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A note on the maximal semilattice of an R*NC-semigroup decomposition. (English) Zbl 0599.20099
An ideal \(P\subseteq S\) is called completely prime if for any a, b of S, ab\(\in P\) implies that either \(a\in P\) or \(b\in P\). A subsemigroup U of S is a filter of S if xy\(\in U\) implies \(x\in U\) and \(y\in U\). We consider the empty set a filter and a completely prime ideal of S. By N(J) we denote the set of all nilpotent elements of S with respect to J. The Luh radical C(J) is the intersection of all completely prime ideals of S which contain J. The Clifford radical R*(J) is the union of all nilideals of S with respect to J. A commutative semigroup, each element of which is idempotent, is called a semilattice. A congruence \(\rho\) on S is a semilattice congruence if the factor semigroup S/\(\rho\) is a semilattice. By a maximal semilattice decomposition of a semigroup S we mean a partition of S belonging to a minimal semilattice congruence on S. A semigroup S is semilattice indecomposable if the only semilattice congruence on S is the universal congruence.
We define a relation \(\eta\) on a semigroup S as follows: \(a\eta\) b if and only if \(a\in N(J(b))\) and \(b\in N(J(a))\). A semigroup S is called an R*NC-semigroup if for each ideal J of S, \(R*(J)=N(J)=C(J)\) holds.
In this note we prove that in an R*NC-semigroup S the relation \(\eta\) is equal to the minimal semilattice congruence and S is a semilattice of archimedean semigroups.

MSC:
20M10 General structure theory for semigroups
20M12 Ideal theory for semigroups
20M15 Mappings of semigroups
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