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.

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.

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.

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].

Normal bands of commutative cancellative semigroups revisited

2014 ◽  
Vol 24 (05) ◽  
pp. 531-551
Author(s):  
Mario Petrich
Keyword(s):  
Free Semigroup ◽  
Abelian Groups ◽  
Normal Band ◽  
Green’S Relations ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Special Cases ◽  
Green's Relations ◽  
Completely Regular Semigroups

A semigroup S is of the type in the class of the title if S has a congruence ρ such that S/ρ is a normal band (i.e. satisfies the identities x2 = x and axya = ayxa) and all ρ-classes are commutative cancellative semigroups. We consider semigroups S with such a congruence first for completely regular semigroups, then characterize the general case in several ways, including some special cases. When S is an order in a normal band of abelian groups Q, we study the restrictions of Green's relations on Q to S. The paper concludes with the discussion of a free semigroup in the title on two generators.

scholarly journals On the semigroup of all partial fence-preserving injections on a finite set

2017 ◽  
Vol 16 (12) ◽  
pp. 1750223 ◽  
Author(s):  
Ilinka Dimitrova ◽  
Jörg Koppitz
Keyword(s):  
Inverse Semigroup ◽  
Transformation Formula ◽  
Green’S Relations ◽  
Generating Set ◽  
Regular Elements ◽  
Green's Relations ◽  
Finite Set ◽  
Symmetric Inverse Semigroup ◽  
Text Element ◽  
Element Set

For [Formula: see text], let [Formula: see text] be an [Formula: see text]-element set and let [Formula: see text] be a fence, also called a zigzag poset. As usual, we denote by [Formula: see text] the symmetric inverse semigroup on [Formula: see text]. We say that a transformation [Formula: see text] is fence-preserving if [Formula: see text] implies that [Formula: see text], for all [Formula: see text] in the domain of [Formula: see text]. In this paper, we study the semigroup [Formula: see text] of all partial fence-preserving injections of [Formula: see text] and its subsemigroup [Formula: see text]. Clearly, [Formula: see text] is an inverse semigroup and contains all regular elements of [Formula: see text] We characterize the Green’s relations for the semigroup [Formula: see text]. Further, we prove that the semigroup [Formula: see text] is generated by its elements with rank [Formula: see text]. Moreover, for [Formula: see text], we find the least generating set and calculate the rank of [Formula: see text].

scholarly journals Green’s Relations on Regular Elements of Semigroup of Relational Hypersubstitutions for Algebraic Systems of Type ((m), (n))

2021 ◽  
Vol 53 ◽  
Author(s):  
Sorasak Leeratanavalee ◽  
Jukkrit Daengsaen
Keyword(s):  
Green’S Relations ◽  
Operation Symbol ◽  
Algebraic Systems ◽  
Regular Elements ◽  
Relational Term ◽  
Green's Relations ◽  
Algebraic Properties

Any relational hypersubstitution for algebraic systems of type (τ,τ′) = ((mi)i∈I,(nj)j∈J) is a mapping which maps any mi-ary operation symbol to an mi-ary term and maps any nj - ary relational symbol to an nj-ary relational term preserving arities, where I,J are indexed sets. Some algebraic properties of the monoid of all relational hypersubstitutions for algebraic systems of a special type, especially the characterization of its order and the set of all regular elements, were first studied by Phusanga and Koppitz[13] in 2018. In this paper, we study the Green’srelationsontheregularpartofthismonoidofaparticulartype(τ,τ′) = ((m),(n)), where m, n ≥ 2.

A generalized superposition of linear tree languages and products of linear tree languages

2018 ◽  
Vol 11 (04) ◽  
pp. 1850048
Author(s):  
Pongsakorn Kitpratyakul ◽  
Bundit Pibaljommee
Keyword(s):  
Green’S Relations ◽  
Linear Superposition ◽  
Tree Language ◽  
Regular Elements ◽  
Type Formula ◽  
Green's Relations ◽  
Tree Languages

A linear tree language of type [Formula: see text] is a set of linear terms, terms containing no multiple occurrences of the same variable, of that type. Instead of the usual generalized superposition of tree languages, we define the generalized linear superposition to deal with linear tree languages and study its properties. Using this superposition, we define the product of linear tree languages. This product is not associative on the collection of all linear tree languages, but it is associative on some subsets of this collection whose products of any element in the subsets are nonempty. We classify such subsets and study properties of the obtained semigroup especially idempotent elements, regular elements, and Green’s relations [Formula: see text] and [Formula: see text].

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