Combinatorics of Nilpotents in Symmetric Inverse Semigroups

Ganyushkin, Olexandr; Mazorchuk, Volodymyr

doi:10.1007/s00026-004-0213-7

Cited by 6 publications

(2 citation statements)

References 13 publications

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“…Note that the Bell numbers [5] are denoted, as a rule, by B k , but we use another notation to distinguish them from the Boolean. In the given case, we are interested in the maximal nilpotent subsemigroup with idempotent 0, i.e., with a nowhere-defined transformation.…”

Section: Proposition 8 For Any Idempotent E E Is Mmentioning

confidence: 99%

Mathematical modeling of nilpotent subsemigroups of semigroups of contracting transformations of a Boolean

Selezneva

2009

Ukr Math J

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We study mathematical models of the structure of nilpotent subsemigroups of the semigroup PTD B n ( ) of partial contracting transformations of a Boolean, the semigroup TD B n ( ) of full contracting transformations of a Boolean, and the inverse semigroup ISD B n ( ) of contracting transformations of a Boolean. We propose a convenient graphical representation of the semigroups considered. For each of these semigroups, the uniqueness of its maximal nilpotent subsemigroup is proved. For PTD B n ( ) and TD B n ( ) , the capacity of a maximal nilpotent subsemigroup is calculated. For ISD B n ( ) , we construct estimates for the capacity of a maximal nilpotent subsemigroup and calculate this capacity for small n. For all indicated semigroups, we describe the structure of nilelements and maximal nilpotent subsemigroups of nilpotency degree k and determine the number of elements and subsemigroups for some special cases. Main NotionsLet S be a semigroup with zero element 0. We call S nilpotent if there exists a number k such that S k = { } 0 . The minimal number k is called the nilpotency degree of the semigroup S. The set of all nilpotent subsemigroups of S is partially ordered by inclusion, and the maximal elements of this set are called the maximal nilpotent subsemigroups of S [1,2].In addition to general nilpotent subsemigroups, nilpotent subsemigroups with fixed nilpotency degree are also considered. The set of these subsemigroups is also partially ordered, and, hence, one can consider the maximal nilpotent subsemigroups of nilpotency degree k. For what follows, we need the propositions [3] presented below.Proposition 1. Let S be a finite semigroup. Then the following assertions are equivalent:(1) S is nilpotent;(2) every element a S ∈ nilpotent;(3) the unique idempotent of S is the zero element.Note that Proposition 1 is not always satisfied in the case of an infinite semigroup. For example, consider the semigroup T = ( , ) n m { n, m N ∈ , 0 < n < m} ∪ 0 { } with multiplication

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Section: Proposition 8 For Any Idempotent E E Is Mmentioning

confidence: 99%

Mathematical modeling of nilpotent subsemigroups of semigroups of contracting transformations of a Boolean

Selezneva

2009

Ukr Math J

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“…Hence, the importance of I n in inverse semigroup theory is similar to the importance of S n in group theory. Although the semigroup I n has been extensively studied (see, for example, [1,5,10,14]), there are still many interesting problems concerning I n to be investigated.…”

Section: Introductionmentioning

confidence: 99%

Quasi-idempotent ranks of the proper ideals in finite symmetric inverse semigroups

Bugay¹

2021

Turk J Math

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Let In and Sn be the symmetric inverse semigroup and the symmetric group on a finite chain Xn = {1,. .. , n} , respectively. Also let In,r = {α ∈ In : |im(α)| ≤ r} for 1 ≤ r ≤ n − 1. For any α ∈ In , if α = α 2 = α 4 then α is called a quasi-idempotent. In this paper, we show that the quasi-idempotent rank of In,r (both as a semigroup and as an inverse semigroup) is n 2 if r = 2 , and n r + 1 if r ≥ 3. The quasi-idempotent rank of In,1 is n (as a semigroup) and n − 1 (as an inverse semigroup).

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Combinatorial results for semigroups of order-preserving or order-reversing subpermutations

Laradji

Umar

2015

Journal of Difference Equations and Applications

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Combinatorics of Nilpotents in Symmetric Inverse Semigroups

Cited by 6 publications

References 13 publications

Mathematical modeling of nilpotent subsemigroups of semigroups of contracting transformations of a Boolean

Mathematical modeling of nilpotent subsemigroups of semigroups of contracting transformations of a Boolean

Quasi-idempotent ranks of the proper ideals in finite symmetric inverse semigroups

Combinatorial results for semigroups of order-preserving or order-reversing subpermutations

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