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- Title
Regular, unit-regular, and idempotent elements of semigroups of transformations that preserve a partition.
- Authors
Sarkar, Mosarof; Singh, Shubh N.
- Abstract
Let X be a nonempty set and T X the full transformation semigroup on X. For a partition P of X, we study semigroups T (X , P) = { f ∈ T X ∣ (∀ X i ∈ P) (∃ X j ∈ P) X i f ⊆ X j } , Σ (X , P) = { f ∈ T (X , P) ∣ (∀ X i ∈ P) X f ∩ X i ≠ ∅ } , and Γ (X , P) = { f ∈ T X ∣ (∀ X i ∈ P) (∃ X j ∈ P) X i f = X j } under composition. We give necessary and sufficient conditions for Γ (X , P) to be the semigroup of all closed selfmaps on X endowed with the topology having P as a basis. For finite X, we characterize unit-regular elements of T (X , P) and Σ (X , P) . We discuss set inclusions between Γ (X , P) and certain semigroups of selfmaps that preserve P . We characterize idempotents and regular elements of Γ (X , P) . For finite X, we also prove that all regular elements of Γ (X , P) are unit-regular. We finally count the number of elements, idempotents, and regular elements of Γ (X , P) for finite X.
- Subjects
IDEMPOTENTS; TOPOLOGY; PARTITIONS (Mathematics); FINITE, The; REGULAR graphs
- Publication
Semigroup Forum, 2022, Vol 104, Issue 1, p148
- ISSN
0037-1912
- Publication type
Article
- DOI
10.1007/s00233-021-10238-2