Quasi-idempotents in certain transformation semigroups

BUGAY L.

Algebra and Discrete Mathematics, vol.37, no.2, pp.181-190, 2024 (ESCI, Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 37 Issue: 2
  • Publication Date: 2024
  • Doi Number: 10.12958/adm2223
  • Journal Name: Algebra and Discrete Mathematics
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus, zbMATH
  • Page Numbers: pp.181-190
  • Keywords: orbit, partial (full) transformations semigroup, quasi-idempotent, rank
  • Çukurova University Affiliated: Yes

Abstract

Let Pn and Tn be the partial transformations se-migroup and the (full) transformations semigroup on the set Xn = {1, …, n}, respectively. In this paper, we first state the orbit structure of quasi-idempotents (non-idempotent element whose square is an idempotent) in Pn. Then, for 2 ≤ r ≤ n − 1, we find the quasi-idempotent ranks of the subsemigroup P K(n, r) = {α ∈ Pn: h (α) ≤ r} of Pn, and the subsemigroup K(n, r) = {α ∈ Tn: h (α) ≤ r} of Tn, where h (α) denotes the cardinality of the image set of α.