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

On Drinfel'd associators

In 1986, in order to study the linear representations of the braid group $B\_n$coming from the monodromy of the Knizhnik-Zamolodchikov differential equations,Drinfel'd introduced a class of formal power series $\Phi$on noncommutative variables. These formal series can be considered as a class of associators. We here give an interpretation of them as well as some new tools over Noncommutative Evolution Equations. Asymptotic phenomena are also discussed

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

Hopf algebras: motivations and examples

This paper provides motivation as well as a method of construction for Hopf algebras, starting from an associative algebra. The dualization technique involved relies heavily on the use of Sweedler's dual

Modeling the early stages of a user-centered process in architectural design through adaptation of the methodologies of New Product Design

In order to reach a degree of quality in architectural buildings that is likely to lead to user satisfaction, architectural design relies on integrating user-related information even before generation of building concepts. However, integrating such information may be seen as a hindrance to architectural creation. It therefore seems necessary to propose a methodological approach that allows integration of a user-centred point of view as well as generation of creative architectural concepts. Our research proposes to apply a collaborative process of New Product Design (NPD) in order to further enrich the traditional process of architectural design. We will present some experimental work carried out as part of an architectural project for the design of emergency shelters, as an alternative to more usual habitats. We will then discuss the possibility of adapting NPD methodology to architectural design, and what potential this offers to improve the integration of user-related information within architectural creativity