692 research outputs found
Enumeration of idempotents in planar diagram monoids
We classify and enumerate the idempotents in several planar diagram monoids:
namely, the Motzkin, Jones (a.k.a. Temperley-Lieb) and Kauffman monoids. The
classification is in terms of certain vertex- and edge-coloured graphs
associated to Motzkin diagrams. The enumeration is necessarily algorithmic in
nature, and is based on parameters associated to cycle components of these
graphs. We compare our algorithms to existing algorithms for enumerating
idempotents in arbitrary (regular *-) semigroups, and give several tables of
calculated values.Comment: Majorly revised (new title, new abstract, one additional author), 24
pages, 6 figures, 8 tables, 5 algorithm
On Kiselman quotients of 0-Hecke monoids
Combining the definition of 0-Hecke monoids with that of Kiselman semigroups,
we define what we call Kiselman quotients of 0-Hecke monoids associated with
simply laced Dynkin diagrams. We classify these monoids up to isomorphism,
determine their idempotents and show that they are -trivial. For
type we show that Catalan numbers appear as the maximal cardinality of our
monoids, in which case the corresponding monoid is isomorphic to the monoid of
all order-preserving and order-decreasing total transformations on a finite
chain. We construct various representations of these monoids by matrices, total
transformations and binary relations. Motivated by these results, with a mixed
graph we associate a monoid, which we call a Hecke-Kiselman monoid, and
classify such monoids up to isomorphism. Both Kiselman semigroups and Kiselman
quotients of 0-Hecke monoids are natural examples of Hecke-Kiselman monoids.Comment: 14 pages; International Electronic Journal of Algebra, 201
Amalgams of Inverse Semigroups and C*-algebras
An amalgam of inverse semigroups [S,T,U] is full if U contains all of the
idempotents of S and T. We show that for a full amalgam [S,T,U], the C*-algebra
of the inverse semigroup amaglam of S and T over U is the C*-algebraic amalgam
of C*(S) and C*(T) over C*(U). Using this result, we describe certain
amalgamated free products of C*-algebras, including finite-dimensional
C*-algebras, the Toeplitz algebra, and the Toeplitz C*-algebras of graphs
On the insertion of n-powers
In algebraic terms, the insertion of -powers in words may be modelled at
the language level by considering the pseudovariety of ordered monoids defined
by the inequality . We compare this pseudovariety with several other
natural pseudovarieties of ordered monoids and of monoids associated with the
Burnside pseudovariety of groups defined by the identity . In
particular, we are interested in determining the pseudovariety of monoids that
it generates, which can be viewed as the problem of determining the Boolean
closure of the class of regular languages closed under -power insertions. We
exhibit a simple upper bound and show that it satisfies all pseudoidentities
which are provable from in which both sides are regular elements
with respect to the upper bound
Profinite Groups Associated to Sofic Shifts are Free
We show that the maximal subgroup of the free profinite semigroup associated
by Almeida to an irreducible sofic shift is a free profinite group,
generalizing an earlier result of the second author for the case of the full
shift (whose corresponding maximal subgroup is the maximal subgroup of the
minimal ideal). A corresponding result is proved for certain relatively free
profinite semigroups. We also establish some other analogies between the kernel
of the free profinite semigroup and the \J-class associated to an irreducible
sofic shift
Semigroups, rings, and Markov chains
We analyze random walks on a class of semigroups called ``left-regular
bands''. These walks include the hyperplane chamber walks of Bidigare, Hanlon,
and Rockmore. Using methods of ring theory, we show that the transition
matrices are diagonalizable and we calculate the eigenvalues and
multiplicities. The methods lead to explicit formulas for the projections onto
the eigenspaces. As examples of these semigroup walks, we construct a random
walk on the maximal chains of any distributive lattice, as well as two random
walks associated with any matroid. The examples include a q-analogue of the
Tsetlin library. The multiplicities of the eigenvalues in the matroid walks are
``generalized derangement numbers'', which may be of independent interest.Comment: To appear in J. Theoret. Proba
- …