The structure of subdirectly irreducible and uniform acts over rectangular bands

Abstract

The aim of this paper is characterizing right subdirectly irreducible completely 0-simple semigroups. We prove that such semigroups are indeed groups with least nontrivial subgroups. On the other hand we prove that right irreducible completely 0-simple semigroups are groups for which nontrivial subgroups have nontrivial intersection. Ultimately, we characterize the class of uniform acts as an overclass of subdirectly irreducible acts over rectangular bands.2010 Mathematics Subject Classification. 20M20, 20M30. Key words and phrases. completely 0-simple semigroup, rectangular band, right congruence, right subdirectly irreducible semigroup, uniform act.Definition 2.1. For a semigroup S, an S-act A is called unif orm if every nonzero subact is large in A. Also a semigroup S is called right (left) uniform if the right (left) S-act S S ( S S) is uniform.Definition 2.2. A subset B of a right S-act A is called separated, provided that for a = b ∈ B there exists s ∈ S\{1} such that as = bs.The next two results, stated below, are preliminary results of uniform acts needed in the next argument.Corollary 2.3. [9, Corollary 3.15] If S is a right uniform semigroup and xy = y for x, y ∈ S, then x is a left identity or y is a left zero.

Acts Over Semigroups

“…In particular, in the case where |Q| = 1, the act X is subdirectly irreducible if and only if |X| ≤ 2. More transparent conditions for subdirect irreducibility, as well as finite subdirect irreducibility and uniformity of acts over rectangular band were obtained in [87]. Then X is subdirectly irreducible if and only if at least one of the following conditions hold:…”

“…In [87], it was proved that if |I| < ∞, then |X| ≤ 2 |I| + 1. In addition, the number of pairwise nonisomorphic subdirectly irreducible acts over finite rectangular band is calculated.…”