ℋ -Trivial Transformation Semibands and Digraphs
Xiuliang Yang, Yang Haobo · Communications in Algebra · 2005
Let Sing n be the semigroup of all singular full transformations on the set X n = {1, 2,…, n} under the composition of functions. Let E(J n − 1) be the set of all idempotents of the top 𝒥-class J n − 1 = {α ∈ Sing n :|im α| = n-1}. For any nonempty subset I of E(J n − 1), the aim of this paper is to find a constructive necessary and sufficient condition for the semiband S(I) = ⟨I⟩ to be ℋ-trivial. Further, the semiband S(I) is locally maximal ℋ-trivial if S(I) is ℋ-trivial and S(I ∪ {e}) is not ℋ-trivial for any e ∈ E(J n − 1 )\I. As applications, we classify locally maximal ℋ-trivial subsemibands and locally maximal regular ℋ-trivial subsemibands of Sing n , respectively. Moreover, the characterization of which S(I) is a band is obtained.