2,621 research outputs found
Graphs with regular monoids
AbstractThis paper is motivated by an open question: which graphs have a regular (endomorphism) monoid? We present an infinite family of graphs, which possess a regular monoid; we also give an approach to construct a nontrivial graph of any order with this property based on a known one, by which the join of two trees with a regular monoid is explicitly described
Application of verification techniques to inverse monoids
The word problem for inverse monoids generated by
a set subject to relations of the form , where and
are both idempotents in the free inverse monoid generated by ,
is investigated. It is
shown that for every fixed monoid of this form the word problem
can be solved in polynomial time which solves an open problem of
Margolis and Meakin. For the uniform word problem, where the presentation is
part of the input, EXPTIME-completeness is shown.
For the Cayley-graphs of these
monoids, it is shown that the first-order theory with regular path
predicates is decidable. Regular path predicates allow to state
that there is a path from a node to a node that is labeled
with a word from some regular language. As a corollary, the decidability
of the generalized word problem is deduced. Finally, some results
on free partially commutative inverse monoids are presented
Algebraic and combinatorial aspects of sandpile monoids on directed graphs
The sandpile group of a graph is a well-studied object that combines ideas
from algebraic graph theory, group theory, dynamical systems, and statistical
physics. A graph's sandpile group is part of a larger algebraic structure on
the graph, known as its sandpile monoid. Most of the work on sandpiles so far
has focused on the sandpile group rather than the sandpile monoid of a graph,
and has also assumed the underlying graph to be undirected. A notable exception
is the recent work of Babai and Toumpakari, which builds up the theory of
sandpile monoids on directed graphs from scratch and provides many connections
between the combinatorics of a graph and the algebraic aspects of its sandpile
monoid.
In this paper we primarily consider sandpile monoids on directed graphs, and
we extend the existing theory in four main ways. First, we give a combinatorial
classification of the maximal subgroups of a sandpile monoid on a directed
graph in terms of the sandpile groups of certain easily-identifiable subgraphs.
Second, we point out certain sandpile results for undirected graphs that are
really results for sandpile monoids on directed graphs that contain exactly two
idempotents. Third, we give a new algebraic constraint that sandpile monoids
must satisfy and exhibit two infinite families of monoids that cannot be
realized as sandpile monoids on any graph. Finally, we give an explicit
combinatorial description of the sandpile group identity for every graph in a
family of directed graphs which generalizes the family of (undirected)
distance-regular graphs. This family includes many other graphs of interest,
including iterated wheels, regular trees, and regular tournaments.Comment: v2: Cleaner presentation, new results in final section. Accepted for
publication in J. Combin. Theory Ser. A. 21 pages, 5 figure
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
Inverse monoids of partial graph automorphisms
A partial automorphism of a finite graph is an isomorphism between its vertex
induced subgraphs. The set of all partial automorphisms of a given finite graph
forms an inverse monoid under composition (of partial maps). We describe the
algebraic structure of such inverse monoids by the means of the standard tools
of inverse semigroup theory, namely Green's relations and some properties of
the natural partial order, and give a characterization of inverse monoids which
arise as inverse monoids of partial graph automorphisms. We extend our results
to digraphs and edge-colored digraphs as well
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
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