On Magnifying Elements in E-Preserving Partial Transformation Semigroups

Let S be a semigroup. An element a of S is called a right [left] magnifying element if there exists a proper subset M of S satisfying S = M a [ S = a M ] . Let E be an equivalence relation on a nonempty set X. In this paper, we consider the semigroup P ( X , E ) consisting of all E-preserving partial transformations, which is a subsemigroup of the partial transformation semigroup P ( X ) . The main propose of this paper is to show the necessary and sufficient conditions for elements in P ( X , E ) to be right or left magnifying.

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Magnifiers in Some Generalization of the Full Transformation Semigroups

Mathematics ◽

10.3390/math8040473 ◽

2020 ◽

Vol 8 (4) ◽

pp. 473

Author(s):

Thananya Kaewnoi ◽

Montakarn Petapirak ◽

Ronnason Chinram

Keyword(s):

Equivalence Relation ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Proper Subset ◽

Transformation Semigroups ◽

Full Transformation ◽

Necessary And Sufficient ◽

Composition Of Functions

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.

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Square roots in finite full transformation semigroups

Glasgow Mathematical Journal ◽

10.1017/s0017089500004912 ◽

1982 ◽

Vol 23 (2) ◽

pp. 137-149 ◽

Cited By ~ 5

Author(s):

Mary Snowden ◽

J. M. Howie

Keyword(s):

Transformation Semigroup ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Necessary Condition ◽

Square Root ◽

Transformation Semigroups ◽

Full Transformation ◽

Square Roots ◽

Finite Set ◽

Necessary And Sufficient

Let X be a finite set and let (X) be the full transformation semigroup on X, i.e. the set of all mappings from X into X, the semigroup operation being composition of mappings. This paper aims to characterize those elements of (X) which have square roots. An easily verifiable necessary condition, that of being quasi-square, is found in Theorem 2, and in Theorems 4 and 5 we find necessary and sufficient conditions for certain special elements of (X). The property of being compatibly amenable is shown in Theorem 7 to be equivalent for all elements of (X) to the possession of a square root.

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Left and Right Magnifying Elements in Generalized Semigroups of Transformations by Using Partitions of a Set

European Journal of Pure and Applied Mathematics ◽

10.29020/nybg.ejpam.v11i3.3260 ◽

2018 ◽

Vol 11 (3) ◽

pp. 580-588

Author(s):

Ronnason Chinram ◽

Pattarawan Petchkaew ◽

Samruam Baupradist

Keyword(s):

Linear Transformation ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Proper Subset ◽

Transformation Semigroups ◽

Left And Right ◽

Necessary And Sufficient ◽

Composition Of Functions

An element a of a semigroup S is called left [right] magnifying if there exists a proper subset M of S such that S = aM [S = Ma]. Let X be a nonempty set and T(X) be the semigroup of all transformation from X into itself under the composition of functions. For a partition P = {X_α | α ∈ I} of the set X, let T(X,P) = {f ∈ T(X) | (X_α)f ⊆ X_α for all α ∈ I}. Then T(X,P) is a subsemigroup of T(X) and if P = {X}, T(X,P) = T(X). Our aim in this paper is to give necessary and suﬃcient conditions for elements in T(X,P) to be left or right magnifying. Moreover, we apply those conditions to give necessary and suﬃcient conditions for elements in some generalized linear transformation semigroups.

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Magnifying Elements in Semigroups of Fixed Point Set Transformations Restricted by an Equivalence Relation

Journal of Mathematics ◽

10.1155/2021/4332094 ◽

2021 ◽

Vol 2021 ◽

pp. 1-7

Author(s):

Thananya Kaewnoi ◽

Ronnason Chinram ◽

Montakarn Petapirak

Keyword(s):

Fixed Point ◽

Equivalence Relation ◽

Nonempty Subset ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Fixed Point Set ◽

Point Set ◽

Necessary And Sufficient ◽

Semigroup Of Transformations

Let X be a nonempty set and ρ be an equivalence relation on X . For a nonempty subset S of X , we denote the semigroup of transformations restricted by an equivalence relation ρ fixing S pointwise by E F S X , ρ . In this paper, magnifying elements in E F S X , ρ will be investigated. Moreover, we will give the necessary and sufficient conditions for elements in E F S X , ρ to be right or left magnifying elements.

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Characterization of finite amenable transformation semigroups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700012799 ◽

1973 ◽

Vol 15 (1) ◽

pp. 86-93 ◽

Cited By ~ 2

Author(s):

Carroll Wilde

Keyword(s):

Transformation Semigroup ◽

Sufficient Conditions ◽

Mean Value ◽

Explicit Description ◽

Transformation Semigroups ◽

Shift Operators ◽

Finite Transformation ◽

Invariant Means ◽

Necessary And Sufficient

Abstract. In this paper we develop necessary and sufficient conditions for a finite transformation semigroup to have a mean value which is invariant under the induced shift operators. The structure of such transformation semigroups is described and an explicit description of all possible invariant means given.

