The monoid of all orientation-preserving and extensive full transformations on a finite chain

2021 ◽  
pp. 2250105
Author(s):  
De Biao Li ◽  
Wen Ting Zhang ◽  
Yan Feng Luo
Keyword(s):  
Finite Chain ◽  
Generating Set ◽  
Maximal Subsemigroups ◽  
Idempotent Rank

Let [Formula: see text] be the monoid of all orientation-preserving and extensive full transformations on [Formula: see text] ordered in the standard way. In this paper, we determine the minimum generating set and the minimum idempotent generating set of [Formula: see text], and so the rank and the idempotent rank of [Formula: see text] are obtained. Moreover, we describe maximal subsemigroups and maximal idempotent generated subsemigroups of [Formula: see text] and completely obtain their classifications.

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].

scholarly journals COMBINATORIAL TECHNIQUES FOR DETERMINING RANK AND IDEMPOTENT RANK OF CERTAIN FINITE SEMIGROUPS

2002 ◽  
Vol 45 (3) ◽  
pp. 617-630 ◽  
Author(s):  
Inessa Levi ◽  
Steve Seif
Keyword(s):  
Positive Integer ◽  
Directed Graph ◽  
Subject Classification ◽  
Generating Set ◽  
Finite Semigroups ◽  
Idempotent Rank ◽  
Strongly Connected ◽  
Primary 20M20

AbstractLet $\tau$ be a partition of the positive integer $n$. A partition of the set $\{1,2,\dots,n\}$ is said to be of type $\tau$ if the sizes of its classes form the partition $\tau$ of $n$. It is known that the semigroup $S(\tau)$, generated by all the transformations with kernels of type $\tau$, is idempotent generated. When $\tau$ has a unique non-singleton class of size $d$, the difficult Middle Levels Conjecture of combinatorics obstructs the application of known techniques for determining the rank and idempotent rank of $S(\tau)$. We further develop existing techniques, associating with a subset $U$ of the set of all idempotents of $S(\tau)$ with kernels of type $\tau$ a directed graph $D(U)$; the directed graph $D(U)$ is strongly connected if and only if $U$ is a generating set for $S(\tau)$, a result which leads to a proof if the fact that the rank and the idempotent rank of $S(\tau)$ are both equal to$$ \max\biggl\{\binom{n}{d},\binom{n}{d+1}\biggr\}. $$AMS 2000 Mathematics subject classification: Primary 20M20; 05A18; 05A17; 05C20

Twisted Brauer monoids

2018 ◽  
Vol 148 (4) ◽  
pp. 731-750 ◽  
Author(s):  
Igor Dolinka ◽  
James East
Keyword(s):  
Principal Ideal ◽  
Sufficient Conditions ◽  
Singular Part ◽  
Necessary And Sufficient Conditions ◽  
Green’S Relations ◽  
Generating Set ◽  
Idempotent Rank ◽  
Singular Ideal ◽  
Necessary And Sufficient ◽  
Lattice Of Ideals

We investigate the structure of the twisted Brauer monoid , comparing and contrasting it with the structure of the (untwisted) Brauer monoid . We characterize Green's relations and pre-orders on , describe the lattice of ideals and give necessary and sufficient conditions for an ideal to be idempotent generated. We obtain formulae for the rank (smallest size of a generating set) and (where applicable) the idempotent rank (smallest size of an idempotent generating set) of each principal ideal; in particular, when an ideal is idempotent generated, its rank and idempotent rank are equal. As an application of our results, we describe the idempotent generated subsemigroup of (which is not an ideal), as well as the singular ideal of (which is neither principal nor idempotent generated), and we deduce that the singular part of the Brauer monoid is idempotent generated, a result previously proved by Maltcev and Mazorchuk.

scholarly journals HALL'S CONDITION AND IDEMPOTENT RANK OF IDEALS OF ENDOMORPHISM MONOIDS

