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.

Green's Relations for Regular Elements of Semigroups of Endomorphisms

1974 ◽  
Vol 26 (6) ◽  
pp. 1484-1497 ◽  
Author(s):  
K. D. Magill ◽  
S. Subbiah
Keyword(s):  
Topological Space ◽  
Vector Space ◽  
Green’S Relations ◽  
Linear Transformations ◽  
Regular Elements ◽  
Special Cases ◽  
Green's Relations ◽  
Composition Of Functions

X is a set and End X is a semigroup, under composition, of functions, which map X into X. We characterize those elements of End X which are regular and then we completely determine Green's relations for these elements. The conditions we place on End X are sufficiently mild to permit such semigroups as S(X), the semigroup of all continuous self maps of a topological space X and L(V), the semigroup of all linear transformations on a vector space V, to be regarded as special cases.

INJECTIVE LINEAR TRANSFORMATIONS WITH EQUAL GAP AND DEFECT

2021 ◽  
pp. 1-11
Author(s):  
C. MENDES ARAÚJO ◽  
S. MENDES-GONÇALVES
Keyword(s):  
Inverse Semigroup ◽  
Vector Space ◽  
Green’S Relations ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Green's Relations ◽  
Partial Linear

Abstract Let V be an infinite-dimensional vector space over a field F and let $I(V)$ be the inverse semigroup of all injective partial linear transformations on V. Given $\alpha \in I(V)$ , we denote the domain and the range of $\alpha $ by ${\mathop {\textrm {dom}}}\,\alpha $ and ${\mathop {\textrm {im}}}\,\alpha $ , and we call the cardinals $g(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {dom}}}\,\alpha $ and $d(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {im}}}\,\alpha $ the ‘gap’ and the ‘defect’ of $\alpha $ . We study the semigroup $A(V)$ of all injective partial linear transformations with equal gap and defect and characterise Green’s relations and ideals in $A(V)$ . This is analogous to work by Sanwong and Sullivan [‘Injective transformations with equal gap and defect’, Bull. Aust. Math. Soc.79 (2009), 327–336] on a similarly defined semigroup for the set case, but we show that these semigroups are never isomorphic.

Semigroups of linear transformations with fixed subspaces: Green’s relations, ideals and finiteness conditions

2019 ◽  
Vol 12 (04) ◽  
pp. 1950062 ◽  
Author(s):  
Yanisa Chaiya ◽  
Chollawat Pookpienlert ◽  
Jintana Sanwong
Keyword(s):  
Vector Space ◽  
Green’S Relations ◽  
Linear Transformations ◽  
Finiteness Conditions ◽  
Green's Relations

Let [Formula: see text] be a vector space and [Formula: see text] denote the semigroup (under the composition of maps) of all linear transformations from [Formula: see text] into itself. For a fixed subspace [Formula: see text] of [Formula: see text], let [Formula: see text] be the subsemigroup of [Formula: see text] consisting of all linear transformations on [Formula: see text] which fix all elements in [Formula: see text]. In this paper, we describe Green’s relations, regularity and ideals of [Formula: see text]; and characterize when [Formula: see text] is factorizable, unit-regular and directly finite, from which the results on [Formula: see text] can be recaptured easily when taking [Formula: see text] as a zero subspace of [Formula: see text].

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.

THE REGULAR PART OF A SEMIGROUP OF LINEAR TRANSFORMATIONS WITH RESTRICTED RANGE

2017 ◽  
Vol 103 (3) ◽  
pp. 402-419 ◽  
Author(s):  
WORACHEAD SOMMANEE ◽  
KRITSADA SANGKHANAN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Restricted Range ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional ◽  
Idempotent Rank ◽  
Image Position

Let$V$be a vector space and let$T(V)$denote the semigroup (under composition) of all linear transformations from$V$into$V$. For a fixed subspace$W$of$V$, let$T(V,W)$be the semigroup consisting of all linear transformations from$V$into$W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’,Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$is the largest regular subsemigroup of$T(V,W)$and characterized Green’s relations on$T(V,W)$. In this paper, we determine all the maximal regular subsemigroups of$Q$when$W$is a finite-dimensional subspace of$V$over a finite field. Moreover, we compute the rank and idempotent rank of$Q$when$W$is an$n$-dimensional subspace of an$m$-dimensional vector space$V$over a finite field$F$.

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.

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