61 research outputs found
Higher-dimensional categories with finite derivation type
We study convergent (terminating and confluent) presentations of
n-categories. Using the notion of polygraph (or computad), we introduce the
homotopical property of finite derivation type for n-categories, generalizing
the one introduced by Squier for word rewriting systems. We characterize this
property by using the notion of critical branching. In particular, we define
sufficient conditions for an n-category to have finite derivation type. Through
examples, we present several techniques based on derivations of 2-categories to
study convergent presentations by 3-polygraphs
Homological Computations for Term Rewriting Systems
An important problem in universal algebra consists in finding
presentations of algebraic theories by generators and relations, which
are as small as possible. Exhibiting lower bounds on the number of
those generators and relations for a given theory is a difficult task
because it a priori requires considering all possible sets of
generators for a theory and no general method exists. In this article,
we explain how homological computations can provide such lower bounds,
in a systematic way, and show how to actually compute those in the
case where a presentation of the theory by a convergent rewriting
system is known. We also introduce the notion of coherent presentation
of a theory in order to consider finer homotopical invariants. In some
aspects, this work generalizes, to term rewriting systems, Squier\u27s
celebrated homological and homotopical invariants for string rewriting
systems
Maurice Janet's algorithms on systems of linear partial differential equations
This article presents the emergence of formal methods in theory of partial differential equations (PDE) in the french school of mathematics through Janet's work in the period 1913-1930. In his thesis and in a series of articles published during this period, M. Janet introduced an original formal approach to deal with the solvability of the problem of initial conditions for finite linear PDE systems. His constructions implicitly used an interpretation of a monomial PDE system as a generating family of a multiplicative set of monomials. He introduced an algorithmic method on multiplicative sets to compute compatibility conditions, and to study the problem of the existence and the unicity of a solution to a linear PDE system with given initial conditions. The compatibility conditions are formulated using a refinement of the division operation on monomials defined with respect to a partition of the set of variables into multiplicative and non-multiplicative variables. M. Janet was a pioneer in the development of these algorithmic methods, and the completion procedure that he introduced on polynomials was the first one in a long and rich series of works on completion methods which appeared independently throughout the 20th century in various algebraic contexts
A Homotopical Completion Procedure with Applications to Coherence of Monoids
International audienceOne of the most used algorithm in rewriting theory is the Knuth-Bendix completion procedure which starts from a terminating rewriting system and iteratively adds rules to it, trying to produce an equivalent convergent rewriting system. It is in particular used to study presentations of monoids, since normal forms of the rewriting system provide canonical representatives of words modulo the congruence generated by the rules. Here, we are interested in extending this procedure in order to retrieve information about the low-dimensional homotopy properties of a monoid. We therefore consider the notion of coherent presentation, which is a generalization of rewriting systems that keeps track of the cells generated by confluence diagrams. We extend the Knuth-Bendix completion procedure to this setting, resulting in a homotopical completion procedure. It is based on a generalization of Tietze transformations, which are operations that can be iteratively applied to relate any two presentations of the same monoid. We also explain how these transformations can be used to remove useless generators, rules, or confluence diagrams in a coherent presentation, thus leading to a homotopical reduction procedure. Finally, we apply these techniques to the study of some examples coming from representation theory, to compute minimal coherent presentations for them: braid, plactic and Chinese monoids
Higher Catoids, Higher Quantales and their Correspondences
We establish modal correspondences between omega-catoids and convolution
omega-quantales. These are related to J\'onsson-Tarski style-dualities between
relational structures and lattices with operators. We introduce omega-catoids
as generalisations of (strict) omega-categories and in particular of the higher
path categories generated by polygraphs (or computads) in higher rewriting.
Convolution omega-quantales generalise the powerset omega-Kleene algebras
recently proposed for algebraic coherence proofs in higher rewriting to
weighted variants. We extend these correspondences to ({\omega},p)-catoids and
convolution ({\omega},p)-quantales suitable for modelling homotopies in higher
rewriting. We also specialise them to finitely decomposable ({\omega},
p)-catoids, an appropriate setting for defining ({\omega}, p)-semirings and
({\omega}, p)-Kleene algebras. These constructions support the systematic
development and justification of higher quantale axioms relative to a previous
ad hoc approach.Comment: 46 pages, 8 figure
Algebraic coherent confluence and higher-dimensional globular Kleene algebras
We extend the formalisation of confluence results in Kleene algebras to a
formalisation of coherent proofs by confluence. To this end, we introduce the
structure of modal higher-dimensional globular Kleene algebra, a
higher-dimensional generalisation of modal and concurrent Kleene algebra. We
give a calculation of a coherent Church-Rosser theorem and Newman's lemma in
higher-dimensional Kleene algebras. We interpret these results in the context
of higher-dimensional rewriting systems described by polygraphs.Comment: Pre-print (second version
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