scholarly journals Green's-Like Relations on Algebras and Varieties

2008 ◽  
Vol 2008 ◽  
pp. 1-12
Author(s):  
K. Denecke ◽  
S. L. Wismath
Keyword(s):  
Binary Operation ◽  
Equivalence Relations ◽  
Green’S Relations ◽  
Green's Relations ◽  
Variety Of Semigroups

There are five equivalence relations known as Green's relations definable on any semigroup or monoid, that is, on any algebra with a binary operation which is associative. In this paper, we examine whether Green's relations can be defined on algebras of any typeτ. Some sort of (super-)associativity is needed for such definitions to work, and we consider algebras which are clones of terms of typeτ, where the clone axioms including superassociativity hold. This allows us to define for any varietyVof typeτtwo Green's-like relationsℒVandℛVon the term clone of typeτ. We prove a number of properties of these two relations, and describe their behaviour whenVis a variety of semigroups.

*-Relations on ternary semigroups

2018 ◽  
Vol 15 (06) ◽  
pp. 1850094
Author(s):  
Nahid Ashrafi ◽  
Zahra Yazdanmehr
Keyword(s):  
Equivalence Relations ◽  
Green’S Relations ◽  
Green's Relations

In this paper, we define certain equivalence relations called *-relations on ternary semigroups and we mention some properties of these relations. We study these relations in respect to Green’s relations in ternary semigroups and by bringing some examples, we show that while some propositions are correct for Green’s relations, they are not necessarily true for these relations. Then we investigate *-relations in certain ternary semigroups.

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.

scholarly journals Green's Relations on HypG(2)

2012 ◽  
Vol 20 (1) ◽  
pp. 249-264
Author(s):  
Wattapong Puninagool ◽  
Sorasak Leeratanavalee
Keyword(s):  
Binary Operation ◽  
Green’S Relations ◽  
Operation Symbol ◽  
Green's Relations ◽  
Natural Way

AbstractA generalized hypersubstitution of type τ = (2) is a mapping which maps the binary operation symbol f to a term σ(f) which does not necessarily preserve the arity. Any such σ can be inductively extended to a map σ̂ on the set of all terms of type τ = (2), and any two such extensions can be composed in a natural way. Thus, the set HypG(2) of all generalized hypersubstitutions of type τ = (2) forms a monoid. Green's relations on the monoid of all hypersubstitutions of type τ = (2) were studied by K. Denecke and Sh.L. Wismath. In this paper we describe the classes of generalized hypersubstitutions of type τ = (2) under Green's relations.

scholarly journals Green's Relations for Hypergroupoids

2018 ◽  
Vol 11 (3) ◽  
pp. 598-611
Author(s):  
Niovi Kehayopulu
Keyword(s):  
Equivalence Relations ◽  
Green’S Relations ◽  
Semilattice Congruence ◽  
Green's Relations ◽  
Left Simple

We give some information concerning the Green's relations $\cal R$ and $\cal L$ in hypergroupoids extending the concepts of right (left) consistent or intra-consistent groupoids in case of hypergroupoids. We prove, for example, that if an hypergroupoid $H$ is right (left) consistent or intra-consistent, then the Green's relations $\cal R$ and $\cal L$ are equivalence relations on $H$ and give some conditions under which in consistent commutative hypergroupoids the relation $\cal R$ (= $\cal L$) is a semilattice congruence. A commutative hypergroupoid is right consistent if and only if it is left consistent and if an hypergroupoid is commutative and right (left) consistent, then it is intra-consistent. A characterization of right (left) consistent (or intra-consistent) right (left) simple hypergroupoids has been also given. Illustrative examples are given.

Green’s relations on ternary semihypergroups and crossed hyperproduct of hypergroups

2020 ◽  
pp. 2150119
Author(s):  
Seda Oğuz Ünal
Keyword(s):  
Equivalence Relations ◽  
Green’S Relations ◽  
Green's Relations

In this paper, we introduce all Green’s relations on the ternary semihypergroup and study some properties of these equivalence relations in view of those obtained in binary and ternary semigroups. We also define crossed hyperproduct of two hypergroups.

Semigroups of an inductive composition of terms

2021 ◽  
pp. 2250038
Author(s):  
Pongsakorn Kitpratyakul ◽  
Bundit Pibaljommee
Keyword(s):  
Regularity Condition ◽  
Binary Operation ◽  
Green’S Relations ◽  
Algebraic Structures ◽  
Type Formula ◽  
Green's Relations ◽  
Composition Of Terms

The set of all [Formula: see text]-ary terms of type [Formula: see text] together with a binary operation derived from a superposition [Formula: see text] forms various forms of semigroups. One may generalize such binary operation by deriving it from an inductive composition of terms and call it an inductive product. However, this operation is not associative on the same base set but it becomes associative when all elements of subterms of a fixed term used in an inductive product except itself are excluded from the base set. Hence, a semigroup is formed. In this paper, we mainly focus on the algebraic structures of this semigroup such as idempotent elements, elements associating with each type of regularity condition, and Green’s relations. The formulae of complexity of inducted terms are also under investigation.

SEMIGROUP PROPERTIES OF N-ARY OPERATIONS ON FINITE SETS

2008 ◽  
Vol 01 (01) ◽  
pp. 27-44
Author(s):  
R. Butkote ◽  
K. Denecke ◽  
Ch. Ratanaprasert
Keyword(s):  
Algebraic Structure ◽  
Complete Lattice ◽  
Binary Operation ◽  
Green’S Relations ◽  
Finite Sets ◽  
Regular Elements ◽  
Green's Relations ◽  
Finite Set

A clone is a set of operations defined on a base set A which is closed under composition and contains all the projection operations. There are several ways to regard a clone as an algebraic structure (see e.g. [3]). If f, g1,…,gn : An → A are n-ary operations defined on A, then by Sn(f, g1 … , gn)(a1 … , an) := f(g1(a1,…,an),…,gn(a1,…,an)) for all a1,…, an ∈ A an (n + 1)-ary operation on the set On(A) of all n-ary operations can be defined. From this operation one can derive a binary operation + defined by f + g := Sn(f, g,…,g) and obtains a semigroup (On(A);+). The collection of all clones of operations on a finite set forms a complete lattice. This lattice is well-described ([4], [5]) if |A| = 2. If |A| > 2, this lattice is uncountably infinite and very complex. In this paper instead of clones we study semigroups of n-ary operations, i.e. subsemigroups of the semigroup (On(A); +) and their properties. We look for idempotent and regular elements of (On(A); +), consider Green's relations for the semigroup (On(A); +), characterize all constant subsemigroups of (On(A);+), all semilattices, rectangular bands and normal bands contained in (On(A);+).

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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