430 research outputs found
M\"obius inversion formula for monoids with zero
The M\"obius inversion formula, introduced during the 19th century in number
theory, was generalized to a wide class of monoids called locally finite such
as the free partially commutative, plactic and hypoplactic monoids for
instance. In this contribution are developed and used some topological and
algebraic notions for monoids with zero, similar to ordinary objects such as
the (total) algebra of a monoid, the augmentation ideal or the star operation
on proper series. The main concern is to extend the study of the M\"obius
function to some monoids with zero, i.e., with an absorbing element, in
particular the so-called Rees quotients of locally finite monoids. Some
relations between the M\"obius functions of a monoid and its Rees quotient are
also provided.Comment: 12 pages, r\'esum\'e \'etendu soumis \`a FPSAC 201
Generalized Results on Monoids as Memory
We show that some results from the theory of group automata and monoid
automata still hold for more general classes of monoids and models. Extending
previous work for finite automata over commutative groups, we demonstrate a
context-free language that can not be recognized by any rational monoid
automaton over a finitely generated permutable monoid. We show that the class
of languages recognized by rational monoid automata over finitely generated
completely simple or completely 0-simple permutable monoids is a semi-linear
full trio. Furthermore, we investigate valence pushdown automata, and prove
that they are only as powerful as (finite) valence automata. We observe that
certain results proven for monoid automata can be easily lifted to the case of
context-free valence grammars.Comment: In Proceedings AFL 2017, arXiv:1708.0622
The Tutte-Grothendieck group of a convergent alphabetic rewriting system
The two operations, deletion and contraction of an edge, on multigraphs
directly lead to the Tutte polynomial which satisfies a universal problem. As
observed by Brylawski in terms of order relations, these operations may be
interpreted as a particular instance of a general theory which involves
universal invariants like the Tutte polynomial, and a universal group, called
the Tutte-Grothendieck group. In this contribution, Brylawski's theory is
extended in two ways: first of all, the order relation is replaced by a string
rewriting system, and secondly, commutativity by partial commutations (that
permits a kind of interpolation between non commutativity and full
commutativity). This allows us to clarify the relations between the semigroup
subject to rewriting and the Tutte-Grothendieck group: the later is actually
the Grothendieck group completion of the former, up to the free adjunction of a
unit (this was even not mention by Brylawski), and normal forms may be seen as
universal invariants. Moreover we prove that such universal constructions are
also possible in case of a non convergent rewriting system, outside the scope
of Brylawski's work.Comment: 17 page
Commutative positive varieties of languages
We study the commutative positive varieties of languages closed under various
operations: shuffle, renaming and product over one-letter alphabets
Automata and rational expressions
This text is an extended version of the chapter 'Automata and rational
expressions' in the AutoMathA Handbook that will appear soon, published by the
European Science Foundation and edited by JeanEricPin
Convolution, Separation and Concurrency
A notion of convolution is presented in the context of formal power series
together with lifting constructions characterising algebras of such series,
which usually are quantales. A number of examples underpin the universality of
these constructions, the most prominent ones being separation logics, where
convolution is separating conjunction in an assertion quantale; interval
logics, where convolution is the chop operation; and stream interval functions,
where convolution is used for analysing the trajectories of dynamical or
real-time systems. A Hoare logic is constructed in a generic fashion on the
power series quantale, which applies to each of these examples. In many cases,
commutative notions of convolution have natural interpretations as concurrency
operations.Comment: 39 page
On the representation theory of finite J-trivial monoids
In 1979, Norton showed that the representation theory of the 0-Hecke algebra
admits a rich combinatorial description. Her constructions rely heavily on some
triangularity property of the product, but do not use explicitly that the
0-Hecke algebra is a monoid algebra.
The thesis of this paper is that considering the general setting of monoids
admitting such a triangularity, namely J-trivial monoids, sheds further light
on the topic. This is a step to use representation theory to automatically
extract combinatorial structures from (monoid) algebras, often in the form of
posets and lattices, both from a theoretical and computational point of view,
and with an implementation in Sage.
Motivated by ongoing work on related monoids associated to Coxeter systems,
and building on well-known results in the semi-group community (such as the
description of the simple modules or the radical), we describe how most of the
data associated to the representation theory (Cartan matrix, quiver) of the
algebra of any J-trivial monoid M can be expressed combinatorially by counting
appropriate elements in M itself. As a consequence, this data does not depend
on the ground field and can be calculated in O(n^2), if not O(nm), where n=|M|
and m is the number of generators. Along the way, we construct a triangular
decomposition of the identity into orthogonal idempotents, using the usual
M\"obius inversion formula in the semi-simple quotient (a lattice), followed by
an algorithmic lifting step.
Applying our results to the 0-Hecke algebra (in all finite types), we recover
previously known results and additionally provide an explicit labeling of the
edges of the quiver. We further explore special classes of J-trivial monoids,
and in particular monoids of order preserving regressive functions on a poset,
generalizing known results on the monoids of nondecreasing parking functions.Comment: 41 pages; 4 figures; added Section 3.7.4 in version 2; incorporated
comments by referee in version
- …