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Periodic monoids over which all flat cyclic right acts satisfy condition (P). (English) Zbl 0868.20051

Let \(S\) be a monoid. A right \(S\)-act \(A_S\) is flat (weakly flat) if the functor \(A\otimes-\) preserves monomorphisms (injections of left ideals of \(S\) into \(S\)). The following condition is denoted by (P): If \(as=a't\) for \(a,a'\in A\), \(s,t\in S\) then there exist \(a''\in A\) and \(u,v\in S\) such that \(a=a''u\), \(a'=a''v\), and \(us=vt\). It is proved that for a periodic monoid \(S\), the following conditions are equivalent: 1) \(S=G\dot\cup N\) where \(G\) is a group and either \(N=\emptyset\) or every element of \(N\) is right nil; 2) every weakly flat cyclic right \(S\)-act satisfies condition (P); 3) every flat cyclic right \(S\)-act satisfies condition (P). It is also shown that the implication \(1)\Rightarrow 2)\) holds for arbitrary monoids.
Reviewer: P.Normak (Tallinn)

MSC:

20M50 Connections of semigroups with homological algebra and category theory
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References:

[1] S. Bulman-Fleming,Flat and Strongly Flat S-systems, Communications in Algebra20(9) (1992), 2553–2567. · Zbl 0792.20065 · doi:10.1080/00927879208824478
[2] S. Bulman-Fleming, and P. Normak,Monoids over which all Flat Cyclic Right Acts are Strongly Flat, Semigroup Forum50 (1995), 223–241. · Zbl 0824.20054 · doi:10.1007/BF02573519
[3] Liu Zhongkui,Characterization of monoids by condition (P) of cyclic left acts, Semigroup Forum49 (1994), 31–39. · Zbl 0804.20048 · doi:10.1007/BF02573468
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