2,532 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
On Drinfel'd associators
In 1986, in order to study the linear representations of the braid group
coming from the monodromy of the Knizhnik-Zamolodchikov differential
equations,Drinfel'd introduced a class of formal power series 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
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