Abundant Semigroups of Transformations Preserving an Equivalence Relation

2011 ◽  
Vol 18 (01) ◽  
pp. 77-82 ◽  
Author(s):  
Huisheng Pei ◽  
Huijuan Zhou
Keyword(s):  
Equivalence Relation ◽  
Transformation Semigroup ◽  
Equivalence Relations ◽  
Full Transformation ◽  
Full Transformation Semigroup ◽  
Abundant Semigroups

Let X be a set with |X| ≥ 3, [Formula: see text] the full transformation semigroup on X, and E an equivalence relation on X. Let TE(X) be the set of transformations f in [Formula: see text] which preserve E, i.e., (x,y) ∈ E implies (f(x),f(y)) ∈ E. It is known that TE(X) is a subsemigroup of [Formula: see text]. In this paper, we describe the equivalence relations E so that the semigroup TE(X) is abundant.

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.

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.

ABUNDANCE OF THE SEMIGROUP OF ALL TRANSFORMATIONS OF A SET THAT REFLECT AN EQUIVALENCE RELATION

2013 ◽  
Vol 13 (02) ◽  
pp. 1350088 ◽  
Author(s):  
LEI SUN ◽  
LIMIN WANG
Keyword(s):  
Equivalence Relation ◽  
Transformation Semigroup ◽  
Full Transformation ◽  
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. We write [Formula: see text] Then T∃(X) is a subsemigroup of [Formula: see text]. In this paper, we proved that the semigroup T∃(X) is not abundant if X/E is infinite.

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 Products of two idempotent transformations over arbitrary sets and vector spaces

1998 ◽  
Vol 57 (1) ◽  
pp. 59-71 ◽  
Author(s):  
Rachel Thomas
Keyword(s):  
Vector Space ◽  
Transformation Semigroup ◽  
Vector Spaces ◽  
Endomorphism Monoid ◽  
Linear Transformations ◽  
Arbitrary Vector ◽  
Full Transformation ◽  
Full Transformation Semigroup

In this paper we consider the characterisation of those elements of a transformation semigroup S which are a product of two proper idempotents. We give a characterisation where S is the endomorphism monoid of a strong independence algebra A, and apply this to the cases where A is an arbitrary set and where A is an arbitrary vector space. The results emphasise the analogy between the idempotent generated subsemigroups of the full transformation semigroup of a set and of the semigroup of linear transformations from a vector space to itself.

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.

Maximal inverse subsemigroups of the semigroup of all full transformations satisfying x ≈ x3

2016 ◽  
Vol 09 (01) ◽  
pp. 1650042
Author(s):  
Somnuek Worawiset
Keyword(s):  
Natural Number ◽  
Transformation Semigroup ◽  
Inverse Semigroups ◽  
Full Transformation ◽  
Full Transformation Semigroup ◽  
Inverse Subsemigroup ◽  
Number Formula ◽  
Inverse Subsemigroups ◽  
Text Element ◽  
Element Set

We classify the maximal Clifford inverse subsemigroups [Formula: see text] of the full transformation semigroup [Formula: see text] on an [Formula: see text]-element set with [Formula: see text] for all [Formula: see text]. This classification differs from the already known classifications of Clifford inverse semigroups, it provides an algorithm for its construction. For a given natural number [Formula: see text], we find also the largest size of an inverse subsemigroup [Formula: see text] of [Formula: see text] satisfying [Formula: see text] with least rank [Formula: see text] for any element in [Formula: see text].

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.

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.

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