scholarly journals Skew Pairs of Idempotents in Partial Transformation Semigroups

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 2
Author(s):  
Panuwat Luangchaisri ◽  
Thawhat Changphas
Keyword(s):  
Regular Semigroup ◽  
Transformation Semigroup ◽  
Acta Math ◽  
Partial Transformation ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Full Transformation Semigroup ◽  
Left Regular

Let S be a regular semigroup. A pair (e,f) of idempotents of S is said to be a skew pair of idempotents if fe is idempotent, but ef is not. T. S. Blyth and M. H. Almeida (T. S. Blyth and M. H. Almeida, skew pair of idempotents in transformation semigroups, Acta Math. Sin. (English Series), 22 (2006), 1705–1714) gave a characterization of four types of skew pairs—those that are strong, left regular, right regular, and discrete—existing in a full transformation semigroup T(X). In this paper, we do in this line for partial transformation semigroups.

scholarly journals On transformation semigroups which areℬ𝒬-semigroups

2006 ◽  
Vol 2006 ◽  
pp. 1-10
Author(s):  
S. Nenthein ◽  
Y. Kemprasit
Keyword(s):  
Vector Space ◽  
Regular Semigroup ◽  
Transformation Semigroup ◽  
Transformation Semigroups ◽  
Linear Transformations ◽  
Full Transformation ◽  
Full Transformation Semigroup

A semigroup whose bi-ideals and quasi-ideals coincide is called aℬ𝒬-semigroup. The full transformation semigroup on a setXand the semigroup of all linear transformations of a vector spaceVover a fieldFinto itself are denoted, respectively, byT(X)andLF(V). It is known that every regular semigroup is aℬ𝒬-semigroup. Then bothT(X)andLF(V)areℬ𝒬-semigroups. In 1966, Magill introduced and studied the subsemigroupT¯(X,Y)ofT(X), where∅≠Y⊆XandT¯(X,Y)={α∈T(X,Y)|Yα⊆Y}. IfWis a subspace ofV, the subsemigroupL¯F(V,W)ofLF(V)will be defined analogously. In this paper, it is shown thatT¯(X,Y)is aℬ𝒬-semigroup if and only ifY=X,|Y|=1, or|X|≤3, andL¯F(V,W)is aℬ𝒬-semigroup if and only if (i)W=V, (ii)W={0}, or (iii)F=ℤ2,dimFV=2, anddimFW=1.

scholarly journals Automorphisms of normal partial transformation semigroups

1987 ◽  
Vol 29 (2) ◽  
pp. 149-157 ◽  
Author(s):  
Inessa Levi
Keyword(s):  
Symmetric Group ◽  
Transformation Semigroup ◽  
Partial Transformation ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Full Transformation Semigroup ◽  
Infinite Set

We let X be an arbitrary infinite set. A semigroup S of total or partial transformations of X is called -normal if hSh-1 = S, for all h in , the symmetric group on X. For example, the full transformation semigroup , the semigroup of all partial transformations , the semigroup of all 1–1 partial transformations and all ideals of and are -normal.

Naturally Ordered Transformation Semigroups Preserving an Equivalence Relation and a Cross-section

2011 ◽  
Vol 18 (03) ◽  
pp. 523-532 ◽  
Author(s):  
Lei Sun ◽  
Weina Deng ◽  
Huisheng Pei
Keyword(s):  
Cross Section ◽  
Equivalence Relation ◽  
Transformation Semigroup ◽  
Natural Order ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Minimal Elements ◽  
Full Transformation Semigroup

The paper is concerned with the so-called natural order on the semigroup [Formula: see text], where [Formula: see text] is the full transformation semigroup on a set X, E is a non-trivial equivalence on X and R is a cross-section of the partition X/E induced by E. We determine when two elements of TE(X,R) are related under this order, find elements of TE(X,R) which are compatible with ≤ on TE(X,R), and observe the maximal and minimal elements and the covering elements.

scholarly journals The structure of elements in finite full transformation semigroups

2005 ◽  
Vol 71 (1) ◽  
pp. 69-74 ◽  
Author(s):  
Gonca Ayik ◽  
Hayrullah Ayik ◽  
Yusuf Ünlü ◽  
John M. Howie
Keyword(s):  
Finite Semigroup ◽  
Transformation Semigroup ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Full Transformation Semigroup

The index and period of an element a of a finite semigroup are the smallest values of m ≥ 1 and r ≥ 1 such that am+r = am. An element with index m and period 1 is called an m-potent element. For an element α of a finite full transformation semigroup with index m and period r, a unique factorisation α = σβ such that Shift(σ) ∩ Shift(β) = ∅ is obtained, where σ is a permutation of order r and β is an m-potent. Some applications of this factorisation are given.

scholarly journals NATURALLY ORDERED TRANSFORMATION SEMIGROUPS PRESERVING AN EQUIVALENCE

2008 ◽  
Vol 78 (1) ◽  
pp. 117-128 ◽  
Author(s):  
LEI SUN ◽  
HUISHENG PEI ◽  
ZHENGXING CHENG
Keyword(s):  
Transformation Semigroup ◽  
Natural Order ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Minimal Elements ◽  
Full Transformation Semigroup ◽  
Image Position

AbstractLet 𝒯X be the full transformation semigroup on a set X and E be a nontrivial equivalence on X. Write then TE(X) is a subsemigroup of 𝒯X. In this paper, we endow TE(X) with the so-called natural order and determine when two elements of TE(X) are related under this order, then find out elements of TE(X) which are compatible with ≤ on TE(X). Also, the maximal and minimal elements and the covering elements are described.

