97 research outputs found
Partial monoids: associativity and confluence
A partial monoid is a set with a partial multiplication (and
total identity ) which satisfies some associativity axiom. The partial
monoid may be embedded in a free monoid and the product is
simulated by a string rewriting system on that consists in evaluating the
concatenation of two letters as a product in , when it is defined, and a
letter as the empty word . In this paper we study the profound
relations between confluence for such a system and associativity of the
multiplication. Moreover we develop a reduction strategy to ensure confluence
and which allows us to define a multiplication on normal forms associative up
to a given congruence of . Finally we show that this operation is
associative if, and only if, the rewriting system under consideration is
confluent
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
The mechanics of shuffle products and their siblings
We carry on the investigation initiated in [15] : we describe new shuffle
products coming from some special functions and group them, along with other
products encountered in the literature, in a class of products, which we name
-shuffle products. Our paper is dedicated to a study of the latter
class, from a combinatorial standpoint. We consider first how to extend
Radford's theorem to the products in that class, then how to construct their
bi-algebras. As some conditions are necessary do carry that out, we study them
closely and simplify them so that they can be seen directly from the definition
of the product. We eventually test these conditions on the products mentioned
above
Sweedler's duals and SchĂŒtzenberger's calculus
We describe the problem of Sweedler's duals for bialgebras as essentially characterizing the domain of the transpose of the multiplication. This domain is the set of what could be called ``representative linear forms'' which are the elements of the algebraic dual which are also representative on the multiplicative semigroup of the algebra. When the algebra is free, this notion is indeed equivalent to that of rational functions of automata theory. For the sake of applications, the range of coefficients has been considerably broadened, i.e. extended to semirings, so that the results could be specialized to the boolean and multiplicity cases. This requires some caution (use of ``positive formulas'', iteration replacing inversion, stable submodules replacing finite-rank families for instance). For the theory and its applications has been created a rational calculus which can, in return, be applied to harness Sweedler's duals. A new theorem of rational closure and application to Hopf algebras of use in Physics and Combinatorics is provided. The concrete use of this ``calculus'' is eventually illustrated on an example
Free quasi-symmetric functions, product actions and quantum field theory of partitions
We examine two associative products over the ring of symmetric functions
related to the intransitive and Cartesian products of permutation groups. As an
application, we give an enumeration of some Feynman type diagrams arising in
Bender's QFT of partitions. We end by exploring possibilities to construct
noncommutative analogues.Comment: Submitted 28.11.0
Dual bases for non commutative symmetric and quasi-symmetric functions via monoidal factorization
In this work, an effective construction, via Sch\"utzenberger's monoidal
factorization, of dual bases for the non commutative symmetric and
quasi-symmetric functions is proposed
Noncommutative Symmetric Functions VII: Free Quasi-Symmetric Functions Revisited
We prove a Cauchy identity for free quasi-symmetric functions and apply it to
the study of various bases. A free Weyl formula and a generalization of the
splitting formula are also discussed.Comment: 21 pages, Latex, 2 figure
Combinatorics of -deformed stuffle Hopf algebras
In order to extend the Sch\"utzenberger's factorization to general
perturbations, the combinatorial aspects of the Hopf algebra of the
-deformed stuffle product is developed systematically in a parallel way
with those of the shuffle product
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