The Subsemigroup Generated By the Idempotents of a Full Transformation Semigroup

Howie, James

doi:10.1112/jlms/s1-41.1.707

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“…In [1], J. M. Howie considered the semigroup of transformations of sets and proved (Theorem 1) that every transformation of a finite set which is not a permutation can be written as a product of idempotents. In view of the analogy between the theories of transformations of finite sets and linear transformations of finite dimensional vector spaces, Howie's theorem suggests a corresponding result for matrices.…”

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On products of idempotent matrices

1967

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In [1], J. M. Howie considered the semigroup of transformations of sets and proved (Theorem 1) that every transformation of a finite set which is not a permutation can be written as a product of idempotents. In view of the analogy between the theories of transformations of finite sets and linear transformations of finite dimensional vector spaces, Howie's theorem suggests a corresponding result for matrices. The purpose of this note is to prove such a result.THEOREM. Every singular square matrix can be written as a product ofidempotent matrices.Proof. If E is an idempotent matrix and P is non-singular, then P ~ 1 EP is also idempotent and hence it is sufficient to prove that every singular matrix is similar to a product of idempotents. Two separate proofs of this are given. The first is an induction proof for matrices that are similar to triangular matrices and hence holds in general only for matrices over an algebraically closed field. For a proof valid for matrices over any field an explicit construction is given in II. It is clear that II is sufficient to establish the result but as it relies on the " rational canonical form " and as I is so simple, it is of interest to include both proofs. I. Proof for matrices that are similar to triangular matrices.We proceed by induction on the order of the matrix, and make the induction hypothesis that the theorem is valid for such matrices of order n -1. Suppose that a matrix A is of order n. As A is singular it is similar to an upper triangular matrix whose first column is zero and so we may partition this matrix as°

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confidence: 99%

On products of idempotent matrices

1967

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“…In [4] Howie showed that the subsemigroup of T n generated by E, the set of all idempotents of T n of defect one, is the semigroup S of all "singular" mappings:…”

Section: Is Denoted By V(t) V(z) Etc the Radius R(t) Of A Tree T Ismentioning

confidence: 99%

“…As another example we find all the 6th roots of the idempotent a e T 4 The full details are left to the reader, with the reminder that offspring tuples involving one-arrow trees may include trivial trees with no non-root points. The square roots of a are:…”

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“…Also T x must contain the topmost tree (containing the point 21), as it has a maximal path whose In the previous construction the total number of choices for P x , P 2 , P 3 was 1 x 2 x 2 = 4, leading to four distinct cube roots of E (although the reader should be aware that different choices of the maximal paths P t does not automatically lead to different roots; e.g. see the example of [4]). It is also conceivable that (T u T 3 , T 2 ) might be an offspring Therefore the total number of cube roots of a is 1 x 4 x 4 = 16.…”

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Digraphs and the semigroup of all functions on a finite set

Higgins

1988

Glasgow Math. J.

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Let T n denote the full transformation semigroup on the finite set n = {1, 2, ..., n), that is the set of all mappings from h to n, with function composition as the semigroup operation. In this paper algorithms are introduced to solve equations such as ax m b = c and ax = xb (a, b, c e T n ), which employ a representation of members of T n as special directed graphs.Algorithms for equations in T n . The object of study here is T n , the full transformation semigroup on n = {1, 2, . . . , n}, which means the semigroup of all mappings from n to itself under composition. This semigroup is to the theory of algebraic semigroups what the symmetric group (the group of all permutations on a set) is to group theory. It is easy to prove a "Cayley Theorem" to the effect that any semigroup S can be faithfully represented as a subsemigroup of T s i, the full transformation semigroup on the set S 1 (the semigroup S 1 is the semigroup S, with an adjoined identity 1, if 5 does not already possess one (see [1])). The full transformation semigroup also enjoys the important property of regularity, meaning that each element a e T n has an inverse x (not necessarily unique) in the sense that a = axa and x = xax.In [3] the author introduced a method for the calculation of all square roots of a given a e T n , the full transformation semigroup on n = {1, 2,. . . , n), as an alternative to the necessary and sufficient conditions given in [6] for or to be a square. The technique, which relies on a representation of a as a special directed graph, is extended here to furnish algorithms for the solution of the equations:(1) and ax =xb (a,b,ceT n , n 3=1).(2) The following graph theoretic definitions and results come from [2]. A digraph is said to be weak if it is connected when viewed as a graph. A functional digraph is a weak digraph in which every point has outdegree one. An in-tree is a digraph with a sink (point of outdegree zero), which is a tree when regarded as a graph.

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“…For a positive integer r n − 1, let K(n, r) be the subsemigroup of T n consisting of all transformations of height at most r. Howie shows in [5] that K(n, r) is idempotent generated, and in [6] he establishes a one-to-one correspondence between minimal generating sets of K(n, n − 1) and the set of strongly connected tournaments, a correspondence we describe in Remark 3.4 below.…”

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Combinatorial Techniques for Determining Rank and Idempotent Rank of Certain Finite Semigroups

Levi¹,

Seif²

2002

Proceedings of the Edinburgh Mathematical Society

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Let τ be a partition of the positive integer n. A partition of the set {1, 2, . . . , n} is said to be of type τ if the sizes of its classes form the partition τ of n. It is known that the semigroup S(τ ), generated by all the transformations with kernels of type τ , is idempotent generated. When τ 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(τ ). We further develop existing techniques, associating with a subset U of the set of all idempotents of S(τ ) with kernels of type τ a directed graph D(U ); the directed graph D(U ) is strongly connected if and only if U is a generating set for S(τ ), a result which leads to a proof if the fact that the rank and the idempotent rank of S(τ ) are both equal to max n d , n d + 1 .

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The Subsemigroup Generated By the Idempotents of a Full Transformation Semigroup

Cited by 211 publications

References 0 publications

On products of idempotent matrices

On products of idempotent matrices

Digraphs and the semigroup of all functions on a finite set

Combinatorial Techniques for Determining Rank and Idempotent Rank of Certain Finite Semigroups

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