×

Cover for \(S\)-acts and condition (A) for a monoid \(S\). (English) Zbl 1327.20058

Let \(S\) be a monoid and \(X\) a property of left \(S\)-acts. Then \(\mathcal IX\) denotes the class of left \(S\)-acts, the indecomposable components of which have property \(X\). A monoid \(S\) satisfies ‘Condition (A)’ if every locally cyclic left \(S\)-act is cyclic. For a class \(\mathcal W\) of left \(S\)-acts, \(S\) is said to be ‘left \(\mathcal W\)-perfect’ if every left \(S\)-act has a \(\mathcal W\)-cover (a cover lying in \(\mathcal W\)).
The general result says that if \(X\) is a property of left \(S\)-acts that is stronger than being locally cyclic, then \(S\) is left \(\mathcal IX\)-perfect if and only if every cyclic \(S\)-act has an \(\mathcal IX\)-cover and Condition (A) holds.
Let now \(B\) be a left \(S\)-act, \(x\in B\) an element and \(C\) the property of left \(S\)-acts of being cyclic. Denote \(L(x)=\{t\in S:tx=x\}\). It is proved (among a number of applications, examples and counterexamples) that if \(S\) is commutative and \(\mathcal IX\subseteq\mathcal IC\), then the following are equivalent: 1) \(S\) is \(\mathcal IX\)-perfect; 2) every strongly flat \(S\)-act is in \(\mathcal IX\); 3) \(S\) satisfies Condition (A) and for any unitary submonoid \(T\) of \(S\), there exists a cyclic \(S\)-act \(Sa\) with property \(X\) such that for any \(p,q\in S\), if \(pa=qa\) then \(pt=qt\) for some \(t\in T\), and \(Su\cap L(a)\neq\emptyset\) for any \(u\in T\).

MSC:

20M50 Connections of semigroups with homological algebra and category theory
20M30 Representation of semigroups; actions of semigroups on sets
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1080/00927870008827000 · Zbl 0953.20050 · doi:10.1080/00927870008827000
[2] DOI: 10.1515/9783110812909 · doi:10.1515/9783110812909
[3] DOI: 10.1007/BF02574099 · Zbl 0844.20051 · doi:10.1007/BF02574099
[4] DOI: 10.1007/BF02572283 · Zbl 0224.20061 · doi:10.1007/BF02572283
[5] DOI: 10.1080/00927870903390660 · Zbl 1222.20046 · doi:10.1080/00927870903390660
[6] DOI: 10.1007/BF02572973 · Zbl 0231.18013 · doi:10.1007/BF02572973
[7] DOI: 10.1017/S0013091500010592 · Zbl 0356.20060 · doi:10.1017/S0013091500010592
[8] DOI: 10.1007/PL00006014 · Zbl 0957.20051 · doi:10.1007/PL00006014
[9] DOI: 10.1017/S0017089511000504 · Zbl 1235.20057 · doi:10.1017/S0017089511000504
[10] DOI: 10.1007/s00233-008-9094-0 · Zbl 1161.20054 · doi:10.1007/s00233-008-9094-0
[11] DOI: 10.1007/s00233-010-9237-y · Zbl 1205.06010 · doi:10.1007/s00233-010-9237-y
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.