2008 ◽  
Vol 51 (1) ◽  
pp. 57-72 ◽  
Author(s):  
R. Gray
Keyword(s):  
Simple Semigroup ◽  
Analogous Result ◽  
Direct Consequence ◽  
Full Transformation ◽  
Generating Set ◽  
Endomorphism Monoids ◽  
Idempotent Rank ◽  
Transformation Monoid ◽  
Hall's Condition ◽  
Property Holding

AbstractIn 1990, Howie and McFadden showed that every proper two-sided ideal of the full transformation monoid $T_n$, the set of all maps from an $n$-set to itself under composition, has a generating set, consisting of idempotents, that is no larger than any other generating set. This fact is a direct consequence of the same property holding in an associated finite $0$-simple semigroup. We show a correspondence between finite $0$-simple semigroups that have this property and bipartite graphs that satisfy a condition that is similar to, but slightly stronger than, Hall's condition. The results are applied in order to recover the above result for the full transformation monoid and to prove the analogous result for the proper two-sided ideals of the monoid of endomorphisms of a finite vector space.

A general approach for generating sets of certain subsemigroups of monotone maps

2019 ◽  
Vol 13 (07) ◽  
pp. 2050132
Author(s):  
Leyla Bugay
Keyword(s):  
Natural Order ◽  
Finite Chain ◽  
Generating Set ◽  
Generating Sets ◽  
Monotone Maps ◽  
Minimal Generating Set

Let [Formula: see text] and [Formula: see text] be the monoids of all (full) transformations and of all partial transformations, on a finite chain [Formula: see text] under its natural order, respectively. Moreover, let [Formula: see text] ([Formula: see text]) be the subsemigroup of [Formula: see text] ([Formula: see text]) consists of all monotone transformations (monotone partial transformations) with height less than or equal to [Formula: see text] for [Formula: see text] ([Formula: see text]). In this paper, we develop a new and general approach to find a (minimal) generating set of [Formula: see text] ([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.

ON A NEW APPROACH TO THE DUAL SYMMETRIC INVERSE MONOID $\mathcal{I}_{X}^{\ast}$

2007 ◽  
Vol 17 (03) ◽  
pp. 567-591 ◽  
Author(s):  
VICTOR MALTCEV
Keyword(s):  
Inverse Semigroup ◽  
Cross Section ◽  
Inverse Monoid ◽  
Finite Sets ◽  
New Approach ◽  
Generating Set ◽  
Maximal Subsemigroups ◽  
Symmetric Inverse Semigroup ◽  
Transitive Representation ◽  
Symmetric Inverse Monoid

We construct the inverse partition semigroup[Formula: see text], isomorphic to the dual symmetric inverse monoid[Formula: see text], introduced in [6]. We give a convenient geometric illustration for elements of [Formula: see text]. We describe all maximal subsemigroups of [Formula: see text] and find a generating set for [Formula: see text] when X is finite. We prove that all the automorphisms of [Formula: see text] are inner. We show how to embed the symmetric inverse semigroup into the inverse partition one. For finite sets X, we establish that, up to equivalence, there is a unique faithful effective transitive representation of [Formula: see text], namely to [Formula: see text]. Finally, we construct an interesting [Formula: see text]-cross-section of [Formula: see text], which is reminiscent of [Formula: see text], the [Formula: see text]-cross-section of [Formula: see text], constructed in [4].

scholarly journals ON THE SINGULAR PART OF THE PARTITION MONOID

2011 ◽  
Vol 21 (01n02) ◽  
pp. 147-178 ◽  
Author(s):  
JAMES EAST
Keyword(s):  
Symmetric Group ◽  
Singular Part ◽  
Minimal Cardinality ◽  
Generating Set ◽  
Generators And Relations ◽  
Ideal Formula ◽  
Idempotent Rank ◽  
The Ideal

We study the singular part of the partition monoid [Formula: see text]; that is, the ideal [Formula: see text], where [Formula: see text] is the symmetric group. Our main results are presentations in terms of generators and relations. We also show that [Formula: see text] is idempotent generated, and that its rank and idempotent-rank are both equal to [Formula: see text]. One of our presentations uses an idempotent generating set of this minimal cardinality.

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