scholarly journals On theS-Invariance Property forS-Flows

2010 ◽  
Vol 2010 ◽  
pp. 1-5
Author(s):  
Amin Saif ◽  
Adem Kılıçman
Keyword(s):  
Topological Space ◽  
Equivalence Relation ◽  
Transformation Semigroup ◽  
Invariance Property ◽  
Topological Monoid

We define an equivalence relation on a topological space which is acted by topological monoidSas a transformation semigroup. Then, we give some results about theS-invariant classes for this relation. We also provide a condition for the existence of relativeS-invariant classes.

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.

scholarly journals On Magnifying Elements in E-Preserving Partial Transformation Semigroups

Mathematics ◽  
2018 ◽  
Vol 6 (9) ◽  
pp. 160
Author(s):  
Thananya Kaewnoi ◽  
Montakarn Petapirak ◽  
Ronnason Chinram
Keyword(s):  
Equivalence Relation ◽  
Transformation Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Proper Subset ◽  
Partial Transformation ◽  
Transformation Semigroups ◽  
Necessary And Sufficient

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.

The Position of Rough Set in Soft Set

2012 ◽  
Vol 3 (3) ◽  
pp. 33-48
Author(s):  
Tutut Herawan
Keyword(s):  
Topological Space ◽  
Set Theory ◽  
Rough Set ◽  
Equivalence Relation ◽  
Rough Set Theory ◽  
Soft Set ◽  
General Topology ◽  
Discrete Topology ◽  
Special Case

In this paper, the author presents the concept of topological space that must be used to show a relation between rough set and soft set. There are two main results presented; firstly, a construction of a quasi-discrete topology using indiscernibility (equivalence) relation in rough set theory is described. Secondly, the paper describes that a “general” topology is a special case of soft set. Hence, it is concluded that every rough set can be considered as a soft set.

scholarly journals Semigroups of continuous selfmaps for which Green's and ℐ relations coincide

1979 ◽  
Vol 20 (1) ◽  
pp. 25-28 ◽  
Author(s):  
K. D. Magill
Keyword(s):  
Topological Space ◽  
Metric Spaces ◽  
Transformation Semigroup ◽  
Doctoral Dissertation ◽  
Discrete Space ◽  
Full Transformation ◽  
Full Transformation Semigroup ◽  
The One ◽  
One Point Compactification ◽  
Countably Infinite

For algebraic terms which are not defined, one may consult [2]. The symbol S(X) denotes the semigroup, under composition, of all continuous selfmaps of the topological space X. When X is discrete, S(X) is simply the full transformation semigroup on the set X. It has long been known that Green's relations and ℐ coincide for [2, p. 52] and F. A. Cezus has shown in his doctoral dissertation [1, p. 34] that and ℐ also coincide for S(X) when X is the one-point compactification of the countably infinite discrete space. Our main purpose here is to point out the fact that among the 0-dimensional metric spaces, Cezus discovered the only nondiscrete space X with the property that and ℐ coincide on the semigroup S(X). Because of a result in a previous paper [6] by S. Subbiah and the author, this property (for 0-dimensional metric spaces) is in turn equivalent to the semigroup being regular. We gather all this together in the following

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.

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.

A quotient ordered space

1968 ◽  
Vol 64 (2) ◽  
pp. 317-322 ◽  
Author(s):  
S. D. McCartan
Keyword(s):  
Topological Space ◽  
Partial Order ◽  
Equivalence Relation ◽  
Quotient Space ◽  
Partial Orders ◽  
Projection Mapping ◽  
Quotient Spaces ◽  
The Family ◽  
Ordered Space ◽  
Image Position

It is well known that, in the study of quotient spaces it suffices to consider a topological space (X, ), an equivalence relation R on X and the projection mapping p: X → X/R (where X/R is the family of R-classes of X) defined by p(x) = Rx (where Rx is the R-class to which x belongs) for each x ∈ X. A topology may be defined for the set X/R by agreeing that U ⊆ X/R is -open if and only if p-1 (U) is -open in X. The topological space is known as the quotient space relative to the space ) and projection p. If (or simply ) since the symbol ≤ denotes all partial orders and no confusion arises) is a topological ordered space (that is, X is a set for which both a topology and a partial order ≤ is defined) then, providing the projection p satisfies the propertya partial order may be defined in X/R by agreeing that p(x) < p(y) if and only if x < y in x. The topological ordered space is known as the quotient ordered space relative to the ordered space and projection p.

Classifying positive equivalence relations

1983 ◽  
Vol 48 (3) ◽  
pp. 529-538 ◽  
Author(s):  
Claudio Bernardi ◽  
Andrea Sorbi
Keyword(s):  
Topological Space ◽  
Equivalence Relation ◽  
Recursive Function ◽  
Equivalence Classes ◽  
Point Of View ◽  
Classification Theorem ◽  
Equivalence Relations ◽  
Natural Numbers ◽  
Uniformity Property

AbstractGiven two (positive) equivalence relations ~1, ~2 on the set ω of natural numbers, we say that ~1 is m-reducible to ~2 if there exists a total recursive function h such that for every x, y ∈ ω, we have x ~1y iff hx ~2hy. We prove that the equivalence relation induced in ω by a positive precomplete numeration is complete with respect to this reducibility (and, moreover, a “uniformity property” holds). This result allows us to state a classification theorem for positive equivalence relations (Theorem 2). We show that there exist nonisomorphic positive equivalence relations which are complete with respect to the above reducibility; in particular, we discuss the provable equivalence of a strong enough theory: this relation is complete with respect to reducibility but it does not correspond to a precomplete numeration.From this fact we deduce that an equivalence relation on ω can be strongly represented by a formula (see Definition 8) iff it is positive. At last, we interpret the situation from a topological point of view. Among other things, we generalize a result of Visser by showing that the topological space corresponding to a partition in e.i. sets is irreducible and we prove that the set of equivalence classes of true sentences is dense in the Lindenbaum algebra of the theory.

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