Autonomization of Monoidal Categories

We define the free autonomous category generated by a monoidal category and study some of its properties. From a linguistic perspective, this expands the range of possible models of meaning within the distributional compositional framework, by allowing nonlinearities in maps. From a categorical point of view, this provides a factorization of the construction in [Preller and Lambek, 2007] of the free autonomous category generated by a category. ; Comment: Under review. Comments welcome!

Links between Probabilistic Convergence Groups Under Triangular Norms and Enriched Lattice-Valued Convergence Groups

We propose here two types of probabilistic convergence groups under triangular norms; present some basic facts, and give some characterizations for both the cases. We look at the possible link from categorical point of view between each of the proposed type and enriched lattice-valued convergence group. We produce several natural examples on probabilistic convergence groups under triangular norms. We also present a notion of probabilistic uniform convergence structure in a new perspective, showing that every probabilistic convergence group is probabilistic uniformizable. Moreover, we prove that this probabilistic uniform structure maintains a close connection with a known enriched lattice-valued uniform convergence structure.

The Category of Bratteli Diagrams

A category structure for Bratteli diagrams is proposed and a functor from the category of AF algebras to the category of Bratteli diagrams is constructed. Since isomorphism of Bratteli diagrams in this category coincides with Bratteli’s notion of equivalence, we obtain in particular a functorial formulation of Bratteli’s classification of AF algebras (and at the same time, of Glimm’s classification of UHF algebras). It is shown that the three approaches to classification of AF algebras, namely, through Bratteli diagrams, K-theory, and a certain natural abstract classifying category, are essentially the same from a categorical point of view.

Categories of(I,I)-Fuzzy Greedoids

The concepts ofI-greedoids, fuzzifying greedoids, and (I,I)-fuzzy greedoids are introduced and feasibility preserving mappings between greedoids are defined. ThenI-feasibility preserving mappings, fuzzifying feasibility preserving mappings, and (I,I)-fuzzy feasibility preserving mappings are given as generalizations of feasibility preserving mappings. We study the relations among greedoids,I-greedoids, fuzzifying greedoids, and (I,I)-fuzzy greedoids from a categorical point of view.