Regularity of Transformation Semigroups Defined by a Partition

Abstract

Abstract. Let X be a nonempty set, and let F = {Y i : i ∈ I} be a family of nonempty subsets of X with the properties that X = i∈I Y i , and Y i ∩ Y j = ∅ for all i, j ∈ I with i = j. Let ∅ = J ⊆ I, and let T (J )Next, we define two congruence relations χ and χ on T (J )We begin this paper by studying the regularity of the quotient semigroups T (J )

Regularity of Semigroups of Transformations Whose Characters Form the Semigroup of a Δ -Structure

“…Theorem 2 (see [11], eorem 2.6). e semigroup T (J) F (X) is regular if and only if |T (J) F (X)| � 1 or T (J) F (X) � T(X).…”

“…ere have been several works involving transformations preserving an equivalence relation (see [3][4][5][7][8][9][10][11][12] for some references). roughout this paper, let X be a nonempty set, and let F � Y i : i ∈ I be a partition of X, which are arbitrarily fixed.…”

“…roughout this paper, let X be a nonempty set, and let F � Y i : i ∈ I be a partition of X, which are arbitrarily fixed. From the definition of a partition, Purisang and Rakbud [11] defined a function χ (α) : I ⟶ I associated with each α ∈ T F (X) called the character of α, by ∀i, j ∈ I,…”

“…It is easy to see that for each α ∈ T F (X), the equivalence class [α] of α under the equivalence relation χ is a subsemigroup of T F (X) if and only if χ (α) is an idempotent element of T(I). In [11], the authors studied the regularity of the semigroup [α] when α is an idempotent element of T(I).…”

Left and Right Magnifying Elements in Generalized Semigroups of Transformations by Using Partitions of a Set

“…In 2015, Araujo, Bentz, Mitchelll and Schneider [1] solved the problem of finding the minimum size of the generating sets of T (X, P ), when P is an arbitrary partition. Next, in 2016, Purisang and Rakbud investigated the regularity of transformation semigroups which defined by a partition in [11]. These are our motivation to do this research.…”