Partial monoids: associativity and confluence

A partial monoid $P$ is a set with a partial multiplication $\times$ (and total identity $1_P$) which satisfies some associativity axiom. The partial monoid $P$ may be embedded in a free monoid $P^*$ and the product $\star$ is simulated by a string rewriting system on $P^*$ that consists in evaluating the concatenation of two letters as a product in $P$, when it is defined, and a letter $1_P$ as the empty word $\epsilon$. 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 $P^*$. 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

On the Expressive Power of Counting

http://www.springerlink.com/content/f713305702708547/?p=e5505b568c714aaab33444b548f67070π=7We investigate the expressive power of various extensions of first-order, inductive, and infinitary logic with counting quantifiers. We consider in particular a LOGSPACE extension of first-order logic, and a PTIME extension of fixpoint logic with counters. Counting is a fundamental tool of algorithms. It is essential in the case of unordered structures. Our aim is to understand the expressive power gained with a limited counting ability. We consider two problems: (i) unnested counters, and (ii) counters with no free variables. We prove a hierarchy result based on the arity of the counters under the first restriction. The proof is based on a game technique that is introduced in the paper. We also establish results on the asymptotic probabilities of sentences with counters under the second restriction. In particular, we show that first-order logic with equality of the cardinalities of relations has a 0/1 law

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 $\varphi$-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 Schützenberger'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

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

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

Hyperdeterminantal computation for the Laughlin wave function

International audienceThe decomposition of the Laughlin wave function in the Slater orthogonal basis appears in the discussion on the second-quantized form of the Laughlin states and is straightforwardly equivalent to the decomposition of the even powers of the Vandermonde determinants in the Schur basis. Such a computation is notoriously difficult and the coefficients of the expansion have not yet been interpreted. In our paper, we give an expression of these coefficients in terms of hyperdeterminants of sparse tensors. We use this result to construct an algorithm allowing to compute one coefficient of the development without computing the others. Thanks to a program in {\tt C}, we performed the calculation for the square of the Vandermonde up to an alphabet of eleven lettres

Dendriform structures for restriction-deletion and restriction-contraction matroid Hopf algebras

International audienceWe endow the set of isomorphism classes of matroids with a new Hopf algebra structure, in which the coproduct is implemented via the combinatorial operations of restriction and deletion. We also initiate the investigation of dendriform coalgebra structures on matroids and introduce a monomial invariant which satisfy a convolution identity with respect to restriction and deletion