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Extending abelian groups to rings

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700036132 ◽

2007 ◽

Vol 82 (3) ◽

pp. 297-314 ◽

Cited By ~ 2

Author(s):

Lynn M. Batten ◽

Robert S. Coulter ◽

Marie Henderson

Keyword(s):

Abelian Group ◽

Commutative Ring ◽

Equivalence Relation ◽

Binary Operation ◽

Sufficient Conditions ◽

Additive Group ◽

Necessary And Sufficient Conditions ◽

Odd Order ◽

Weak Isomorphism ◽

Necessary And Sufficient

AbstractFor any abelian group G and any function f: G → G we define a commutative binary operation or ‘multiplication’ on G in terms of f. We give necessary and sufficient conditions on f for G to extend to a commutative ring with the new multiplication. In the case where G is an elementary abelian p–group of odd order, we classify those functions which extend G to a ring and show, under an equivalence relation we call weak isomorphism, that there are precisely six distinct classes of rings constructed using this method with additive group the elementary abelian p–group of odd order p2.

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NEUTRAL ELEMENTS, FUNDAMENTAL RELATIONS AND n-ARY HYPERSEMIGROUPS

International Journal of Algebra and Computation ◽

10.1142/s0218196709005226 ◽

2009 ◽

Vol 19 (04) ◽

pp. 567-583 ◽

Cited By ~ 30

Author(s):

B. DAVVAZ ◽

W. A. DUDEK ◽

S. MIRVAKILI

Keyword(s):

Equivalence Relation ◽

Identity Element ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Fundamental Relation ◽

Necessary And Sufficient

The main tools in the theory of n-ary hyperstructures are the fundamental relations. The fundamental relation on an n-ary hypersemigroup is defined as the smallest equivalence relation so that the quotient would be the n-ary semigroup. In this paper we study neutral elements in n-ary hypersemigroups and introduce a new strongly compatible equivalence relation on an n-ary hypersemigroup so that the quotient is a commutative n-ary semigroup. Also we determine some necessary and sufficient conditions so that this relation is transitive. Finally, we prove that this relation is transitive on an n-ary hypergroup with neutral (identity) element.

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Toeplitz Matrices in the Problem of Semiscalar Equivalence of Second-Order Polynomial Matrices

International Journal of Analysis ◽

10.1155/2017/6701078 ◽

2017 ◽

Vol 2017 ◽

pp. 1-14 ◽

Cited By ~ 4

Author(s):

B. Z. Shavarovskii

Keyword(s):

Equivalence Relation ◽

Polynomial Matrix ◽

Toeplitz Matrices ◽

Sufficient Conditions ◽

Second Order ◽

Necessary And Sufficient Conditions ◽

Polynomial Matrices ◽

Order Polynomial ◽

Necessary And Sufficient

We consider the problem of determining whether two polynomial matrices can be transformed to one another by left multiplying with some nonsingular numerical matrix and right multiplying by some invertible polynomial matrix. Thus the equivalence relation arises. This equivalence relation is known as semiscalar equivalence. Large difficulties in this problem arise already for 2-by-2 matrices. In this paper the semiscalar equivalence of polynomial matrices of second order is investigated. In particular, necessary and sufficient conditions are found for two matrices of second order being semiscalarly equivalent. The main result is stated in terms of determinants of Toeplitz matrices.

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Left and Right Magnifying Elements in the Semigroup of all Binary Relations

European Journal of Pure and Applied Mathematics ◽

10.29020/nybg.ejpam.v13i4.3870 ◽

2020 ◽

Vol 13 (4) ◽

pp. 987-994

Author(s):

Watchara Teparos ◽

Soontorn Boonta ◽

Thitiya Theparod

Keyword(s):

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Proper Subset ◽

Binary Relations ◽

Left And Right ◽

Necessary And Sufficient

An element a of a semigroup S is called left [right] magnifying if there exists a proper subset M of S such that S = aM [S = M a]. Let X be a nonempty set and BX the semigroup of binary relations on X. In this paper, we give necessary and sufficient conditions for elements in BX to be left or right magnifying.

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Left (right) magnifying elements of a partial transformation semigroup

Asian-European Journal of Mathematics ◽

10.1142/s1793557120500163 ◽

2018 ◽

Vol 13 (01) ◽

pp. 2050016

Author(s):

Panuwat Luangchaisri ◽

Thawhat Changphas ◽

Chalida Phanlert

Keyword(s):

Semigroup Forum ◽

Transformation Semigroup ◽

Proper Subset ◽

Partial Transformation ◽

Transformation Semigroups

An element [Formula: see text] of a semigroup [Formula: see text] is a called a left (respectively right) magnifying element of [Formula: see text] if [Formula: see text] (respectively [Formula: see text]) for some proper subset [Formula: see text] of [Formula: see text]. In this paper, left magnifying elements and right magnifying elements of a partial transformation semigroup will be characterized. The results obtained generalize the results of Magill [K. D. Magill, Magnifying elements of transformation semigroups, Semigroup Forum, 48 (1994) 119–126].

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