tailieunhanh - On subdirectly irreducible regular bands

Subdirectly irreducible regular bands whose structural semilattices are finite chains are characterized in terms of a refined semilattice of semigroups. | Turk J Math (2017) 41: 1337 – 1343 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article On subdirectly irreducible regular bands Zheng-Pan WANG∗, Jing LENG, Hou-Yi YU School of Mathematics and Statistics, Southwest University, Chongqing, . China Received: • Accepted/Published Online: • Final Version: Abstract: Subdirectly irreducible regular bands whose structural semilattices are finite chains are characterized in terms of a refined semilattice of semigroups. Key words: Subdirectly irreducible semigroups, regular bands, refined semilattices of semigroups 1. Introduction and preliminaries Every nontrivial semigroup is a subdirect product of some subdirectly irreducible semigroups. Therefore, it is certainly of great importance to describe kinds of subdirectly irreducible semigroups. It is known that a nontrivial semigroup S is subdirectly irreducible if and only if S contains the least nontrivial congruence. In [1], Gerhard gave a representation of subdirectly irreducible bands (also called idempotent semigroups) in terms of transformations. However, it is not easy to construct an arbitrary subdirectly irreducible band according to Gerhard’s representation. In this paper, we give a construction for a special kind of subdirectly irreducible regular bands, whose structural semilattices form finite chains, by using a structure theorem of regular bands. To date, we are not able to give a construction of a general subdirectly irreducible regular band. First we introduce some notations and concepts. Let X be a nonempty set. Then we write the identity relation on X as εX and write the universal relation on X as ωX . If X is a partially ordered set, then for any x, y ∈ X , x is said to immediately cover y if whenever x ≥ z ≥ y , one has x = z or y = z for any z ∈ X , written as x ≻ y . Let A, B be nonempty sets. Usually, a mapping from A to the power set .