# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  BOSÁK J.: On radicals of semigroups. Mat. časop., 1968, 204-212. · Zbl 0169.33301  KMEŤ F.: On radicals in semigroups. Math. Slovaca, 1982, 183-188. · Zbl 0491.20050  KUCZKOWSKI J. E.: On radicals in certain class of semigroups. Mat. časop., 1970, 278-280. · Zbl 0228.20044  LAL H.: Radicals in quasi-commutative semigroups. Mat. časop., 1975, 287-288. · Zbl 0312.20035  MUKHERJEE N. P.: Quasicommutative semigroups I. Czech. Math. J. 22 (97), 1972,449-453. · Zbl 0245.20056  PETRICH M.: Introduction to semigroups. Merrill Publ. Comp., 1973. · Zbl 0321.20037  PETRICH M.: The maximal semilattice decomposition of a semigroup. Math. Z., 1964, 68-82. · Zbl 0124.25801  ŠULKA R.: О нильпотєнтных элємєнтах, идєалах и радикалах полугруппы. Mat. fyz. časop., 1963, 209-222.  ŠULKA R.: Радикалы и топология в полугруппах. Mat. fyz. časop., 1965, 3-14.  ŠULKA R.: The maximal semilattice decomposition of a semigroup, radicals and nilpotency. Mat. časop., 1970, 172-180. · Zbl 0249.20031  ŠULKA R.: The maximal semilattice decomposition of a semigroup. Mat. časop., 1971, 269-276. · Zbl 0228.20048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.