3,116 research outputs found
Syntactic Monoids in a Category
The syntactic monoid of a language is generalized to the level of a symmetric
monoidal closed category D. This allows for a uniform treatment of several
notions of syntactic algebras known in the literature, including the syntactic
monoids of Rabin and Scott (D = sets), the syntactic semirings of Polak (D =
semilattices), and the syntactic associative algebras of Reutenauer (D = vector
spaces). Assuming that D is an entropic variety of algebras, we prove that the
syntactic D-monoid of a language L can be constructed as a quotient of a free
D-monoid modulo the syntactic congruence of L, and that it is isomorphic to the
transition D-monoid of the minimal automaton for L in D. Furthermore, in case
the variety D is locally finite, we characterize the regular languages as
precisely the languages with finite syntactic D-monoids
Gr\"obner-Shirshov bases for categories
In this paper we establish Composition-Diamond lemma for small categories. We
give Gr\"obner-Shirshov bases for simplicial category and cyclic category.Comment: 20 page
Manin products, Koszul duality, Loday algebras and Deligne conjecture
In this article we give a conceptual definition of Manin products in any
category endowed with two coherent monoidal products. This construction can be
applied to associative algebras, non-symmetric operads, operads, colored
operads, and properads presented by generators and relations. These two
products, called black and white, are dual to each other under Koszul duality
functor. We study their properties and compute several examples of black and
white products for operads. These products allow us to define natural
operations on the chain complex defining cohomology theories. With these
operations, we are able to prove that Deligne's conjecture holds for a general
class of operads and is not specific to the case of associative algebras.
Finally, we prove generalized versions of a few conjectures raised by M. Aguiar
and J.-L. Loday related to the Koszul property of operads defined by black
products. These operads provide infinitely many examples for this generalized
Deligne's conjecture.Comment: Final version, a few references adde
Constructions of free commutative integro-differential algebras
In this survey, we outline two recent constructions of free commutative
integro-differential algebras. They are based on the construction of free
commutative Rota-Baxter algebras by mixable shuffles. The first is by
evaluations. The second is by the method of Gr\"obner-Shirshov bases.Comment: arXiv admin note: substantial text overlap with arXiv:1302.004
Algebraic geometry over algebraic structures II: Foundations
In this paper we introduce elements of algebraic geometry over an arbitrary
algebraic structure. We prove Unification Theorems which gather the description
of coordinate algebras by several ways.Comment: 55 page
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