A PARTIAL ORDER ON TRANSFORMATION SEMIGROUPS THAT PRESERVE DOUBLE DIRECTION EQUIVALENCE RELATION

2013 ◽  
Vol 12 (08) ◽  
pp. 1350041 ◽  
Author(s):  
LEI SUN ◽  
JUNLING SUN
Keyword(s):  
Lower Bound ◽  
Partial Order ◽  
Equivalence Relation ◽  
Transformation Semigroup ◽  
Natural Partial Order ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Minimal Elements ◽  
Full Transformation Semigroup

Let [Formula: see text] be the full transformation semigroup on a nonempty set X and E be an equivalence relation on X. Then [Formula: see text] is a subsemigroup of [Formula: see text]. In this paper, we endow it with the natural partial order. With respect to this partial order, we determine when two elements are related, find the elements which are compatible and describe the maximal (minimal) elements. Also, we investigate the greatest lower bound of two elements.

THE NATURAL ORDER FOR THE E-ORDER-PRESERVING TRANSFORMATION SEMIGROUPS

2012 ◽  
Vol 05 (03) ◽  
pp. 1250035 ◽  
Author(s):  
Huisheng Pei ◽  
Weina Deng
Keyword(s):  
Transformation Semigroup ◽  
Natural Order ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Minimal Elements ◽  
Full Transformation Semigroup ◽  
Finite Set

Let (X, ≤) be a totally ordered finite set, [Formula: see text] be the full transformation semigroup on X and E be an arbitrary equivalence on X. We consider a subsemigroup of [Formula: see text] defined by [Formula: see text] and call it the E-order-preserving transformation semigroup on X. In this paper, we endow EOPX with the so-called natural order ≤ and discuss when two elements in EOPX are related under this order, then determine those elements of EOPX which are compatible with ≤. Also, the maximal (minimal) elements are described.

Some subsemigroups of infinite full transformation semigroups

1981 ◽  
Vol 88 (1-2) ◽  
pp. 159-167 ◽  
Author(s):  
J. M. Howie
Keyword(s):  
Transformation Semigroup ◽  
Infinite Cardinal ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Full Transformation Semigroup ◽  
Cardinal Numbers ◽  
Infinite Set

SynopsisAs in an earlier paper by the author, three cardinal numbers, the shift, the defect and the collapse, are associated with each element of the full transformation semigroup ℑ(X), where X is an infinite set. It is shown that the elements of finite shift and non-zero defect form a subsemigroup F of ℑ(X). Moreover, if E(F) denotes the set of idempotents in F then 〈E(F)〉 = F, but (E(F))n ⊂F for every finite n. For each infinite cardinal m not exceeding ∣X∣ the set Qm of balanced elements of weight m, i.e. those with shift, defect and collapse all equal to m, forms a subsemigroup of ℑ(X). Moreover, (E(Qm))4=Qm,(E(Qm))3⊂Qm.

scholarly journals Computation of the transformation semigroups on three letters

1972 ◽  
Vol 14 (3) ◽  
pp. 335-335 ◽  
Author(s):  
Carroll Wilde ◽  
Sharon Raney
Keyword(s):  
Transformation Semigroup ◽  
Functional Composition ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Full Transformation Semigroup

Let H denote a set with three elements, and T3 the full transformation semigroup on X, i.e. T3 consists of the twenty-seven self maps of X under functional composition. A transformation semigroup (briefly a τ-semigroup) on three letters is an ordered pair (X, S), where S is any subsemigroup of T3.

scholarly journals Finite full transformation semigroups as collections of random functions

1988 ◽  
Vol 30 (2) ◽  
pp. 203-211 ◽  
Author(s):  
B. Brown ◽  
P. M. Higgins
Keyword(s):  
Semigroup Theory ◽  
Transformation Semigroup ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Random Functions ◽  
Full Transformation Semigroup ◽  
Finite Set ◽  
Algebraic Semigroup

The collection of all self-maps on a non-empty set X under composition is known in algebraic semigroup theory as the full transformation semigroup on X and is written x. Its importance lies in the fact that any semigroup S can be embedded in the full transformation semigroup (where S1 is the semigroup S with identity 1 adjoined, if S does not already possess one). The proof is similar to Cayley's Theorem that a group G can be embedded in SG, the group of all bijections of G to itself. In this paper X will be a finite set of order n, which we take to be and so we shall write Tn for X.

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