Idempotent rank in finite full transformation semigroups

Howie, James; McFadden, R.

doi:10.1017/s0308210500024355

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“…Partitions P, Q of X are said to be of the same type (denoted by P = Q) if they have the same number of classes of each size. We show that (a:S n ) is idem potent-generated and consists of all transformations b in T n for which n(b) contains a partition of the same type as n(a).The idempotent rank of an idempotent-generated semigroup S is the cardinality of a minimal generating set of idempotents of S [2]. It was shown in [2] that the idempotent rank of the S B -normal semigroup K(n,r), consisting of all transformations a with |im(a)|^r, is S(n,r), the Stirling number of the second kind.…”

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confidence: 99%

“…We present a recursive formula for T(n,r) and show that the S n -idempotent rank of K(n,r) is T(n,r). Moreover, we can choose a minimal S n -generating set of idempotents in a single L-class of both T n and S.For each r such that 2 g r g « , the principal factor K(n,r)/K(n,r-1) of T n is denoted by P r in [2]. Each P r is a completely 0-simple semigroup whose non-zero elements may be thought of as the elements a of T n having |im(a)| = r. Then P r is a band of T (n,r) 471…”

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confidence: 99%

“…For each r such that 2 g r g « , the principal factor K(n,r)/K(n,r-1) of T n is denoted by P r in [2]. Each P r is a completely 0-simple semigroup whose non-zero elements may be thought of as the elements a of T n having |im(a)| = r. Then P r is a band of T(n,r) subsemigroups, each of which is a quotient semigroup of an S n -normal semigroup of S n -idempotent rank 1 (Theorem 8).…”

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S_n-normal semigroups

Levi

McFadden

1994

Proceedings of the Edinburgh Mathematical Society

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Certain subsemigroups of the full transformation semigroup T n on a finite set of cardinality n are investigated, namely those subsemigroups S of T n which are normalised by the symmetric group on n elements, the group of units of T n . The S n -normal closure of an element of T n is determined, and the structure o f the S.-normal ideals consisting of the members of T n whose image contains at most r elements is studied.1991 Mathematics subject classification: 20M20.Let T n denote the full transformation semigroup on a set of finite cardinality n, and let S n denote the symmetric group on n elements, the group of units of T n . A subsemigroup S of Tj, is defined to be S n -normal if for each a in S and for each h in S n , the element h~lah is in S. Both T n and S n themselves are S n -normal; so are the ideals K(n,r) = Given aeT n , denote by the smallest S n -normal subsemigroup of T n containing a. Thus is the subsemigroup S of T n generated by {g~lag:geS n }. If a is a permutation then is a normal subgroup of S n and we know what that is. Assuming for the rest of this paper that a is not a permutation, associate with a the partition n(a) of X such that x and y are in the same class of n(a) if and only if xa = ya. Partitions P, Q of X are said to be of the same type (denoted by P = Q) if they have the same number of classes of each size. We show that (a:S n ) is idem potent-generated and consists of all transformations b in T n for which n(b) contains a partition of the same type as n(a).The idempotent rank of an idempotent-generated semigroup S is the cardinality of a minimal generating set of idempotents of S [2]. It was shown in [2] that the idempotent rank of the S B -normal semigroup K(n,r), consisting of all transformations a with |im(a)|^r, is S(n,r), the Stirling number of the second kind. We define the S n -idempotent rank of an S n -normal semigroup S to be the cardinality of a minimal generating set A of idempotents of S such that S = ( = <{g~lag:aeA,geS n }y). Given l g r^n , let T(n,r) denote the number of different types of partitions of an n-element set into r subsets. We present a recursive formula for T(n,r) and show that the S n -idempotent rank of K(n,r) is T(n,r). Moreover, we can choose a minimal S n -generating set of idempotents in a single L-class of both T n and S.For each r such that 2 g r g « , the principal factor K(n,r)/K(n,r-1) of T n is denoted by P r in [2]. Each P r is a completely 0-simple semigroup whose non-zero elements may be thought of as the elements a of T n having |im(a)| = r. Then P r is a band of T (n,r) 471

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S_n-normal semigroups

Levi

McFadden

1994

Proceedings of the Edinburgh Mathematical Society

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“…Our interest in orthogonally labeled lists of the r-sets of X n stems from the J. M. Howie and R. B. McFadden paper [12], where the authors proved the following theorem. THEOREM 1.3 [12].…”

Section: {12}134|2{23}124|3{13}14|23{34}123|4{24}12|34{14}234|1mentioning

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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“…In fact, for τ = r1 r−1 , the existence of an orthogonally r1 r−1 -labelled list is equivalent to the validity of the difficult Middle Levels Conjecture for r, as we show in Remark 3.10. In order to determine the rank and idempotent rank of S(d1 r−1 ) semigroups without proving the Middle Levels Conjecture, we further develop the techniques of [6] and [7], by associating a certain directed graph with a subset of idempotents of a semigroup. Our techniques can be applied to a broad range of semigroups of transformations of finite sets, including certain semigroups of endomorphisms [15].…”

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confidence: 99%

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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Idempotent rank in finite full transformation semigroups

Cited by 97 publications

References 2 publications

S_n-normal semigroups

S_n-normal semigroups

Algorithms for Labeling Constant Weight Gray Codes

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

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Idempotent rank in finite full transformation semigroups

Cited by 97 publications

References 2 publications

Sn-normal semigroups

Sn-normal semigroups

Algorithms for Labeling Constant Weight Gray Codes

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

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S_n-normal semigroups

S_n-normal semigroups