Products of idempotents in finite full transformation semigroups

1980 ◽  
Vol 86 (3-4) ◽  
pp. 243-254 ◽  
Author(s):  
J. M. Howie
Keyword(s):  
Lower Bound ◽  
Fixed Points ◽  
Transformation Semigroups ◽  
Singular Element ◽  
Full Transformation

SynopsisIf |X| = n and α is a singular mapping in J(X), define c(α) to be the number of cyclic orbits of α and f(α) to be the number of fixed points. Then α is expressible as a product of n + c(α)−f(α) idempotents of rank n − 1, and no smaller number of idempotents of rank n − 1 will suffice. The maximum possible value of n + c(α)–f(α) is [3/2(n − 1)], which is thus a best possible global lower bound for the number of idempotents required to generate a singular element of J(X).

Products of idempotents in finite full transformation semigroups: some improved bounds

1984 ◽  
Vol 98 (1-2) ◽  
pp. 25-35 ◽  
Author(s):  
John M. Howie
Keyword(s):  
Lower Bound ◽  
Fixed Points ◽  
Upper Bound ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Arbitrary Rank

SynopsisLet E be the set of idempotents in Sn, the semigroup of all singular selfmaps of {1,…, n}. For each α in Sn, there is a unique (κ(α)≧1 such that αψEκ,(α). It is known that κ(α)≦ n + cycl α -fix α, where cyclα is the number of cyclic orbits of a and fix α is the number of fixed points. Equality holds only in the case where a is of rank n – 1. An improved upper bound is obtained for κ(α), applying to elements of arbitrary rank. A lower bound is obtained also.

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.

scholarly journals Lower Bounds on Words Separation: Are There Short Identities in Transformation Semigroups?

10.37236/6450 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Andrei A. Bulatov ◽  
Olga Karpova ◽  
Arseny M. Shur ◽  
Konstantin Startsev
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Symmetric Groups ◽  
Computer Search ◽  
Transformation Semigroups ◽  
Deterministic Finite Automaton ◽  
Separation Problem ◽  
Full Transformation ◽  
Multiplicative Constant ◽  
Minimum Number

The words separation problem, originally formulated by Goralcik and Koubek (1986), is stated as follows. Let $Sep(n)$ be the minimum number such that for any two words of length $\le n$ there is a deterministic finite automaton with $Sep(n)$ states, accepting exactly one of them. The problem is to find the asymptotics of the function $Sep$. This problem is inverse to finding the asymptotics of the length of the shortest identity in full transformation semigroups $T_k$. The known lower bound on $Sep$ stems from the unary identity in $T_k$. We find the first series of identities in $T_k$ which are shorter than the corresponding unary identity for infinitely many values of $k$, and thus slightly improve the lower bound on $Sep(n)$. Then we present some short positive identities in symmetric groups, improving the lower bound on separating words by permutational automata by a multiplicative constant. Finally, we present the results of computer search for short identities for small $k$.

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.

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