On products of idempotent matrices

Erdös, J. A.

doi:10.1017/s0017089500000173

Cited by 138 publications

(106 citation statements)

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“…The unit group of M, if non-trivial, is never idempotent generated. Both the full transformation semigroup of a finite set and the multiplicative monoid of n x n matrices over a field have the property that the non-units are products of idempotents, see, for example, [3,6].…”

Section: Proof Let a E (E(m)) Then A M Jfa Lmmentioning

confidence: 99%

Products of idempotents in algebraic monoids

Putcha¹

2006

J. Aust. Math. Soc.

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Section: Proof Let a E (E(m)) Then A M Jfa Lmmentioning

confidence: 99%

Products of idempotents in algebraic monoids

Putcha¹

2006

J. Aust. Math. Soc.

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“…J. A. Erdos [6] proved that the semigroup of singular endomorphisms of a finite dimensional vector space is idempotent-generated. J.…”

Section: Efficient Labeling Algorithms (Theorem 15)mentioning

confidence: 99%

Algorithms for Labeling Constant Weight Gray Codes

Levi¹,

McFadden²,

Seif³

2005

Glasgow Math. J.

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Abstract. Let n and r be positive integers with 1 < r < n, and let X n = {1, 2, . . . , n}. An r-set A and a partition π of X n are said to be orthogonal if every class of π meets A in exactly one element. We prove that if A 1 , A 2 , . . . , A ( n r ) is a list of the distinct r-sets of X n with |A i ∩ A i+1 | = r − 1 for i = 1, 2, . . . , n r taken modulo n r , then there exists a list of distinct partitions π 1 , π 2 , . . . , π ( n r ) such that π i is orthogonal to both A i and A i+1 . This result states that any constant weight Gray code admits a labeling by distinct orthogonal partitions. Using an algorithm from the literature on Gray codes, we provide a surprisingly efficient algorithm that on input (n, r) outputs an orthogonally labeled constant weight Gray code. We also prove a two-fold Gray enumeration result, presenting an orthogonally labeled constant weight Gray code in which the partition labels form a cycle in the covering graph of the lattice of all partitions of X n . This leads to a conjecture related to the Middle Levels Conjecture. Finally, we provide an application of our results to calculating minimal generating sets of idempotents for finite semigroups.2000 Mathematics Subject Classification. 94B25, 05A18, 20M20. Introduction and background.We prove a combinatorics result concerning constant weight Gray codes, with application to minimal generating sets of finite semigroups. The paper contributes techniques, algorithms, and problems aimed at understanding the combinatorics of functions on finite sets and their efficient listing.Let X n = {1, 2 . . . , n}. In the 1950's Frank Gray [9] developed algorithms that for a positive integer n, list the 2 n subsets of X n in such a way that successive subsets differ minimally, having a singleton symmetric difference (including the first and last set). The lists, now known as Gray codes, were used by Gray to minimize errors in certain analog computer operations.For a given positive integer r with 1 ≤ r < n, refer to an r-element subset of X n as an r-set. Gray [9] also constructed listings of the r-sets of X n so that successive r-sets differ minimally, having two element symmetric difference (equivalently, successive rsets intersect in an (r − 1)-set). These r-set listings became known as constant weight Gray codes; we specify parameters by calling such a code a constant weight (n, r)-Gray code. A partition π of X n is said to be a weight-r partition if π has r classes. In the present paper, we prove that every (n, r)-Gray code admits what we call an orthogonal labeling by weight-r partitions. An orthogonal labeling of a constant weight (n, r)-Gray code leads to a listing of a minimal generating set for the finite semigroup K(n, r), consisting of all transformations on X n with image sets of r or fewer elements. We

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“…A result of J. A. Erdos 6 states that if A is a singular n × n matrix with entries in field F then A can be written as the product of idempotents over F. Krishnamoorthy, Rajagopalan and Vijayakumar 7 have studied the basic concepts of kidempotent matrices as generalization of idempotent matrices. The concept of circulant matrices was introduced and some of their properties were discussed in 8,9 .…”

Section: Introductionmentioning

confidence: 99%

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2018

jusps-A

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In this paper we introduce the concept of kidempotent circulant matrices and discuss some of its basic characterizations. We obtain the necessary and sufficient conditions for the sum of two kidempotent circulant matrices to be kidempotent circulant and then it is generalized for the sum of 'n' kidempotent circulant matrices and also obtain the necessary and sufficient conditions for the product of two kidempotent circulant matrices to be kidempotent circulant and then it is generalized for the sum of 'n' kidempotent circulant matrices.

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On products of idempotent matrices

Cited by 138 publications

References 1 publication

Products of idempotents in algebraic monoids

Products of idempotents in algebraic monoids

Algorithms for Labeling Constant Weight Gray Codes

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