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.

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.

THE NATURAL ORDER FOR THE E-ORDER-PRESERVING TRANSFORMATION SEMIGROUPS

2012 ◽  
Vol 05 (03) ◽  
pp. 1250035 ◽  
Author(s):  
Huisheng Pei ◽  
Weina Deng
Keyword(s):  
Transformation Semigroup ◽  
Natural Order ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Minimal Elements ◽  
Full Transformation Semigroup ◽  
Finite Set

Let (X, ≤) be a totally ordered finite set, [Formula: see text] be the full transformation semigroup on X and E be an arbitrary equivalence on X. We consider a subsemigroup of [Formula: see text] defined by [Formula: see text] and call it the E-order-preserving transformation semigroup on X. In this paper, we endow EOPX with the so-called natural order ≤ and discuss when two elements in EOPX are related under this order, then determine those elements of EOPX which are compatible with ≤. Also, the maximal (minimal) elements are described.

scholarly journals Finite full transformation semigroups as collections of random functions

1988 ◽  
Vol 30 (2) ◽  
pp. 203-211 ◽  
Author(s):  
B. Brown ◽  
P. M. Higgins
Keyword(s):  
Semigroup Theory ◽  
Transformation Semigroup ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Random Functions ◽  
Full Transformation Semigroup ◽  
Finite Set ◽  
Algebraic Semigroup

The collection of all self-maps on a non-empty set X under composition is known in algebraic semigroup theory as the full transformation semigroup on X and is written x. Its importance lies in the fact that any semigroup S can be embedded in the full transformation semigroup (where S1 is the semigroup S with identity 1 adjoined, if S does not already possess one). The proof is similar to Cayley's Theorem that a group G can be embedded in SG, the group of all bijections of G to itself. In this paper X will be a finite set of order n, which we take to be and so we shall write Tn for X.

Combinatorial results relating to products of idempotents in finite full transformation semigroups

1990 ◽  
Vol 115 (3-4) ◽  
pp. 289-299 ◽  
Author(s):  
John M. Howie ◽  
Ewing L. Lusk ◽  
Robert B. McFadden
Keyword(s):  
Logic Programming ◽  
Programming Language ◽  
Transformation Semigroup ◽  
Transformation Semigroups ◽  
Singular Element ◽  
Full Transformation ◽  
Logic Programming Language ◽  
Full Transformation Semigroup ◽  
Finite Set

SynopsisEach singular element α of the full transformation semigroup on a finite set is generated by the idempotents of defect one. The length of the shortest expression of α as a product of such idempotents is given by the gravity function g(α).We use certain consequences of a result by Tatsuhiko Saito to explore connections between the defect and the gravity of α, and then determine the number of elements that have maximum gravity. Finally, we obtain formulae for the number of elements of small gravity. Such elements must have defect 1, and we determine their number within each ℋ-class. Many of the results obtained were suggested, and all have been verified, by programs written in PROLOG, a logic programming language very well suited for algebraic calculations.

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.

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.

scholarly journals Square roots in finite full transformation semigroups

1982 ◽  
Vol 23 (2) ◽  
pp. 137-149 ◽  
Author(s):  
Mary Snowden ◽  
J. M. Howie
Keyword(s):  
Transformation Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Square Root ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Square Roots ◽  
Finite Set ◽  
Necessary And Sufficient

Let X be a finite set and let (X) be the full transformation semigroup on X, i.e. the set of all mappings from X into X, the semigroup operation being composition of mappings. This paper aims to characterize those elements of (X) which have square roots. An easily verifiable necessary condition, that of being quasi-square, is found in Theorem 2, and in Theorems 4 and 5 we find necessary and sufficient conditions for certain special elements of (X). The property of being compatibly amenable is shown in Theorem 7 to be equivalent for all elements of (X) to the possession of a square root.

scholarly journals NATURALLY ORDERED TRANSFORMATION SEMIGROUPS PRESERVING AN EQUIVALENCE

2008 ◽  
Vol 78 (1) ◽  
pp. 117-128 ◽  
Author(s):  
LEI SUN ◽  
HUISHENG PEI ◽  
ZHENGXING CHENG
Keyword(s):  
Transformation Semigroup ◽  
Natural Order ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Minimal Elements ◽  
Full Transformation Semigroup ◽  
Image Position

AbstractLet 𝒯X be the full transformation semigroup on a set X and E be a nontrivial equivalence on X. Write then TE(X) is a subsemigroup of 𝒯X. In this paper, we endow TE(X) with the so-called natural order and determine when two elements of TE(X) are related under this order, then find out elements of TE(X) which are compatible with ≤ on TE(X). Also, the maximal and minimal elements and the covering elements are described.

A PARTIAL ORDER ON TRANSFORMATION SEMIGROUPS THAT PRESERVE DOUBLE DIRECTION EQUIVALENCE RELATION

2013 ◽  
Vol 12 (08) ◽  
pp. 1350041 ◽  
Author(s):  
LEI SUN ◽  
JUNLING SUN
Keyword(s):  
Lower Bound ◽  
Partial Order ◽  
Equivalence Relation ◽  
Transformation Semigroup ◽  
Natural Partial Order ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Minimal Elements ◽  
Full Transformation Semigroup

Let [Formula: see text] be the full transformation semigroup on a nonempty set X and E be an equivalence relation on X. Then [Formula: see text] is a subsemigroup of [Formula: see text]. In this paper, we endow it with the natural partial order. With respect to this partial order, we determine when two elements are related, find the elements which are compatible and describe the maximal (minimal) elements. Also, we investigate the greatest lower bound of two elements.

Some subsemigroups of infinite full transformation semigroups

1981 ◽  
Vol 88 (1-2) ◽  
pp. 159-167 ◽  
Author(s):  
J. M. Howie
Keyword(s):  
Transformation Semigroup ◽  
Infinite Cardinal ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Full Transformation Semigroup ◽  
Cardinal Numbers ◽  
Infinite Set

SynopsisAs in an earlier paper by the author, three cardinal numbers, the shift, the defect and the collapse, are associated with each element of the full transformation semigroup ℑ(X), where X is an infinite set. It is shown that the elements of finite shift and non-zero defect form a subsemigroup F of ℑ(X). Moreover, if E(F) denotes the set of idempotents in F then 〈E(F)〉 = F, but (E(F))n ⊂F for every finite n. For each infinite cardinal m not exceeding ∣X∣ the set Qm of balanced elements of weight m, i.e. those with shift, defect and collapse all equal to m, forms a subsemigroup of ℑ(X). Moreover, (E(Qm))4=Qm,(E(Qm))3⊂Qm.

Sign in / Sign up

Export Citation Format

Share Document