Idempotent Of Rank Research Articles

On the semigroups of order-preserving transformations generated by idempotents of rank n −1

Let [Formula: see text] be the semigroup of all singular order-preserving mappings on the finite set [Formula: see text]. It is known that [Formula: see text] is generated by its set of idempotents of rank [Formula: see text], and its rank and idempotent rank are [Formula: see text] and [Formula: see text], respectively. In this paper, we study the structure of the semigroup generated by any nonempty subset of idempotents of rank [Formula: see text] in [Formula: see text]. We also calculate its rank and idempotent rank.

Lengths of words in transformation semigroups generated by digraphs

Given a simple digraph D on n vertices (with nge 2), there is a natural construction of a semigroup of transformations langle Drangle . For any edge (a, b) of D, let arightarrow b be the idempotent of rank n-1 mapping a to b and fixing all vertices other than a; then, define langle Drangle to be the semigroup generated by a rightarrow b for all (a,b) in E(D). For alpha in langle Drangle , let ell (D,alpha ) be the minimal length of a word in E(D) expressing alpha . It is well known that the semigroup mathrm {Sing}_n of all transformations of rank at most n-1 is generated by its idempotents of rank n-1. When D=K_n is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate ell (K_n,alpha ), for any alpha in langle K_nrangle = mathrm {Sing}_n; however, no analogous non-trivial results are known when D ne K_n. In this paper, we characterise all simple digraphs D such that either ell (D,alpha ) is equal to Howie–Iwahori’s formula for all alpha in langle Drangle , or ell (D,alpha ) = n - mathrm {fix}(alpha ) for all alpha in langle Drangle , or ell (D,alpha ) = n - mathrm {rk}(alpha ) for all alpha in langle Drangle . We also obtain bounds for ell (D,alpha ) when D is an acyclic digraph or a strong tournament (the latter case corresponds to a smallest generating set of idempotents of rank n-1 of mathrm {Sing}_n). We finish the paper with a list of conjectures and open problems.

Reducibility of semigroups and nilpotent commutators with idempotents of rank two

Let f be a noncommutative polynomial in two variables. Let S be a multiplicative semigroup of linear operators on a finite-dimensional vector space and T a fixed linear operator such that f ( T , S ) is nilpotent for all S in S. In [H. Radjavi, M. Omladič, Nilpotent commutators and reducibility of semigroups , Lin. and Multilin. Alg. 57 (2009), 307-317] the authors proposed questions, what one can say about the invariant subspace structure of S under this and other related conditions. In particular, they study the question under the condition that [ S , T ] 2 = 0, where T is a given idempotent of rank one. In this paper we extend some of the results given there to the case that T is a given idempotent of rank two.

Fully invariant transformations and associated groups

It is well known that the symmetric group S ntogether with one idempotent of rank n- 1 on a finite n-element set Nserves as a set of generators for the semigroup T nof all the total transformations on N. It is also well known that the singular part Sing n of T n can be generated by a set of idempotents of rank n- 1. The purpose of this paper is to begin an investigation of the way in which Singnand its subsemigroups can be generated by the conjugates of a subset of elements of T n by a subgroup of S n . We look for the smallest subset of elements of T n that will serve and, correspondingly, for a characterization of those subgroups of S n that will serve. Using some techniques from graph theory we prove our main result:the conjugates of a single transformation of rank n- 1 under Gsuffice to generate Singnif and only if Gis what we define to be a 2-block transitive subgroup of S n .

Semigroups generated by a group and an idempotent

It is well known that the semigroup of all transformations on a finite set X of order n is generated by its group of units, the symmetric group, and any idempotent of rank n − 1. Similarly, the symmetric inverse semigroup on X is generated by its group of units and any idempotent of rank n − 1 while the analogous result is true for the semigroup of all n × n matrices over a field. In this paper we begin a systematic study of the structure of a semigroup S generated by its group G of units and an idempotent ϵ . The first section consists of preliminaries while the second contains some general results which provide the setting for those which follow. In the third section we shall investigate the situation where G is a permutation group on a set X of order n and ϵ is an idempotent of rank n − 1. In particular, we shall show that any such semigroup S is regular. Furthermore we shall determine when S is an inverse or orthodox semigroup or completely regular semigroup. The fourth section deals with a special case, that in which G is cyclic. The fifth, and last, deals with the situation where G is dihedral. In both cases, the resulting semigroup has a particularly delicate structure which is of interest in its own right. Both situations are replete with interesting combinatorial gems. The author was led to the results of this paper by considering the output of a computer program he was writing for generating and analyzing semigroups.

Large Sample Asymptotic Theory of Tests for Uniformity on the Grassmann Manifold

The Grassmann manifold Gk,m − k consists of k-dimensional linear subspaces V in Rm. To each V in Gk,m − k, corresponds a unique m × m orthogonal projection matrix P idempotent of rank k. Let Pk,m − k denote the set of all such orthogonal projection matrices. We discuss distribution theory on Pk,m − k, presenting the differential form for the invariant measure and properties of the uniform distribution, and suggest a general family F(P) of non-uniform distributions. We are mainly concerned with large sample asymptotic theory of tests for uniformity on Pk,m − k. We investigate the asymptotic distribution of the standardized sample mean matrix U taken from the family F(P) under a sequence of local alternatives for large sample size n. For tests of uniformity versus the matrix Langevin distribution which belongs to the family F(P), we consider three optimal tests-the Rayleigh-style, the likelihood ratio, and the locally best invariant tests. They are discussed in relation to the statistic U, and are shown to be approximately, near uniformity, equivalent to one another. Zonal and invariant polynomials in matrix arguments are utilized in derivations.

Powers of matrices and idempotency

In this paper, we establish the following results: Let A be a square matrix of rank r. Then (a) (A+A ∗) 2 is idempotent of rank r, and tr r A (defined as the sum of the principal minors of order r in A) is one iff A is Hermitian idempotent. (b) A s = A t for some positive integers s≠ t, and tr A= rank A iff A is idempotent. (c) A(A∗A) s= A(AA ∗) t for some integers s≠ t iff AA ∗=A ∗A is idempotent, while A(A ∗A) s= A(AA ∗) s for some integers s≠0 iff AA ∗=A ∗A . (d) A(A ∗A) s=A ∗ (AA ∗) t for some integers s≠ t and rank A= tr A iff A is Hermitian idempotent, while A(A ∗A) s= A ∗(AA ∗) s for some integer s iff A is Hermitian. Here A ∗ indicates the conjugate transpose of A, and P - α is defined iff ( P +) α =( P α ) + for all positive integers α and P + is the Moore-Penrose inverse of P.