scholarly journals GREEN'S RELATIONS ON SEMIGROUPS OF REGRESSIVE TRANSFORMATIONS WITH RESTRICTED RANGE

2016 ◽  
Vol 108 (2) ◽  
Author(s):  
K. Sangkhanan
Keyword(s):  
Green’S Relations ◽  
Restricted Range ◽  
Green's Relations

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 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.

On the Ranks of Semigroups of Transformations on a Finite Set with Restricted Range

2014 ◽  
Vol 21 (03) ◽  
pp. 497-510 ◽  
Author(s):  
Vítor H. Fernandes ◽  
Jintana Sanwong
Keyword(s):  
Green’S Relations ◽  
Restricted Range ◽  
Green's Relations ◽  
Finite Set ◽  
Empty Subset

Let [Formula: see text] be the semigroup of all partial transformations on X, [Formula: see text] and [Formula: see text] be the subsemigroups of [Formula: see text] of all full transformations on X and of all injective partial transformations on X, respectively. Given a non-empty subset Y of X, let [Formula: see text], [Formula: see text] and [Formula: see text]. In 2008, Sanwong and Sommanee described the largest regular subsemigroup and determined Green's relations of [Formula: see text]. In this paper, we present analogous results for both [Formula: see text] and [Formula: see text]. For a finite set X with |X| ≥ 3, the ranks of [Formula: see text], [Formula: see text] and [Formula: see text] are well known to be 4, 3 and 3, respectively. In this paper, we also compute the ranks of [Formula: see text], [Formula: see text] and [Formula: see text] for any proper non-empty subset Y of X.

scholarly journals SEMIGROUPS OF LINEAR TRANSFORMATIONS WITH RESTRICTED RANGE

2008 ◽  
Vol 77 (3) ◽  
pp. 441-453 ◽  
Author(s):  
R. P. SULLIVAN
Keyword(s):  
Vector Space ◽  
Green’S Relations ◽  
Restricted Range ◽  
Linear Transformations ◽  
Fixed Subset ◽  
Green's Relations

AbstractIn 1975, Symons described the automorphisms of the semigroup T(X,Y ) consisting of all total transformations from a set X into a fixed subset Y of X. Recently Sanwong, Singha and Sullivan determined all maximal (and all minimal) congruences on T(X,Y ), and Sommanee studied Green’s relations in T(X,Y ). Here, we describe Green’s relations and ideals for the semigroup T(V,W) consisting of all linear transformations from a vector space V into a fixed subspace W of V.

scholarly journals THE IDEAL STRUCTURE OF SEMIGROUPS OF TRANSFORMATIONS WITH RESTRICTED RANGE

2010 ◽  
Vol 83 (2) ◽  
pp. 289-300 ◽  
Author(s):  
SUZANA MENDES-GONÇALVES ◽  
R. P. SULLIVAN
Keyword(s):  
Nonempty Subset ◽  
Green’S Relations ◽  
Restricted Range ◽  
Ideal Structure ◽  
Regular Elements ◽  
Green's Relations ◽  
The Ideal

AbstractLet Y be a fixed nonempty subset of a set X and let T(X,Y ) denote the semigroup of all total transformations from X into Y. In 1975, Symons described the automorphisms of T(X,Y ). Three decades later, Nenthein, Youngkhong and Kemprasit determined its regular elements, and more recently Sanwong, Singha and Sullivan characterized all maximal and minimal congruences on T(X,Y ). In 2008, Sanwong and Sommanee determined the largest regular subsemigroup of T(X,Y ) when |Y |≠1 and Y ≠ X; and using this, they described the Green’s relations on T(X,Y ) . Here, we use their work to describe the ideal structure of T(X,Y ) . We also correct the proof of the corresponding result for a linear analogue of T(X,Y ) .

scholarly journals Certain characterizations of varieties of bands

1988 ◽  
Vol 31 (2) ◽  
pp. 301-319 ◽  
Author(s):  
J. A. Gerhard ◽  
Mario Petrich
Keyword(s):  
Rectangular Band ◽  
Simple System ◽  
Left Zero ◽  
Green’S Relations ◽  
Transformation Semigroups ◽  
Lattice Of Varieties ◽  
Green's Relations ◽  
The World ◽  
New System

The lattice of varieties of bands was constructed in [1] by providing a simple system of invariants yielding a solution of the world problem for varieties of bands including a new system of inequivalent identities for these varieties. References [3] and [5] contain characterizations of varieties of bands determined by identities with up to three variables in terms of Green's relations and the functions figuring in a construction of a general band. In this construction, the band is expressed as a semilattice of rectangular bands and the multiplication is written in terms of functions among these rectangular band components and transformation semigroups on the corresponding left zero and right zero direct factors.

scholarly journals Green's relations for regular elements of sandwich semigroups, II; semigroups of continuous functions

1978 ◽  
Vol 25 (1) ◽  
pp. 45-65 ◽  
Author(s):  
K. D. Magill ◽  
S. Subbiah
Keyword(s):  
Continuous Functions ◽  
Maximal Subgroups ◽  
Green’S Relations ◽  
Boolean Ring ◽  
Regular Elements ◽  
Ring Homomorphisms ◽  
Special Cases ◽  
Green's Relations ◽  
Sandwich Semigroup ◽  
Sandwich Semigroups

AbstractA sandwich semigroup of continuous functions consists of continuous functions with domains all in some space X and ranges all in some space Y with multiplication defined by fg = foαog where α is a fixed continuous function from a subspace of Y into X. These semigroups include, as special cases, a number of semigroups previously studied by various people. In this paper, we characterize the regular elements of such semigroups and we completely determine Green's relations for the regular elements. We also determine the maximal subgroups and, finally, we apply some of these results to semigroups of Boolean ring homomorphisms.

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