3,360 research outputs found
Semiring and semimodule issues in MV-algebras
In this paper we propose a semiring-theoretic approach to MV-algebras based
on the connection between such algebras and idempotent semirings - such an
approach naturally imposing the introduction and study of a suitable
corresponding class of semimodules, called MV-semimodules.
We present several results addressed toward a semiring theory for
MV-algebras. In particular we show a representation of MV-algebras as a
subsemiring of the endomorphism semiring of a semilattice, the construction of
the Grothendieck group of a semiring and its functorial nature, and the effect
of Mundici categorical equivalence between MV-algebras and lattice-ordered
Abelian groups with a distinguished strong order unit upon the relationship
between MV-semimodules and semimodules over idempotent semifields.Comment: This version contains some corrections to some results at the end of
Section
Homomorphisms of Gray-categories as pseudo algebras
Given Gray-categories and , there is a Gray-category
of locally strict trihomomorphisms with
domain and codomain , tritransformations, trimodifications, and
perturbations. If the domain is small and the codomain is cocomplete,
we show that this Gray-category is isomorphic as a Gray-category to the
Gray-category -- of pseudo algebras, pseudo
functors, transformations, and modifications for a Gray-monad derived from
left Kan extension.
Inspired by a similar situation in two-dimensional monad theory, we apply the
coherence theory of three-dimensional monad theory and prove that the the
inclusion of the functor category in the enriched sense into this Gray-category
of locally strict trihomomorphisms has a left adjoint such that the components
of the unit of the adjunction are internal biequivalences. This proves that any
locally strict trihomomorphism between Gray-categories with small domain and
cocomplete codomain is biequivalent to a Gray-functor. Moreover, the hom
Gray-adjunction gives an isomorphism of the hom 2-categories of
tritransformations between a locally strict trihomomorphism and a Gray-functor
with the corresponding hom 2-categories in the functor Gray-category. A notable
example is given by locally strict Gray-valued presheafs with small domain. Our
results have applications in three-dimensional descent theory and point into
the direction of a Yoneda lemma for tricategories.Comment: 62 page
- …