The (p,q)-potent ranks of certain semigroups of transformations

2016 ◽  
Vol 16 (07) ◽  
pp. 1750138
Author(s):  
Ping Zhao ◽  
Taijie You ◽  
Huabi Hu
Keyword(s):  
Partial Transformation ◽  
Transformation Semigroups ◽  
Generating Set ◽  
Idempotent Rank ◽  
Finite Set ◽  
Special Case

Let [Formula: see text] and [Formula: see text] be the partial transformation and the strictly partial transformation semigroups on the finite set [Formula: see text]. It is well known that the ranks of the semigroups [Formula: see text] and [Formula: see text] are [Formula: see text], for [Formula: see text], and [Formula: see text], for [Formula: see text], respectively. The idempotent rank, defined as the smallest number of idempotents generating set, of the semigroup [Formula: see text] has the same value as the rank. Idempotent can be seen as a special case (with [Formula: see text]) of [Formula: see text]-potent. In this paper, we determine the [Formula: see text]-potent ranks, defined as the smallest number of [Formula: see text]-potents generating set, of the semigroups [Formula: see text], for [Formula: see text], and [Formula: see text], for [Formula: see text].

Idempotents in partial transformation semigroups

1990 ◽  
Vol 116 (3-4) ◽  
pp. 359-366 ◽  
Author(s):  
G. U. Garba
Keyword(s):  
Real Number ◽  
Stirling Number ◽  
Partial Transformation ◽  
Transformation Semigroups ◽  
Generating Set ◽  
Idempotent Rank ◽  
Image Position ◽  
Minimal Generating Set

SynopsisAn element α of Pn, the semigroup of all partial transformations of {1,2,…, n}, is said to have projection characteristic (r, s), or to belong to the set [r, s], if dom α= r, im α = s. Let E be the set of all idempotents in Pn\[n, n] and E1, the set of those idempotents with projection characteristic (n, n − 1) or (n − 1, n − 1). For α in Pn\[n, n], we define a number g(α), called the gravity of α and closely related to the number denned in Howie [5] for full transformations, and we obtain the result thatLet d(α) be the defect of α, and for any real number x let [x] be the least integer m such that m ≧ x. Then by analogy with the results of Saito [9] we have thatα ϵ Ek(α) and α ∉ Ek(α)where k(α) = [g(α)/d(α)] or [g(α)/d(α)+ 1. Following Howie, Lusk and McFadden [6] we then explore connections between the defect and the gravity of α. Letting K(n, r) be the subsemigroup of Pn consisting of all elements of rank r or less, we prove a result, corresponding to that of Howie and McFadden [7] for total transformations, that the idempotent rank, defined as the cardinality of a minimal generating set of idempotents, of K(n, r) is S(n + 1, r + 1), the Stirling number of the second kind.

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.

Idempotent generators in finite full transformation semigroups

1978 ◽  
Vol 81 (3-4) ◽  
pp. 317-323 ◽  
Author(s):  
J. M. Howie
Keyword(s):  
Transformation Semigroups ◽  
Minimal Set ◽  
Full Transformation ◽  
Generating Set ◽  
Finite Set

SynopsisIt was proved by Howie in 1966 that , the semigroup of all singular mappings of a finite set X into itself, is generated by its idempotents. Implicit in the method of proof, though not formally stated, is the result that if |X| = n then the n(n – 1) idempotents whose range has cardinal n – 1 form a generating set for. Here it is shown that if n ≧ 3 then a minimal set M of idempotent generators for contains ½n(n–1) members. A formula is given for the number of distinct sets M.

scholarly journals Generalised partial transformation semigroups

1975 ◽  
Vol 19 (4) ◽  
pp. 470-473 ◽  
Author(s):  
R. P. Sullivan
Keyword(s):  
Partial Transformation ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Special Case

It is well-known that for any set X, px, the semigroup of all partial transformations on X, can be embedded in Jx∪a for some a ∉ X (see for example Clifford and Preston (1967) and Ljapin (1963)). Recently Magill (1967) has considered a special case of what we call ‘generalised partial transformation semigroups’. We show here that any such semigroup can always be embedded in a full transformation smigroup in which the operation is not in general equal to the usual composition of mappinas. We then examine conditions under which such a semigroup, (J x, θ), is isomorphic to the semigroup, under composition, of all transformations on the same set X.

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.

Multiplicative invertibility characterization on star-like cyclicpoid CyPω*n finite partial transformation semigroups

2021 ◽  
Vol 10 (1) ◽  
pp. 45-55
Author(s):  
Sulaiman Awwal Akinwunmi ◽  
Morufu Mogbolagade Mogbonju ◽  
Adenike Olusola Adeniji
Keyword(s):  
Partial Transformation ◽  
Transformation Semigroups

SH pulse in an elastic wedge

1973 ◽  
Vol 63 (5) ◽  
pp. 1571-1582
Author(s):  
A. M. Abo-Zena ◽  
Chi-Yu King
Keyword(s):  
Integral Transform ◽  
Wedge Angle ◽  
Integral Form ◽  
Diffraction Effect ◽  
Elastic Wedge ◽  
Sh Waves ◽  
Wedge Surface ◽  
Line Parallel ◽  
Finite Set ◽  
Special Case

abstract This paper gives an analysis of the response of an elastic wedge of arbitrary angle to an impulsive SH source applied on the wedge surface along a line parallel to the edge of the wedge. A two-dimensional time-dependent Green's function for SH waves is constructed from an integral-transform approach. The result is given in a closed form for the incident and the reflected pulses and in an integral form for the diffracted pulse from the edge. For the special case that the wedge angle is an integral fraction of π, the result is interpretable in terms of a finite set of image sources with no diffraction effect. Numerical examples are given for illustration.

scholarly journals On the density of Cayley graphs of R.Thompson’s group F in symmetric generators

2021 ◽  
pp. 1-13
Author(s):  
V. S. Guba
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Finite Graph ◽  
Vertex Degree ◽  
Doubling Property ◽  
Generating Set ◽  
Finite Set ◽  
Standard Set ◽  
Thompson’S Group ◽  
Average Vertex

By the density of a finite graph we mean its average vertex degree. For an [Formula: see text]-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with [Formula: see text] generators is amenable if and only if the density of the corresponding Cayley graph equals [Formula: see text]. A famous problem on the amenability of R. Thompson’s group [Formula: see text] is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators [Formula: see text], is at least [Formula: see text]. This estimate has not been exceeded so far. For the set of symmetric generators [Formula: see text], where [Formula: see text], the same example only gave an estimate of [Formula: see text]. There was a conjecture that for this generating set equality holds. If so, [Formula: see text] would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set [Formula: see text], the inequality [Formula: see text] holds. In this paper, we disprove this conjecture showing that the density of the Cayley graph of [Formula: see text] in symmetric generators [Formula: see text] strictly exceeds [Formula: see text]. Moreover, we show that even larger generating set [Formula: see text] does not have doubling property.

scholarly journals Relative ranks of some partial transformation semigroups

2019 ◽  
Vol 43 (5) ◽  
pp. 2218-2225
Author(s):  
Ebru YİĞİT ◽  
Gonca AYIK ◽  
Hayrullah AYIK
Keyword(s):  
Partial Transformation ◽  
Transformation Semigroups
Sign in / Sign up

Export Citation Format

Share Document