On the Rank of Generalized Transformation Semigroups

AYIK G., Abusarris H. D. M.

BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY, cilt.45, sa.4, ss.1777-1787, 2022 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 45 Sayı: 4
  • Basım Tarihi: 2022
  • Doi Numarası: 10.1007/s40840-022-01267-3
  • Dergi Adı: BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.1777-1787
  • Anahtar Kelimeler: Generalized transformation semigroup, The semigroup of transformations with restricted range, Generating set, Rank
  • Çukurova Üniversitesi Adresli: Evet

Özet

Let X and Y be two non-empty (disjoint) sets, and let GT(X, Y) denote the set of all (full) transformations from X to Y. For any fixed transformation theta: Y -> X, define a sandwich operation (*) on GT(X, Y) by alpha (*) beta = alpha o theta o beta for all alpha, beta is an element of GT(X, Y). Then, the set GT(X, Y) together with the sandwich operation (*) is a semigroup which is called a generalized transformation semigroup, and it is denoted by GT(X, Y; theta). Let T(Z) denote the full transformation semigroup on the set Z = X boolean OR Y. In this paper, first we show that GT (X, Y; theta) can be embedded in T(Z, Y) = {alpha is an element of T(Z) : Z alpha subset of Y}, the semigroup of transformations with restricted range, which is a subsemigroup of T(Z). If theta : Y -> X is one to one, then we show that GT(X, Y; theta) and T (X, im (theta)) are isomorphic semigroups. Moreover, if vertical bar X vertical bar = m, vertical bar Y vertical bar = n, and vertical bar im(theta)vertical bar = r where m, n, r is an element of N, then we show that