Presentations for rook partition monoids and algebras and their singular ideals

East, James

doi:10.1016/j.jpaa.2018.05.016

Cited by 10 publications

(7 citation statements)

References 67 publications

(117 reference statements)

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“…The rook partition monoid, denoted RP X , is a diagram monoid containing P X . It has been studied in a number of settings, and under a variety of different names; see for example [22,42,43,67,68].…”

Section: Rook Partition Monoidsmentioning

confidence: 99%

Ehresmann theory and partition monoids

East

Gray

2021

Journal of Algebra

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Section: Rook Partition Monoidsmentioning

confidence: 99%

Ehresmann theory and partition monoids

East

Gray

2021

Journal of Algebra

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“…The original study of cellular semigroup algebras may be found in [35]; see also [73,74,112,113] for some recent developments. Another example of how the study of these semigroups can give information about the associated algebras may be found in work of the first author [36], who gives presentations for the partition monoid and shows how these presentations give rise to presentations for the partition algebra; see also [37,43,44]. A further example is given in the paper [27] where idempotents in the partition, Brauer and partial Brauer monoids are described and enumerated, and then the results are applied to determine the number of idempotent basis elements in the finite dimensional partition, Brauer and partial Brauer algebras; see also [28].…”

Section: Introductionmentioning

confidence: 99%

Diagram monoids and Graham–Houghton graphs: Idempotents and generating sets of ideals

East

Gray

2017

Journal of Combinatorial Theory, Series A

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Abstract. We study the ideals of the partition, Brauer, and Jones monoid, establishing various combinatorial results on generating sets and idempotent generating sets via an analysis of their Graham-Houghton graphs. We show that each proper ideal of the partition monoid Pn is an idempotent generated semigroup, and obtain a formula for the minimal number of elements (and the minimal number of idempotent elements) needed to generate these semigroups. In particular, we show that these two numbers, which are called the rank and idempotent rank (respectively) of the semigroup, are equal to each other, and we characterize the generating sets of this minimal cardinality. We also characterize and enumerate the minimal idempotent generating sets for the largest proper ideal of Pn, which coincides with the singular part of Pn. Analogous results are proved for the ideals of the Brauer and Jones monoids; in each case, the rank and idempotent rank turn out to be equal, and all the minimal generating sets are described. We also show how the rank and idempotent rank results obtained, when applied to the corresponding twisted semigroup algebras (the partition, Brauer, and Temperley-Lieb algebras), allow one to recover formulae for the dimensions of their cell modules (viewed as cellular algebras) which, in the semisimple case, are formulae for the dimensions of the irreducible representations of the algebras. As well as being of algebraic interest, our results relate to several well-studied topics in graph theory including the problem of counting perfect matchings (which relates to the problem of computing permanents of {0, 1}-matrices and the theory of Pfaffian orientations), and the problem of finding factorizations of Johnson graphs. Our results also bring together several well-known number sequences such as Stirling, Bell, Catalan and Fibonacci numbers.

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“…It would be interesting to attempt to apply the methods of this paper to other natural diagram categories, such as rook partition categories; cf. [35,54].…”

Section: Partial Brauer Categoriesmentioning

confidence: 99%

Congruence lattices of ideals in categories and (partial) semigroups

East¹,

Ruškuc²

2020

Preprint

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This paper presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley-Lieb and Jones partitions; linear and projective linear transformations; and partial braids. Special considerations are needed for certain small ideals, and technically more intricate theoretical underpinnings for the linear and partial braid categories.

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Presentations for rook partition monoids and algebras and their singular ideals

Cited by 10 publications

References 67 publications

Ehresmann theory and partition monoids

Ehresmann theory and partition monoids

Diagram monoids and Graham–Houghton graphs: Idempotents and generating sets of ideals

Congruence lattices of ideals in categories and (partial) semigroups

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