Completing commutative actions.
A commutative monoid \(S\) is complete when every Archimedian component of \(S\) contains an idempotent, and when there is for every \(s\in S\) a least idempotent \(e\) such that \(es=s\). A commutative monoid \(S\) is subcomplete when it can be embedded into a complete commutative monoid \(T\) such that every Archimedian component of \(T\) intersects \(S\); in this case, \(T\) is called a completion of \(S\). A completion \(T\) of \(S\) is tight when \(S\) and \(T\) have the same identity element, every Archimedian component of \(S\) is contained in a unique Archimedian component of \(T\), and every \(\mathcal H\)-class of \(T\) intersects \(S\). A Ponizovsky family is a certain partition of \(S\) induced by the set of all Archimedian components of \(S\). A tight completion of \(S\) which arises from a Ponizovsky family \(\mathcal P\) of \(S\) is denoted by \(\widehat S_{\mathcal P}\). A tight \(\mathcal P\)-completion of an \(S\)-act \(X\) is a complete \(\widehat S_{\mathcal P}\)-act \(Y\) such that \(X\subseteq Y\), the inclusion mapping \(X\to Y\) is a homomorphism of \(X\) into \(Y\), and every transitivity class of \(Y\) intersects \(X\). Given a Ponizovsky family \(\mathcal P\) of a subcomplete monoid \(S\), the author defines a Prostak \(\mathcal P\)-family \(Q\) of an \(S\)-act \(X\) and constructs an \(\widehat S_{\mathcal P}\)-act \(\widehat X_{\mathcal Q}\). It is proved that if \(S\) is a subcomplete monoid (for instance, a finitely generated commutative monoid), \(\mathcal P\) a Ponizovsky family of \(S\), and \(X\) an \(S\)-act, then every tight \(\mathcal P\)-completion of \(X\) induces a cancellative Prostak \(\mathcal P\)-family of \(X\). If conversely \(\mathcal Q\) is a cancellative Prostak \(\mathcal P\)-family of \(X\), then \(\widehat X_{\mathcal Q}\) is a tight \(\mathcal P\)-completion of \(X\), which induces \(\mathcal Q\). Moreover, it is proved that \(X\) has a tight \(\mathcal P\)-completion if and only if the canonical map \(X\to\widehat S_{\mathcal P}\otimes_SX\) is injective. The canonical Prostak \(\mathcal P\)-family turns out to be the least Prostak \(\mathcal P\)-family.
scientific article; zbMATH DE number 5150260