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The maximal semigroup of quotients of a finite semilattice. (English) Zbl 0542.20034

Let S be a subsemigroup of a semigroup \(T=(T,\cdot)\). Then S is called dense in T and T a semigroup of right quotients of S (in the meaning of McMorris) iff for any \(t_ 1,t_ 2,t\in T\) such that \(t_ 1\neq t_ 2\) there is an \(s\in S\) satisfying \(t_ 1s\neq t_ 2s\) and t\(s\in S\). Denoting this by \(S\leq T\), for each semigroup S such that \(S\leq S\) there is, unique up to isomorphisms, a maximal semigroup \(Q_ r(S)\) with \(S\leq Q_ r(S)\), and \(S\leq T\) implies \(Q_ r(S)=Q_ r(T)\) [for details and references cf. H. J. Weinert, Semigroup Forum 19, 1-78 (1980; Zbl 0404.20051)]. If \(S=(S,\cdot)=(S,\wedge)\) is a semilattice, one easily checks \(S\leq S\) and that \(Q_ r(S)\) is a semilattice, too. Investigating finite semilattices, the author proves that for each S there is a unique minimal subsemigroup \(S_ I\) of S satisfying \(S_ I\leq S\), which is at the same time the minimal dense ideal of S, and \(S\leq T\) holds iff \(S_ I=T_ I\). Moreover, interesting explicit descriptions of \(S_ I\) and \(Q_ r(S)\) are given. The latter allows to prove a handy necessary and sufficient condition for \(S=Q_ r(S)\), called fractional completeness of S. As an application we mention: The variety of distributive lattices is the unique variety of lattices all of whose finite members are fractionally complete. Turning to rings in the last section, each finite commutative ring R is shown to be fractionally complete, which results in the problem whether the variety of commutative rings is also characterized by this property.
Reviewer: H.J.Weinert

MSC:

20M10 General structure theory for semigroups
06A12 Semilattices
06D05 Structure and representation theory of distributive lattices
06B20 Varieties of lattices
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References:

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