5,933 research outputs found
On a special monoid with a single defining relation
AbstractWe show that no finite union of congruence classes [w], w being an arbitrary element of the free monoid {a, b}∗ with unit 1, is a context-free language if the congruence is defined by the single pair (abbaab, 1). This congruence is neither confluent nor even preperfect. The monoid formed by its congruence classes is a group which has infinitely many isomorphic proper subgroups
Decidability and Independence of Conjugacy Problems in Finitely Presented Monoids
There have been several attempts to extend the notion of conjugacy from
groups to monoids. The aim of this paper is study the decidability and
independence of conjugacy problems for three of these notions (which we will
denote by , , and ) in certain classes of finitely
presented monoids. We will show that in the class of polycyclic monoids,
-conjugacy is "almost" transitive, is strictly included in
, and the - and -conjugacy problems are decidable with linear
compexity. For other classes of monoids, the situation is more complicated. We
show that there exists a monoid defined by a finite complete presentation
such that the -conjugacy problem for is undecidable, and that for
finitely presented monoids, the -conjugacy problem and the word problem are
independent, as are the -conjugacy and -conjugacy problems.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1503.0091
Crystallizing the hypoplactic monoid: from quasi-Kashiwara operators to the Robinson--Schensted--Knuth-type correspondence for quasi-ribbon tableaux
Crystal graphs, in the sense of Kashiwara, carry a natural monoid structure
given by identifying words labelling vertices that appear in the same position
of isomorphic components of the crystal. In the particular case of the crystal
graph for the -analogue of the special linear Lie algebra
, this monoid is the celebrated plactic monoid, whose
elements can be identified with Young tableaux. The crystal graph and the
so-called Kashiwara operators interact beautifully with the combinatorics of
Young tableaux and with the Robinson--Schensted--Knuth correspondence and so
provide powerful combinatorial tools to work with them. This paper constructs
an analogous `quasi-crystal' structure for the hypoplactic monoid, whose
elements can be identified with quasi-ribbon tableaux and whose connection with
the theory of quasi-symmetric functions echoes the connection of the plactic
monoid with the theory of symmetric functions. This quasi-crystal structure and
the associated quasi-Kashiwara operators are shown to interact just as neatly
with the combinatorics of quasi-ribbon tableaux and with the hypoplactic
version of the Robinson--Schensted--Knuth correspondence. A study is then made
of the interaction of the crystal graph of the plactic monoid and the
quasi-crystal graph for the hypoplactic monoid. Finally, the quasi-crystal
structure is applied to prove some new results about the hypoplactic monoid.Comment: 55 pages. Minor revision to fix typos, add references, and discuss an
open questio
Groupoids, Frobenius algebras and Poisson sigma models
In this paper we discuss some connections between groupoids and Frobenius
algebras specialized in the case of Poisson sigma models with boundary. We
prove a correspondence between groupoids in the category Set and relative
Frobenius algebras in the category Rel, as well as an adjunction between a
special type of semigroupoids and relative H*-algebras. The connection between
groupoids and Frobenius algebras is made explicit by introducing what we called
weak monoids and relational symplectic groupoids, in the context of Poisson
sigma models with boundary and in particular, describing such structures in the
ex- tended symplectic category and the category of Hilbert spaces. This is part
of a joint work with Alberto Cattaneo and Chris Heunen.Comment: 12 pages, 1 figure. To appear in "Mathematical Aspects of Quantum
Field Theories". Mathematical Physical Studies, Springer. Proceedings of the
Winter School in Mathematical Physics, Les Houges, 201
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