Rank and idempotent rank of finite full transformation semigroups with restricted range

2013 ◽  
Vol 87 (1) ◽  
pp. 230-242 ◽  
Author(s):  
Worachead Sommanee ◽  
Jintana Sanwong
Keyword(s):  
Restricted Range ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Idempotent Rank

Idempotent rank in finite full transformation semigroups

1990 ◽  
Vol 114 (3-4) ◽  
pp. 161-167 ◽  
Author(s):  
John M. Howie ◽  
Robert B. McFadden
Keyword(s):  
Transformation Semigroup ◽  
Stirling Number ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Generating Set ◽  
Full Transformation Semigroup ◽  
Idempotent Rank ◽  
Finite Set ◽  
Minimal Generating Set

SynopsisThe subsemigroup Singn of singular elements of the full transformation semigroup on a finite set is generated by n(n − l)/2 idempotents of defect one. In this paper we extend this result to the subsemigroup K(n, r) consisting of all elements of rank r or less. We prove that the idempotent rank, defined as the cardinality of a minimal generating set of idempotents, of K(n, r) is S(n, r), the Stirling number of the second kind.

scholarly journals Structure of principal one-sided ideals

2021 ◽  
pp. 1-53
Author(s):  
James East
Keyword(s):  
Structural Analysis ◽  
Restricted Range ◽  
Transformation Semigroups ◽  
Inverse Monoids ◽  
Full Transformation

We give a thorough structural analysis of the principal one-sided ideals of arbitrary semigroups, and then apply this to full transformation semigroups and symmetric inverse monoids. One-sided ideals of these semigroups naturally occur as semigroups of transformations with restricted range or kernel.

scholarly journals Variants of finite full transformation semigroups

2015 ◽  
Vol 25 (08) ◽  
pp. 1187-1222 ◽  
Author(s):  
Igor Dolinka ◽  
James East
Keyword(s):  
Transformation Semigroup ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Regular Elements ◽  
Full Transformation Semigroup ◽  
Generating Sets ◽  
Idempotent Rank ◽  
Finite Set

The variant of a semigroup [Formula: see text] with respect to an element [Formula: see text], denoted [Formula: see text], is the semigroup with underlying set [Formula: see text] and operation ⋆ defined by [Formula: see text] for [Formula: see text]. In this paper, we study variants [Formula: see text] of the full transformation semigroup [Formula: see text] on a finite set [Formula: see text]. We explore the structure of [Formula: see text] as well as its subsemigroups [Formula: see text] (consisting of all regular elements) and [Formula: see text] (consisting of all products of idempotents), and the ideals of [Formula: see text]. Among other results, we calculate the rank and idempotent rank (if applicable) of each semigroup, and (where possible) the number of (idempotent) generating sets of the minimal possible size.

Naturally Ordered Transformation Semigroups Preserving an Equivalence Relation and a Cross-section

2011 ◽  
Vol 18 (03) ◽  
pp. 523-532 ◽  
Author(s):  
Lei Sun ◽  
Weina Deng ◽  
Huisheng Pei
Keyword(s):  
Cross Section ◽  
Equivalence Relation ◽  
Transformation Semigroup ◽  
Natural Order ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Minimal Elements ◽  
Full Transformation Semigroup

The paper is concerned with the so-called natural order on the semigroup [Formula: see text], where [Formula: see text] is the full transformation semigroup on a set X, E is a non-trivial equivalence on X and R is a cross-section of the partition X/E induced by E. We determine when two elements of TE(X,R) are related under this order, find elements of TE(X,R) which are compatible with ≤ on TE(X,R), and observe the maximal and minimal elements and the covering elements.

THE REGULAR PART OF A SEMIGROUP OF LINEAR TRANSFORMATIONS WITH RESTRICTED RANGE

2017 ◽  
Vol 103 (3) ◽  
pp. 402-419 ◽  
Author(s):  
WORACHEAD SOMMANEE ◽  
KRITSADA SANGKHANAN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Restricted Range ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional ◽  
Idempotent Rank ◽  
Image Position

Let$V$be a vector space and let$T(V)$denote the semigroup (under composition) of all linear transformations from$V$into$V$. For a fixed subspace$W$of$V$, let$T(V,W)$be the semigroup consisting of all linear transformations from$V$into$W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’,Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$is the largest regular subsemigroup of$T(V,W)$and characterized Green’s relations on$T(V,W)$. In this paper, we determine all the maximal regular subsemigroups of$Q$when$W$is a finite-dimensional subspace of$V$over a finite field. Moreover, we compute the rank and idempotent rank of$Q$when$W$is an$n$-dimensional subspace of an$m$-dimensional vector space$V$over a finite field$F$.

scholarly journals The structure of elements in finite full transformation semigroups

2005 ◽  
Vol 71 (1) ◽  
pp. 69-74 ◽  
Author(s):  
Gonca Ayik ◽  
Hayrullah Ayik ◽  
Yusuf Ünlü ◽  
John M. Howie
Keyword(s):  
Finite Semigroup ◽  
Transformation Semigroup ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Full Transformation Semigroup

The index and period of an element a of a finite semigroup are the smallest values of m ≥ 1 and r ≥ 1 such that am+r = am. An element with index m and period 1 is called an m-potent element. For an element α of a finite full transformation semigroup with index m and period r, a unique factorisation α = σβ such that Shift(σ) ∩ Shift(β) = ∅ is obtained, where σ is a permutation of order r and β is an m-potent. Some applications of this factorisation are given.

Sign in / Sign up

Export Citation Format

Share Document