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The local semilattices of chains of idempotents. (English) Zbl 0752.06002

A set \(E\) is called dually-ordered if \(E\) admits two reflexive and transitive relations \(\to\) and \(\dualrightarrow\) such that, for all \(e,f\in E\), if \(e\to f\) and \(f\dualrightarrow e\) then \(e=f\). The pair \((\to,\dualrightarrow)\) is then called a dual-ordering of \(E\). This concept arises naturally in (and comes from) the theory of semigroups as the set \(E_ S\) of idempotents of a semigroup \(S\) by defining for \(e,f\in E_ S: e\to f\) if \(fe=e\), and \(e\dualrightarrow f\) if \(ef=e\). The first author [Proc. Workshop on monoids, Berkeley (1989)] gave necessary and sufficient conditions for a dually-ordered set \(E\) to come from a semigroup, in which case a partial multiplication may be defined on \(E\) making it into a biordered set [see K. Nambooripad, Structure of regular semigroups. I, Mem. Am. Math. Soc. 224 (1979; Zbl 0457.20051)]. A biordered set \(E\) is a local semilattice iff all the sandwich sets of \(E\) are singletons iff \(E=E_ S\) for some locally inverse semigroup \(S\). In this paper the set \(C(E)\) of all subsets \(A\) of a biordered set \(E\) such that for all \(e,f\in A\) either \(e\rightarrowtail f\) or \(f\rightarrowtail e\) (i.e. chains of idempotents) is shown to be a local semilattice. Furthermore, it is proved that \(C(E)\) contains a biordered subset of which \(E\) is a dually-ordered homomorphic image. Finally, a sufficient condition is given for the partial multiplication on a biordered set \(E\) to be uniquely determined by the dual-ordering of \(E\). For example, this condition is satisfied on locally reduced biordered sets (like local semilattices).
Reviewer: H.Mitsch (Wien)

MSC:

06A12 Semilattices
20M10 General structure theory for semigroups

Citations:

Zbl 0457.20051
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References:

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