scholarly journals NATURAL PARTIAL ORDER IN SEMIGROUPS OF TRANSFORMATIONS WITH INVARIANT SET

2012 ◽  
Vol 87 (1) ◽  
pp. 94-107 ◽  
Author(s):  
LEI SUN ◽  
LIMIN WANG
Keyword(s):  
Lower Bound ◽  
Partial Order ◽  
Nonempty Subset ◽  
Transformation Semigroup ◽  
Invariant Set ◽  
Natural Partial Order ◽  
Maximal Elements ◽  
Full Transformation ◽  
Minimal Elements ◽  
Image Position

AbstractLet 𝒯X be the full transformation semigroup on the nonempty set X. We fix a nonempty subset Y of X and consider the semigroup of transformations that leave Y invariant, and endow it with the so-called natural partial order. Under this partial order, we determine when two elements of S(X,Y ) are related, find the elements which are compatible and describe the maximal elements, the minimal elements and the greatest lower bound of two elements. Also, we show that the semigroup S(X,Y ) is abundant.

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.

scholarly journals THE NATURAL PARTIAL ORDER ON THE SEMIGROUP OF ALL TRANSFORMATIONS OF A SET THAT REFLECT AN EQUIVALENCE RELATION

2013 ◽  
Vol 88 (3) ◽  
pp. 359-368
Author(s):  
LEI SUN ◽  
XIANGJUN XIN
Keyword(s):  
Partial Order ◽  
Equivalence Relation ◽  
Transformation Semigroup ◽  
Natural Partial Order ◽  
Maximal Elements ◽  
Full Transformation ◽  
Full Transformation Semigroup ◽  
Image Position

AbstractLet ${ \mathcal{T} }_{X} $ be the full transformation semigroup on a set $X$ and $E$ be a nontrivial equivalence relation on $X$. Denote $$\begin{eqnarray*}{T}_{\exists } (X)= \{ f\in { \mathcal{T} }_{X} : \forall x, y\in X, (f(x), f(y))\in E\Rightarrow (x, y)\in E\} ,\end{eqnarray*}$$ so that ${T}_{\exists } (X)$ is a subsemigroup of ${ \mathcal{T} }_{X} $. In this paper, we endow ${T}_{\exists } (X)$ with the natural partial order and investigate when two elements are related, then find elements which are compatible. Also, we characterise the minimal and maximal elements.

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 NOTE ON NATURALLY ORDERED SEMIGROUPS OF TRANSFORMATIONS WITH INVARIANT SET

2014 ◽  
Vol 91 (2) ◽  
pp. 264-267 ◽  
Author(s):  
LEI SUN ◽  
JUNLING SUN
Keyword(s):  
Partial Order ◽  
Short Note ◽  
Invariant Set ◽  
Natural Partial Order ◽  
Ordered Semigroups ◽  
Image Position

AbstractIn this short note, we describe all the elements in the semigroup $$\begin{eqnarray}S(X,Y)=\{f\in {\mathcal{T}}_{X}:f(Y)\subseteq Y\}\end{eqnarray}$$ which are left compatible with respect to the so-called natural partial order. This result corrects an error in a paper by Sun and Wang [‘Natural partial order in semigroups of transformations with invariant set’, Bull. Aust. Math. Soc.87 (2013), 94–107].

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.

THE NATURAL PARTIAL ORDER ON SOME TRANSFORMATION SEMIGROUPS

2013 ◽  
Vol 89 (2) ◽  
pp. 279-292 ◽  
Author(s):  
SUREEPORN CHAOPRAKNOI ◽  
TEERAPHONG PHONGPATTANACHAROEN ◽  
PATTANACHAI RAWIWAN
Keyword(s):  
Partial Order ◽  
Partially Ordered Sets ◽  
Natural Partial Order ◽  
Transformation Semigroups ◽  
Ordered Sets ◽  
Maximal Elements ◽  
Partially Ordered

