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Semigroups containing covered two-sided ideals. (English) Zbl 0601.20056

A proper two-sided ideal M of a semigroup S is said to be covered if \(M\subseteq S(S\setminus M)S\). It is shown that every non-simple semigroup has covered ideals and the latter form a sublattice in the lattice of ideals. A maximal ideal M of S is covered if and only if it is the greatest ideal of S and it holds \(S^ 2=S^ 3\). A (non-simple) semigroup S has a greatest covered ideal if and only if S as an ideal has a two-sided base \((=\) irreducible generating system). Finally, a necessary and sufficient condition is given in order that every proper ideal of a semigroup be covered.
Reviewer: L.Márki

MSC:

20M12 Ideal theory for semigroups
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References:

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