Generators and relations for partition monoids and algebras

East, James

doi:10.1016/j.jalgebra.2011.04.008

Cited by 69 publications

(93 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 13. It follows from [8,Proposition 39] and its proof that rank(P X : S X ) = 2 for any finite set X with |X| ≥ 2.…”

Section: Theorem 12mentioning

confidence: 99%

“…The partition algebras, originally introduced in the context of statistical mechanics [35], contain all of the above diagram algebras and so provide a unified framework in which to study diagram algebras more generally. Partition algebras may be thought of as twisted semigroup algebras of partition monoids, and many properties of the partition algebras may be deduced from corresponding properties of the associated monoids [8,9,19,39]. Recent studies have also recognized partition monoids and some of their submonoids as key objects in the pseudovarieties of finite aperiodic monoids and semigroups with involution [2][3][4].…”

Section: Introductionmentioning

confidence: 99%

“…As noted in [8,11], the partition monoids contain a number of important transformation semigroups as submonoids, including the symmetric groups, the full transformation semigroups, and the symmetric and dual symmetric inverse monoids; see [15,17,20,26,[31][32][33] for background on these subsemigroups. Many studies of infinite transformation semigroups have concentrated on features concerning generation.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Infinite partition monoids

East

2014

Int. J. Algebra Comput.

Self Cite

View full text Add to dashboard Cite

Let $\mathcal P_X$ and $\mathcal S_X$ be the partition monoid and symmetric group on an infinite set $X$. We show that $\mathcal P_X$ may be generated by $\mathcal S_X$ together with two (but no fewer) additional partitions, and we classify the pairs $\alpha,\beta\in\mathcal P_X$ for which $\mathcal P_X$ is generated by $\mathcal S_X\cup\{\alpha,\beta\}$. We also show that $\mathcal P_X$ may be generated by the set $\mathcal E_X$ of all idempotent partitions together with two (but no fewer) additional partitions. In fact, $\mathcal P_X$ is generated by $\mathcal E_X\cup\{\alpha,\beta\}$ if and only if it is generated by $\mathcal E_X\cup\mathcal S_X\cup\{\alpha,\beta\}$. We also classify the pairs $\alpha,\beta\in\mathcal P_X$ for which $\mathcal P_X$ is generated by $\mathcal E_X\cup\{\alpha,\beta\}$. Among other results, we show that any countable subset of $\mathcal P_X$ is contained in a $4$-generated subsemigroup of $\mathcal P_X$, and that the length function on $\mathcal P_X$ is bounded with respect to any generating set

show abstract

“…Remark 13. It follows from [8,Proposition 39] and its proof that rank(P X : S X ) = 2 for any finite set X with |X| ≥ 2.…”

Section: Theorem 12mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Infinite partition monoids

East

2014

Int. J. Algebra Comput.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Study of P X qua semigroup is more recent and scanty. The reader is referred to the articles [13], [3], [4], which include full descriptions and examples; here we give just a concise description, which suffices for the present purpose. A partition over X is a quotient object (or equivalently, a partition) of the coproduct (disjoint union) X ⊔ X of two copies of the set X.…”

Section: Dualising: Partition Monoidsmentioning

confidence: 99%

Mitsch’s order and inclusion for binary relations and partitions

FitzGerald

2012

Semigroup Forum

View full text Add to dashboard Cite

Abstract. Mitsch's natural partial order on the semigroup of binary relations is here characterised by equations in the theory of relation algebras. The natural partial order has a complex relationship with the compatible partial order of inclusion, which is explored by means of a sublattice of the lattice of preorders on the semigroup. The corresponding sublattice for the partition monoid is also described. Natural partial ordersIn [10], Heinz Mitsch formulated a characterisation of the natural partial order ≤ on the full transformation semigroup T X which did not use inverses or idempotents, and went on to define the natural partial order ≤ on any semigroup S by(The discovery was also made independently by P. M. Higgins, but remained unpublished.) Observe that a = ya follows. Mitsch's natural partial order has now been characterised, and its properties investigated, for several concrete classes of non-regular semigroups-in [8,12] for some semigroups of (partial) transformations, and by Namnak and Preechasilp [11] for the semigroup B X of all binary relations on the set X. The partial order of inclusion which is carried by B X may also be thought of as 'natural', and it is the broad purpose of this note to discuss the relationship between these two partial orders on B X . Moreover, the same questions are addressed for the partition monoid P X , which also carries two 'natural' partial orders. So we shall use a slightly different nomenclature here for the sake of clarity, mostly referring to partial orders as just orders, and the natural partial order as Mitsch's order. We begin by collecting some information about B X . Binary relationsThe notation used here for binary relations follows that found in, for example, Clifford and Preston [1], with the addition of complementation of relations defined by x α c y ⇐⇒ (x, y) ∈ α for x, y ∈ X. Note that the symbol • for composition will be suppressed, except for the composites of order relations on B X . We will make use of the identity relation on X, ι = {(x, x) : x ∈ X} , and the universal relation ω = X × X.The following logical equivalence will also be required; it is the 'Theorem K' of De Morgan [2, p. xxx ] (see also e.g. [6]).

show abstract

“…Since then, Popova's presentation has been re-proved in a variety of different ways, and several other presentations have been discovered [4,6,8,10,27,29]. Presentations for other semigroups containing the symmetric groups have been obtained by a number of different authors, and by many different methods [1,5,9,13,21,25].…”

Section: Introductionmentioning

confidence: 99%