Combinatorial results for semigroups of orientation-preserving transformations

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DAĞDEVİREN A., AYIK G.

Turkish Journal of Mathematics, cilt.48, sa.2, ss.106-117, 2024 (SCI-Expanded) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 48 Sayı: 2
  • Basım Tarihi: 2024
  • Doi Numarası: 10.55730/1300-0098.3496
  • Dergi Adı: Turkish Journal of Mathematics
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.106-117
  • Anahtar Kelimeler: order-decreasing transformation, Order-preserving transformation, orientation-preserving transformation
  • Çukurova Üniversitesi Adresli: Evet

Özet

Let Xn denote the chain {1, 2, …, n} under its natural order. We denote the semigroups consisting of all order-preserving transformations and all orientation-preserving transformations on Xn by On and OPn, respectively. We denote by E(U) the set of all idempotents of a subset U of a semigroup S. In this paper, we first determine the cardinalities of Er(On) = {α ∈ E(On) : |im(α)| = |fix(α)| = r}, $$(On) = {α ∈ Er(On) : 1, n ∈ fix(α)}, Er(OPn) = {α ∈ E(OPn) : |fix(α)| = r}, n(OPn) = {α ∈ Er(OPn) : n ∈ fix(α)} (1 ≤ r ≤ n) and then, by using these results, we determine the numbers of idempotents in On and OPn by a new method. Let OPn denote the semigroup of all orientation-preserving and order-decreasing transformations on Xn. Moreover, we determine the cardinalities of OPn, n = {α ∈ n : fix(α) = Y} for any nonempty subset Y of Xn and OPn = {α ∈ OPn: |fix(α)| = r} for 1 ≤ r ≤ n. Also, we determine the number of idempotents in OPn and the number of nilpotents in OPn.