3,584 research outputs found
Maximal subsemigroups of the semigroup of all mappings on an infinite set
In this paper we classify the maximal subsemigroups of the \emph{full
transformation semigroup} , which consists of all mappings on
the infinite set , containing certain subgroups of the symmetric group
\sym(\Omega) on . In 1965 Gavrilov showed that there are five maximal
subsemigroups of containing \sym(\Omega) when is
countable and in 2005 Pinsker extended Gavrilov's result to sets of arbitrary
cardinality.
We classify the maximal subsemigroups of on a set of
arbitrary infinite cardinality containing one of the following subgroups of
\sym(\Omega): the pointwise stabiliser of a non-empty finite subset of
, the stabiliser of an ultrafilter on , or the stabiliser of a
partition of into finitely many subsets of equal cardinality. If
is any of these subgroups, then we deduce a characterisation of the mappings
such that the semigroup generated by
equals .Comment: Revised according to comments by the referee, 29 pages, 11 figures,
to appear in Trans. American Mathematical Societ
Idempotent rank in the endomorphism monoid of a non-uniform partition
We calculate the rank and idempotent rank of the semigroup generated
by the idempotents of the semigroup , which consists of all
transformations of the finite set preserving a non-uniform partition .
We also classify and enumerate the idempotent generating sets of this minimal
possible size. This extends results of the first two authors in the uniform
case.Comment: 12 pages; 4 figure
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 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
Braids and factorizable inverse monoids
What is the untangling effect on a braid if one is allowed to snip a string, or if two specified strings are allowed to pass through each other, or even allowed to merge and part as newly reconstituted strings? To calculate the effects, one works in an appropriate factorizable inverse
monoid, some aspects of a general theory of which are discussed in this
paper. The coset monoid of a group arises, and turns out to have a universal
property within a certain class of factorizable inverse monoids. This theory
is dual to the classical construction of fundamental inverse semigroups from
semilattices. In our braid examples, we will focus mainly on the ``merge and
part'' alternative, and introduce a monoid which is a natural preimage of
the largest factorizable inverse submonoid of the dual symmetric inverse
monoid on a finite set, and prove that it embeds in the coset monoid of the
braid group
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