Decompositions of semigroups induced by identities

Radicals of Green's relations

“…S is an intra-π-regular semigroup. On the other hand, for any a, b ∈ S there exist n ∈ Z + such that (ab) n ∈ Sa 2 S. By Theorem 1, [4] S is a semilattice of Archimedean semigroups. Now by Theorem 2.12 [12] we have that S is a semilattice of nil-extensions of simple semigroups.…”

“…Then for all a, b ∈ S, (ab, a 2 b) ∈ J . From this it follows that for every a, b ∈ S there exists n ∈ + such that (ab) n ∈ Sa 2 S, and by Theorem 1 [4], S is a semilattice of Archimedean semigroups. Now by Theorem 3 [5] we have that → is a quasi-order.…”

“…Some other characterizations of these semigroups are given by M.Ćirić and S. Bogdanović [4] (see also the survey [3]). By the following theorem we give some new characterizations using the radicals of Green's relations.…”

On the radicals of linear algebraic monoids

“…Kmeť [14] prove that R * (I ) = √ I = C(I ) if and only if √ I is an ideal of S. And S is a semilattice of archimedean semigroups (i.e., b ∈ S 1 aS 1 ⇒ b i ∈ S 1 a 2 S 1 for some i ∈ Z + , for all a, b ∈ S), if and only if for any ideal I of S, √ I is an ideal of S (see [5] and [15]). …”

Bands of λ-simple semigroups

“…(ab) n ∈ a, ab a 2 b, for some n ∈ Z + . Now by Theorem 1 [9] S is a semilattice Y of Archimedean semigroups S α , α ∈ Y . Further, assume α ∈ Y , a, b ∈ S α .…”