AbstractFor a semigroup $S$, let ${S}^{1} $ be the semigroup obtained from $S$ by adding a new symbol 1 as its identity if $S$ has no identity; otherwise let ${S}^{1} = S$. Mitsch defined the natural partial order $\leqslant $ on a semigroup $S$ as follows: for $a, b\in S$, $a\leqslant b$ if and only if $a= xb= by$ and $a= ay$ for some $x, y\in {S}^{1} $. In this paper, we characterise the natural partial order on some transformation semigroups. In these partially ordered sets, we determine the compatibility of their elements, and find all minimal and maximal elements.

scholarly journals Regularity and Green's Relations on a Semigroup of Transformations with Restricted Range

2008 ◽  
Vol 2008 ◽  
pp. 1-11 ◽  
Author(s):  
Jintana Sanwong ◽  
Worachead Sommanee
Keyword(s):  
Nonempty Subset ◽  
Transformation Semigroup ◽  
Green’S Relations ◽  
Sufficient Condition ◽  
Restricted Range ◽  
Necessary And Sufficient Condition ◽  
Full Transformation ◽  
Green's Relations ◽  
Necessary And Sufficient ◽  
Semigroup Of Transformations

LetT(X)be the full transformation semigroup on the setXand letT(X,Y)={α∈T(X):Xα⊆Y}. ThenT(X,Y)is a sub-semigroup ofT(X)determined by a nonempty subsetYofX. In this paper, we give a necessary and sufficient condition forT(X,Y)to be regular. In the case thatT(X,Y)is not regular, the largest regular sub-semigroup is obtained and this sub-semigroup is shown to determine the Green's relations onT(X,Y). Also, a class of maximal inverse sub-semigroups ofT(X,Y)is obtained.

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.

scholarly journals The Regular Part of a Semigroup of Full Transformations with Restricted Range: Maximal Inverse Subsemigroups and Maximal Regular Subsemigroups of Its Ideals

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Worachead Sommanee
Keyword(s):  
Nonempty Subset ◽  
Transformation Semigroup ◽  
Regular Part ◽  
Restricted Range ◽  
Full Transformation ◽  
Full Transformation Semigroup ◽  
Inverse Subsemigroups

Let TX be the full transformation semigroup on a set X. For a fixed nonempty subset Y of a set X, let TX,Y be the semigroup consisting of all full transformations from X into Y. In a paper published in 2008, Sanwong and Sommanee proved that the set FX,Y=α∈TX,Y:Xα=Yα is the largest regular subsemigroup of TX,Y. In this paper, we describe the maximal inverse subsemigroups of FX,Y and completely determine all the maximal regular subsemigroups of its ideals.

scholarly journals PARTIAL ORDERS ON PARTIAL BAER–LEVI SEMIGROUPS

2010 ◽  
Vol 81 (2) ◽  
pp. 195-207 ◽  
Author(s):  
BOORAPA SINGHA ◽  
JINTANA SANWONG ◽  
R. P. SULLIVAN
Keyword(s):  
Inverse Semigroup ◽  
Partial Order ◽  
Partial Orders ◽  
The Other ◽  
Natural Order ◽  
Natural Partial Order ◽  
Transformation Semigroups ◽  
Minimal Elements ◽  
Symmetric Inverse Semigroup ◽  
Containment Order

AbstractMarques-Smith and Sullivan [‘Partial orders on transformation semigroups’, Monatsh. Math.140 (2003), 103–118] studied various properties of two partial orders on P(X), the semigroup (under composition) consisting of all partial transformations of an arbitrary set X. One partial order was the ‘containment order’: namely, if α,β∈P(X) then α⊆β means xα=xβ for all x∈dom α, the domain of α. The other order was the so-called ‘natural order’ defined by Mitsch [‘A natural partial order for semigroups’, Proc. Amer. Math. Soc.97(3) (1986), 384–388] for any semigroup. In this paper, we consider these and other orders defined on the symmetric inverse semigroup I(X) and the partial Baer–Levi semigroup PS(q). We show that there are surprising differences between the orders on these semigroups, concerned with their compatibility with respect to composition and the existence of maximal and minimal elements.

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