Regularity in the Semigroup of Transformations Preserving a Zig-Zag Order

2019 ◽  
Vol 43 (2) ◽  
pp. 1761-1773
Author(s):  
Ratana Srithus ◽  
Ronnason Chinram ◽  
Chompunutch Khongthat
Keyword(s):  
Semigroup Of Transformations

scholarly journals Green’s Relations on a Semigroup of Transformations with Restricted Range that Preserves an Equivalence Relation and a Cross-Section

Mathematics ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 134
Author(s):  
Chollawat Pookpienlert ◽  
Preeyanuch Honyam ◽  
Jintana Sanwong
Keyword(s):  
Cross Section ◽  
Equivalence Relation ◽  
Nonempty Subset ◽  
Identity Relation ◽  
Green’S Relations ◽  
Restricted Range ◽  
Green's Relations ◽  
Semigroup Of Transformations

Let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed nonempty subset Y of X. For an equivalence relation ρ on X, let ρ^ be the restriction of ρ on Y, R a cross-section of Y/ρ^ and define T(X,Y,ρ,R) to be the set of all total transformations α from X into Y such that α preserves both ρ (if (a,b)∈ρ, then (aα,bα)∈ρ) and R (if r∈R, then rα∈R). T(X,Y,ρ,R) is then a subsemigroup of T(X,Y). In this paper, we give descriptions of Green’s relations on T(X,Y,ρ,R), and these results extend the results on T(X,Y) and T(X,ρ,R) when taking ρ to be the identity relation and Y=X, respectively.

scholarly journals ON THE PARTITION MONOID AND SOME RELATED SEMIGROUPS

2010 ◽  
Vol 83 (2) ◽  
pp. 273-288 ◽  
Author(s):  
D. G. FITZGERALD ◽  
KWOK WAI LAU
Keyword(s):  
Transformation Semigroup ◽  
Galois Connection ◽  
Inverse Semigroups ◽  
Natural Order ◽  
Binary Relations ◽  
Ideal Lattices ◽  
New Interpretation ◽  
Semigroup Of Transformations ◽  
The Ideal

AbstractThe partition monoid is a salient natural example of a *-regular semigroup. We find a Galois connection between elements of the partition monoid and binary relations, and use it to show that the partition monoid contains copies of the semigroup of transformations and the symmetric and dual-symmetric inverse semigroups on the underlying set. We characterize the divisibility preorders and the natural order on the (straight) partition monoid, using certain graphical structures associated with each element. This gives a simpler characterization of Green’s relations. We also derive a new interpretation of the natural order on the transformation semigroup. The results are also used to describe the ideal lattices of the straight and twisted partition monoids and the Brauer monoid.

scholarly journals Random semigroup acts on a finite set

1977 ◽  
Vol 23 (4) ◽  
pp. 481-498 ◽  
Author(s):  
Göran Högnäs
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Finite Set ◽  
Semigroup Of Transformations

AbstractLet X be a finite set and S a semigroup of transformations of X. We investigate the trace on X of a random walk on S. We relate the structure of the trace process, which turns out to be a Markov chain, to that of the random walk. We show, for example, that all periods of the trace process divide the period of the random walk.

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.

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

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.

scholarly journals Embedding a semigroup of transformations

1975 ◽  
Vol 20 (2) ◽  
pp. 222-224
Author(s):  
J. S. V. Symons
Keyword(s):  
Associative Operation ◽  
Image Position ◽  
Semigroup Of Transformations

Let X be an arbitrary set and θ a transformation of X. One may use θ to induce an associative operation in Jx, the set of all mappings of X to itself as follows: . We denote the resulting semigroup by {Jx;θ) Magill (1967) introduced this structure and it has been studied by Sullivan and by myself.

Examples for the Theory of Infinite Iteration of Summability Methods

1977 ◽  
Vol 29 (3) ◽  
pp. 489-497 ◽  
Author(s):  
Persi Diaconis
Keyword(s):  
Summability Methods ◽  
Summable Sequence ◽  
Infinite Iteration ◽  
Special Case ◽  
Semigroup Of Transformations

Garten and Knopp [7] introduced the notion of infinite iteration of Césaro (C1 ) averages, which they called H∞ summability. Flehinger [6] (apparently unaware of [7]) produced the first nontrivial example of an H∞ summable sequence: the sequence ﹛ai ﹜ ∞i=1 where at is 1 or 0 as the lead digit of the integer i is one or not. Duran [2] has provided an elegant treatment of H∞ summability as a special case of summability with respect to an ergodic semigroup of transformations.

REGULAR ELEMENTS AND GREEN'S RELATIONS IN GENERALIZED TRANSFORMATION SEMIGROUPS

2013 ◽  
Vol 06 (01) ◽  
pp. 1350006 ◽  
Author(s):  
Suzana Mendes-Gonçalves ◽  
R. P. Sullivan
Keyword(s):  
Equivalence Relations ◽  
Regularity Conditions ◽  
Green’S Relations ◽  
Transformation Semigroups ◽  
Regular Elements ◽  
Green's Relations ◽  
Semigroup Of Transformations

If X and Y are sets, we let P(X, Y) denote the set of all partial transformations from X into Y (that is, all mappings whose domain and range are subsets of X and Y, respectively). If θ ∈ P(Y, X), then P(X, Y) is a so-called "generalized semigroup" of transformations under the "sandwich operation": α * β = α ◦ θ ◦ β, for each α, β ∈ P(X, Y). We denote this semigroup by P(X, Y, θ) and, in this paper, we characterize Green's relations on it: that is, we study equivalence relations which determine when principal left (or right, or 2-sided) ideals in P(X, Y, θ) are equal. This solves a problem raised by Magill and Subbiah in 1975. We also discuss the same idea for important subsemigroups of P(X, Y, θ) and characterize when these semigroups satisfy certain regularity conditions.

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