scholarly journals Divisibility in categories of a class which includes the category of binary relations

1976 ◽  
Vol 17 (1) ◽  
pp. 22-30 ◽  
Author(s):  
D. G. Fitz-Gerald
Keyword(s):  
Full Subcategory ◽  
Slight Modification ◽  
Green’S Relations ◽  
Binary Relations ◽  
Adjoint Functors ◽  
Green's Relations

Green's relations are essential for “co-ordinatizing” semigroups. Jacqueline Klasa, in applying cognate ideas to categories [4, 5], has shown that divisibility in suitably-behaved categories may be described in terms of subobjects and quotients.Here it is shown that adjoint functors which are onto objects preserve divisibility (in a certain sense). The inclusion functor of the category of sets into the category R of binary relations is such a functor. A slight modification of its right adjoint allows the representation of R as a full subcategory in a category CSL of complete semilattice morphisms.

scholarly journals Solvability of a triple of binary relations with an application to green’s relations

2021 ◽  
pp. 2250013
Author(s):  
Thanakorn Prinyasart ◽  
Jittisak Rakbud ◽  
Suchat Samphavat
Keyword(s):  
Green’S Relations ◽  
Binary Relations ◽  
Green's Relations ◽  
Regular Relation

In this paper, we introduce the notion of a solvable triple of binary relations on a set. This notion generalizes the notion of a regular relation and all other notions that are variants of the notion of the regularity, defined previously by many people. We also give some characterizations of the solvability of a triple of relations and use this to study Green’s relations on the monoid of binary relations on a set.

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.

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 Congruences on bicyclic extensions of a linearly ordered group

2020 ◽  
Vol 15 (2) ◽  
pp. 61-80
Author(s):  
Oleg Gutik ◽  
Dušan Pagon ◽  
Kateryna Pavlyk
Keyword(s):  
Positive Cone ◽  
Inverse Semigroups ◽  
Green’S Relations ◽  
Ordered Group ◽  
Green's Relations ◽  
Linearly Ordered Group

In the paper we study inverse semigroups B(G), B^+(G), \overline{B}(G) and \overline{B}^+(G) which are generated by partial monotone injective translations of a positive cone of a linearly ordered group G. We describe Green’s relations on the semigroups B(G), B^+(G), \overline{B}(G) and \overline{B}^+(G), their bands and show that they are simple, and moreover, the semigroups B(G) and B^+(G) are bisimple. We show that for a commutative linearly ordered group G all non-trivial congruences on the semigroup B(G) (and B^+(G)) are group congruences if and only if the group G is archimedean. Also we describe the structure of group congruences on the semigroups B(G), B^+(G), \overline{B}(G) and \overline{B}^+(G).

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