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.

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 The ideal structure of nilpotent-generated transformation semigroups

1999 ◽  
Vol 60 (2) ◽  
pp. 303-318 ◽  
Author(s):  
M. Paula O. Marques-Smith ◽  
R.P. Sullivan
Keyword(s):  
Green’S Relations ◽  
Ideal Structure ◽  
Transformation Semigroups ◽  
Green's Relations ◽  
Algebraic Properties ◽  
Rees Factor ◽  
Free Semigroups ◽  
One To One ◽  
Infinite Set ◽  
The Ideal

In 1987 Sullivan determined the elements of the semigroup N(X) generated by all nilpotent partial transformations of an infinite set X; and later in 1997 he studied subsemigroups of N(X) defined by restricting the index of the nilpotents and the cardinality of the set. Here, we describe the ideals and Green's relations on such semigroups, like Reynolds and Sullivan did in 1985 for the semigroup generated by all idempotent total transformations of X. We then use this information to describe the congruences on certain Rees factor semigroups and to construct families of congruence-free semigroups with interesting algebraic properties. We also study analogous questions for X finite and for one-to-one partial transformations.

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].

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].

SEMIGROUPS OF TREE LANGUAGES

2008 ◽  
Vol 01 (04) ◽  
pp. 489-507 ◽  
Author(s):  
K. Denecke ◽  
N. Sarasit
Keyword(s):  
Left Zero ◽  
Green’S Relations ◽  
Regular Elements ◽  
Green's Relations ◽  
Tree Languages

Sets of terms of type τ are called tree languages (see [6]). There are several possibilities to define superposition operations on sets of tree languages. On the basis of such superposition operations we define binary associative operations on tree languages and investigate the properties of the arising semigroups. We characterize idempotent and regular elements and Green's relations [Formula: see text] and [Formula: see text]. Moreover, we determine constant, left-zero and right-zero subsemigroups and rectangular bands.

Green's Relations on the Monoid of Regular Hypersubstitutions

2006 ◽  
Vol 13 (04) ◽  
pp. 623-632 ◽  
Author(s):  
Th. Changphas ◽  
S. L. Wismath
Keyword(s):  
Galois Connection ◽  
Green’S Relations ◽  
Operation Symbol ◽  
Green's Relations ◽  
Fixed Type

The theory of hyperidentities and hypervarieties is based on the fact that the set Hyp (τ) of all hypersubstitutions of a fixed type τ forms a monoid, with a Galois connection between submonoids of this monoid and complete sublattices of the lattice of all varieties of type τ. For this reason, there is interest in studying the semigroup or monoid properties of Hyp (τ) and its submonoids. One approach is to study the five relations known as Green's relations definable on any semigroup. In this paper, we consider the type τ = (n) with one n-ary operation symbol for n≥ 1, and the submonoid Reg (n) of regular hypersubstitutions. We characterize Green's relations on every subsemigroup of Reg (n); then using this characterization we describe which subsemigroups of Reg (n) are 𝒢-subsemigroups of Reg (n) defined by Levi.

Semigroup properties of linear terms

2017 ◽  
Vol 10 (01) ◽  
pp. 1750051 ◽  
Author(s):  
Laddawan Lohapan ◽  
Prakit Jampachon
Keyword(s):  
Galois Theory ◽  
Green’S Relations ◽  
Regular Elements ◽  
Type Formula ◽  
Green's Relations ◽  
Partial Algebras

The concept of linear terms of a given type [Formula: see text] (terms in which each variable occurs at most once) was introduced by M. Couceiro and E. Lethonen in [Galois theory for sets of operations closed under permutation, cylindrification and composition, Algebr. Univ. 67 (2012) 273–297, doi:10.1007/s00012-012-0184-1] (see also [T. Changphas, K. Denecke and B. Pibaljommee, Linear terms and linear hypersubstitutions, SEAMS Bull. Math. 40 (2016) (to be published).]). In this paper, we introduce binary partial operations [Formula: see text] and [Formula: see text] on the set [Formula: see text] of all linear terms of type [Formula: see text] Then we characterize regular elements and Green’s relations in these partial algebras.

REGULARITY AND GREEN'S RELATIONS OF GENERALIZED PARTIAL TRANSFORMATION SEMIGROUPS

2008 ◽  
Vol 01 (03) ◽  
pp. 295-302 ◽  
Author(s):  
Ronnason Chinram
Keyword(s):  
Transformation Semigroup ◽  
Partial Transformation ◽  
Green’S Relations ◽  
Transformation Semigroups ◽  
Regular Elements ◽  
Green's Relations

Let X be any set and P(X) be the partial transformation semigroup on X. It is well-known that P(X) is regular. To generalize this, let X and Y be any sets and P(X, Y) be the set of all partial transformations from X to Y. For θ ∈ P(Y, X), let (P(X, Y), θ) be a semigroup (P(X, Y), *) where α * β = αθβ for all α, β ∈ P(X, Y). In this paper, we characterize the semigroup (P(X, Y), θ) to be regular, regular elements of the semigroup (P(X, Y), θ), [Formula: see text]-classes, [Formula: see text]-classes, [Formula: see text]-classes and [Formula: see text]-classes of the semigroup (P(X, Y), θ).

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