The Semigroup of Surjective Transformations on an Infinite Set

2019 ◽  
Vol 26 (01) ◽  
pp. 9-22
Author(s):  
Janusz Konieczny
Keyword(s):  
Green’S Relations ◽  
Transformation Semigroups ◽  
Unique Minimum ◽  
Green's Relations ◽  
Countably Infinite ◽  
Infinite Set

For an infinite set X, denote by Ω(X) the semigroup of all surjective mappings from X to X. We determine Green’s relations in Ω(X), show that the kernel (unique minimum ideal) of Ω(X) exists and determine its elements and cardinality. For a countably infinite set X, we describe the elements of Ω(X) for which the 𝒟-class and 𝒥-class coincide. We compare the results for Ω(X) with the corresponding results for other transformation semigroups on X.

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 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 CENTRALIZERS IN THE SEMIGROUP OF INJECTIVE TRANSFORMATIONS ON AN INFINITE SET

2010 ◽  
Vol 82 (2) ◽  
pp. 305-321 ◽  
Author(s):  
JANUSZ KONIECZNY
Keyword(s):  
Partial Orders ◽  
Green’S Relations ◽  
Green's Relations ◽  
Infinite Set

AbstractFor an infinite set X, denote by Γ(X) the semigroup of all injective mappings from X to X. For α∈Γ(X), let C(α)={β∈Γ(X):αβ=βα} be the centralizer of α in Γ(X). For an arbitrary α∈Γ(X), we characterize the elements of C(α) and determine Green’s relations in C(α), including the partial orders of ℒ-, ℛ-, and 𝒥-classes.

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).

scholarly journals Baer-Levi semigroups of partial transformations

2004 ◽  
Vol 69 (1) ◽  
pp. 87-106 ◽  
Author(s):  
Fernanda A. Pinto ◽  
R.P. Sullivan
Keyword(s):  
Inverse Semigroup ◽  
Cancellative Semigroup ◽  
Green’S Relations ◽  
Green's Relations ◽  
Algebraic Properties ◽  
Symmetric Inverse Semigroup ◽  
Infinite Set

Let X be an infinite set and suppose א0 ≤ q ≤ |X|. The Baer-Levi semigroup on X is the set of all injective ‘total’ transformations α: X → X such that |X\Xα| = q. It is known to be a right simple, right cancellative semigroup without idempotents, its automorphisms are “inner”, and some of its congruences are restrictions of Malcev congruences on I(X), the symmetric inverse semigroup on X. Here we consider algebraic properties of the semigroup consisting of all injective ‘partial’ transformations α of X such that |X\Xα| = q: in particular, we descried the ideals and Green's relations of it and some of its subsemigroups.

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 INJECTIVE TRANSFORMATIONS WITH EQUAL GAP AND DEFECT

2009 ◽  
Vol 79 (2) ◽  
pp. 327-336 ◽  
Author(s):  
JINTANA SANWONG ◽  
R. P. SULLIVAN
Keyword(s):  
Inverse Semigroup ◽  
Green’S Relations ◽  
Maximal Subsemigroups ◽  
Green's Relations ◽  
Algebraic Properties ◽  
Symmetric Inverse Semigroup ◽  
Infinite Set

AbstractSuppose that X is an infinite set and I(X) is the symmetric inverse semigroup defined on X. If α∈I(X), we let dom α and ran α denote the domain and range of α, respectively, and we say that g(α)=|X/dom α| and d(α)=|X/ran α| is the ‘gap’ and the ‘defect’ of α, respectively. In this paper, we study algebraic properties of the semigroup $A(X)=\{\alpha \in I(X)\mid g(\alpha )=d(\alpha )\}$. For example, we describe Green’s relations and ideals in A(X), and determine all maximal subsemigroups of A(X) when X is uncountable.

Sign in / Sign up

Export Citation Format

Share Document