대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제61권1호
  • /
  • Pages.117-133
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  • 2024
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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ON THE SEMIGROUP OF PARTITION-PRESERVING TRANSFORMATIONS WHOSE CHARACTERS ARE BIJECTIVE

  • Mosarof Sarkar (Department of Mathematics Central University of South Bihar) ;
  • Shubh N. Singh (Department of Mathematics Central University of South Bihar)
  • 투고 : 2023.01.31
  • 심사 : 2023.07.20
  • 발행 : 2024.01.31
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초록

Let 𝓟 = {Xi : i ∈ I} be a partition of a set X. We say that a transformation f : X → X preserves 𝓟 if for every Xi ∈ 𝓟, there exists Xj ∈ 𝓟 such that Xif ⊆ Xj. Consider the semigroup 𝓑(X, 𝓟) of all transformations f of X such that f preserves 𝓟 and the character (map) χ(f): I → I defined by iχ(f) = j whenever Xif ⊆ Xj is bijective. We describe Green's relations on 𝓑(X, 𝓟), and prove that 𝒟 = 𝒥 on 𝓑(X, 𝓟) if 𝓟 is finite. We give a necessary and sufficient condition for 𝒟 = 𝒥 on 𝓑(X, 𝓟). We characterize unit-regular elements in 𝓑(X, 𝓟), and determine when 𝓑(X, 𝓟) is a unit-regular semigroup. We alternatively prove that 𝓑(X, 𝓟) is a regular semigroup. We end the paper with a conjecture.

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참고문헌

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