Decompositions of semigroups induced by identities
The authors consider decompositions of semigroups induced by identities. First they give new characterizations of a semilattice of archimedean semigroups. For example, this condition is equivalent to one of the following: (1) The radical of every ideal of \(S\) is an ideal, (2) For every \(a, b \in S\) and for every positive integer \(k\) there is a positive integer \(n\) such that \((ab)^ n \in Sa^ k S\). (Two more conditions are considered i.e. (i) replace \((ab)^ n \in Sa^ k S\) by \((ab)^ n \in Sb^ kS\) or (ii) in each case above, let \(k = 2\).) By using this result, the authors describe (1) all identities which induce decompositions into a semilattice of archimedean semigroups, (2) all identities which induce decompositions of \(\pi\)-regular semigroups into a semilattice of completely archimedean semigroups, (3) all identities which induce decompositions of \(\pi\)-regular semigroups into a semilattice of nil-extensions of left groups, (4) study of identities which induce decompositions into bands of nil-extensions of groups. (A completely archimedean semigroup is an archimedean semigroup having a primitive idempotent.) Let \(u = v\) where the words \(u\) and \(v\) are homotype, namely both \(u\) and \(v\) contain the same letters. For example, `u = v' generates a semigroup which is a semilattice of archimedean semigroups if and only if \(u = v\) is not satisfied by the particular semigroup \(B_ 2 = \langle a,b \mid a^ 2 = b^ 2 = 0,\;aba = a,\;bab = b\rangle\).
scientific article; zbMATH DE number 164971
scientific article