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An extended class of exponential semigroups. (English) Zbl 0738.20056

For a given positive integer \(m\) a semigroup \(S\) is called left weakly exponential of type \(m/k\) (written l-WE-\(m/k\)) if \(S\) satisfies the identity \((xy)^{m+k}=x^ my^ m(xy)^ k\), or at least this is what I believe the author intends by his definition. The class then includes all medial (or normal) semigroups and the author claims the class of nil semigroups is included, but it would seem that this would require that the integer \(k\) depended on the values of the variables \(x\) and \(y\). The variety of exponential semigroups (those satisfying the identities \((xy)^ n=x^ ny^ n\)) is certainly included in the class.
The paper derives some basic results concerning this class, including that every l-WE-\(m/k\) semigroup is a semilattice of archimedean semigroups of the same type, and other results concerning completely 0- simple semigroups and retract extensions within the class, thus generalizing corresponding results of A. Nagy [Semigroup Forum 32, 241-250 (1985; Zbl 0571.20061)].

MSC:

20M10 General structure theory for semigroups

Citations:

Zbl 0571.20061
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