Magnifiers in Some Generalization of the Full Transformation Semigroups

Thananya Kaewnoi,Montakarn Petapirak,Ronnason Chinram

doi:10.3390/math8040473

Thananya Kaewnoi, Montakarn Petapirak + Show 1 more

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https://doi.org/10.3390/math8040473

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Abstract

An element a of a semigroup S is called a left [right] magnifier if there exists a proper subset M of S such that a M = S ( M a = S ) . Let T ( X ) denote the semigroup of all transformations on a nonempty set X under the composition of functions, P = { X i ∣ i ∈ Λ } be a partition, and ρ be an equivalence relation on the set X. In this paper, we focus on the properties of magnifiers of the set T ρ ( X , P ) = { f ∈ T ( X ) ∣ ∀ ( x , y ) ∈ ρ , ( x f , y f ) ∈ ρ and X i f ⊆ X i for all i ∈ Λ } , which is a subsemigroup of T ( X ) , and provide the necessary and sufficient conditions for elements in T ρ ( X , P ) to be left or right magnifiers.

Highlights

Let T ( X ) denote the semigroup of all transformations on a nonempty set X
Let P = { Xi | i ∈ Λ} be a partition on a set X such that Xi is infinite for some i ∈ Λ and X/ρ is a refinement of P
Xi f, f is injective, which is a contradiction. It is noticeable in Lemma 9 that, if a right magnifier exists in Tρ ( X, P ), Xi is infinite for some i ∈ Λ

Summary

Introduction

Let T ( X ) denote the semigroup of all transformations on a nonempty set X. The authors published the conditions of elements being left and right magnifiers in the full and partial transformation semigroups which preserve an equivalence relation in [15] and [16], respectively. Xi = X, f is surjective which is a contradiction It is noticeable in Lemma 4 that if a left magnifier exists in Tρ ( X, P ), Xi is infinite for some i ∈ Λ. A function f is a left magnifier of Tρ ( X, P ) if and only if f ∈ Tρ ( X, P ) is injective but not surjective and for any x, y ∈ X, ( x f , y f ) ∈ ρ implies ( x, y) ∈ ρ. For all A[∈ K, there exists a bijective function γ A [

Define a function f by

Conclusions