James East

ON THE SINGULAR PART OF THE PARTITION MONOID

We study the singular part of the partition monoid ; that is, the ideal , where is the symmetric group. Our main results are presentations in terms of generators and relations. We also show that is idempotent generated, and that its rank and idempotent-rank are both equal to . One of our presentations uses an idempotent generating set of this minimal cardinality.

A presentation of the dual symmetric inverse monoid

The dual symmetric inverse monoid is the inverse monoid of all isomorphisms between quotients of an n-set. We give a monoid presentation of and, along the way, establish criteria for a monoid to be inverse when it is generated by completely regular elements.

A Presentation of the Singular Part of the Symmetric Inverse Monoid

We give a semigroup presentation of the singular part of the symmetric inverse monoid on a finite set. Along the way, we derive a monoid presentation of the monoid of all order-preserving injective partial transformations on a finite chain, which differs from the presentation discovered by Fernandes.

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

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 P_n 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 P_n, which coincides with the singular part of P_n. 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.

Enumeration of idempotents in diagram semigroups and algebras

We give a characterisation of the idempotents of the partition monoid, and use this to enumerate the idempotents in the finite partition, Brauer and partial Brauer monoids, giving several formulae and recursions for the number of idempotents in each monoid as well as various R -, L - and D -classes. We also apply our results to determine the number of idempotent basis elements in the finite dimensional partition, Brauer and partial Brauer algebras.

PRESENTATIONS FOR SINGULAR SUBSEMIGROUPS OF THE PARTIAL TRANSFORMATION SEMIGROUP

The partial transformation semigroup

Variants of finite full transformation semigroups

\mathcal{PT}_n

Maximal subsemigroups of the semigroup of all mappings on an infinite set

is the semigroup of all partial transformations on the finite set n = {1,…, n}. The transformation semigroup

Motzkin monoids and partial Brauer monoids

\mathcal{T}_n\subseteq\mathcal{PT...

Idempotent Generation in the Endomorphism Monoid of a Uniform Partition

The variant of a semigroup S with respect to an element a in S, denoted S^a, is the semigroup with underlying set S and operation * defined by x*y=xay for x,y in S. In this article, we study variants T_X^a of the full transformation semigroup T_X on a finite set X. We explore the structure of T_X^a as well as its subsemigroups Reg(T_X^a) (consisting of all regular elements) and E_X^a (consisting of all products of idempotents), and the ideals of Reg(T_X^a). Among other results, we calculate the rank and idempotent rank (if applicable) of each semigroup, and (where possible) the number of (idempotent) generating sets of the minimal possible size.