq X^ X\). The author calls these actions \textit{exponent actions}; the reviewer's preference would be \textit{homomorphic actions}; if \(X\) is a vector space and \(\Hom(X,X)\) denotes the monoid of all linear self-maps, then one would speak of \textit{linear actions} and \textit{linear representations} anyhow. We set \(\Hom_{\text{Sur}}(X,X)=\text{Sur}(X)\cap\Hom(X,X)\). The author calls an action \textit{divisible}, respectively, \textit{uniquely divisible} if for all \(s\in S\) and \(a\in X\) the equation \(sx=a\) has a solution, respectively, a unique solution. We note that an action is divisible in this sense if the associated representation maps \(S\) into the subsemigroup \(\text{Sur}(X)\) of surjective self-maps of \(X\). [This subsemigroup is Green's \(\mathcal R\)-class \({\mathcal R}(\text{id}_ X)\) of the identity (see e.g.\ \textit{A. H. Clifford} and \textit{G. B. Preston}, The algebraic theory of semigroups, Vol. I (1961; Zbl 0111.03403), pp. 52, 53, Theorem 2.9(iv)). We also note that for a vector space \(X\) the monoid is again the \(\mathcal R\)-class of \(\text{id}_ X\) in \(\Hom(X,X)\) (see loc. cit., Exercise 6(b), p. 57).] The action is uniquely divisible if the associated representation maps \(S\) into the group of bijections of \(X\), i.e., into \({\mathcal H}(\text{id}_ X)\). The author characterizes various properties of monoids in terms of the divisibility properties of all possible actions. Theorem A: For a monoid \(S\) the following statements are equivalent: (1) \(S\) is a group. (2) For any set \(X\), any representation \(S\to X^ X\) maps \(S\) into the group \({\mathcal H}(\text{id}_ X)\) of invertible elements. (3) Every action of \(S\) is uniquely divisible. (4) Every action of \(S\) is divisible. (5) For any set \(X\), any monoid morphism \(S\to X^ X\) maps \(S\) into \(\text{Sur}(X)\). (6) For any monoid \(X\), any homomorphic representation \(S\to\Hom(X,X)\) maps \(S\) into \(\Hom_{\text{Sur}}(X,X)\). (7) For any vector space \(X\), any linear representation \(S\to\Hom(X,X)\) maps \(S\) into \(\Hom_{\text{Sur}}(X,X)\). [Proof: Clearly \((1)\Rightarrow(2)\Rightarrow\cdots\Rightarrow(7)\). Remains \((7)\Rightarrow(1)\): Fix a field \(F\) and let \(X=F[S]\) denote the semigroup ring of \(S\) over \(F\) considered as a vector space. Write \(S\subseteq F[S]\) as usual and define \(f\colon S\to\Hom(X,X)\) by \(f(s)(\sum_{t\in S} c_ t{\cdot}t)=\sum_{t\in S} c_ t{\cdot}st\). Obviously, \(f\) is a linear representation. Claim: \(S\) is right simple. For if not, then there is a proper right ideal \(R\). Then \(J=F[R]\subset X\) is a \textit{proper} right ideal of rings. Take \(r\in R\). Then \(f(r)(X)=F[rS]\subseteq J\neq X\). Hence \(f(r)\) is not surjective and thus \(f(r)\notin\Hom_{\text{Sur}}(X,X)\): The claim is true. A right simple semigroup with an idempotent is a right group (see loc. cit., p. 38, Theorem 1.27), and a monoid which is a right group is a group (loc.\ cit., same Theorem). QED] Results implicit in this theorem are scattered over the paper (cf. Proposition 3.3, Corollary 4.3.) One observes that every representation \(S\to X^ X\) maps \({\mathcal R}_ S(1)\) into \({\mathcal R}(\text{id}_ X)\), i.e., any action of \(S\) on \(X\) will always induce a divisible action of \({\mathcal R}_ S(1)\). Let \(E(S)\) denote the set of idempotents of \(S\). Then \(E({\mathcal R}(\text{id}_ X))=\{\text{id}_ X\}\). Any semigroup morphism \(S\to T\) maps \(E(S)\) into \(E(T)\). In particular, any representation \(\pi\colon S\to \text{Sur}(X)={\mathcal R}(\text{id}_ X)\) maps \(E(S)\) to \(\{\text{id}_ X\}\). If \(S\) has a completely simple minimal ideal \(M(S)\) then it follows that \(\pi(S)=\pi(M(S))\subseteq{\mathcal H}(\text{id}_ X)\) and that \(\pi(S)\) is in fact a subgroup. In particular, for a semigroup with completely simple ideal every divisible action is uniquely divisible. (Proposition 2.3.) If \(S\) is the bicyclic semigroup, (see loc. cit., pp. 43ff.) then any homomorphism collapsing all idempotents has a group image. Thus every divisible action of the bicyclic semigroup is uniquely divisible. The author calls a monoid \(S\) \textit{singular} if the trivial action is the only divisible action of \(S\), i.e., if each representation \(\pi\colon S\to {\mathcal R}(\text{id}_ X)\) is constant. This is certainly the case if \(S\) is idempotent (Corollary 1.2). It suffices that \(S\) has an ideal consisting of idempotents (Corollary 1.4), notably if \(S\) has a zero (Corollary 1.5). The author's proof of \((6)\Rightarrow(1)\) in Theorem A above uses the assertion that in a monoid \(S\) the set \(S\setminus{\mathcal H}(1)\) is a prime ideal (p. 142, lines 7 ff.). This claim is not true in general, as the bicyclic semigroup shows. [It is true e.g.\ in all compact topological semigroups, but for topological semigroups other problems arise: Actions and representations should be continuous.] We have seen above that the author's results themselves are not in question. [Small typos: In the 4th line of Section 3 read ``\(S\times T\to S\)'' instead of ``\(S\times T\to T\)''. In two places read ``verify'' instead of ``varify''. In line 5 of 4.3 read ``homomorphism''].
scientific article; zbMATH DE number 167671