A note on nuclei of quantale modules
Categorical structures enriched in a quantaloid: categories, distributors and functors.
Categorical structures enriched in a quantaloid : regular presheaves, regular semicategories
Categorical structures enriched in a quantaloid: tensored and cotensored categories.
Covariant presheaves and subalgebras.
Idempotent versions of Haar’s Lemma: links between comparison of discrete event systems with different state spaces and control
Haar's Lemma (1918) deals with the algebraic characterization of the inclusion of polyhedral sets. This Lemma has been involved many times in automatic control of linear dynamical systems via positive invariance of polyhedrons. More recently, it has been used to characterize stochastic comparison w.r.t. linear/integral ordering of Markov (reward) chains. In this paper we develop a state space oriented approach to the control of Discrete Event Systems (DES) based on the remark that most of control...
Inversions on complete lattices and Girard quantales
On derivations of quantales
A quantale is a complete lattice equipped with an associative binary multiplication distributing over arbitrary joins. We define the notions of right (left, two) sided derivation and idempotent derivation and investigate the properties of them. It’s well known that quantic nucleus and quantic conucleus play important roles in a quantale. In this paper, the relationships between derivation and quantic nucleus (conucleus) are studied via introducing the concept of pre-derivation.
On fuzzification of the notion of quantaloid
The paper considers a fuzzification of the notion of quantaloid of K. I. Rosenthal, which replaces enrichment in the category of -semilattices with that in the category of modules over a given unital commutative quantale. The resulting structures are called quantale algebroids. We show that their constitute a monadic category and prove a representation theorem for them using the notion of nucleus adjusted for our needs. We also characterize the lattice of nuclei on a free quantale algebroid. At...
On monadic quantale algebras: basic properties and representation theorems
Motivated by the concept of quantifier (in the sense of P. Halmos) on different algebraic structures (Boolean algebras, Heyting algebras, MV-algebras, orthomodular lattices, bounded distributive lattices) and the resulting notion of monadic algebra, the paper introduces the concept of a monadic quantale algebra, considers its properties and provides several representation theorems for the new structures.
On the -valued categories of --ordered sets
The aim of this paper is to construct an -valued category whose objects are --ordered sets. To reach the goal, first, we construct a category whose objects are --ordered sets and morphisms are order-preserving mappings (in a fuzzy sense). For the morphisms of the category we define the degree to which each morphism is an order-preserving mapping and as a result we obtain an -valued category. Further we investigate the properties of this category, namely, we observe some special objects, special...
Quantum B-algebras
The concept of quantale was created in 1984 to develop a framework for non-commutative spaces and quantum mechanics with a view toward non-commutative logic. The logic of quantales and its algebraic semantics manifests itself in a class of partially ordered algebras with a pair of implicational operations recently introduced as quantum B-algebras. Implicational algebras like pseudo-effect algebras, generalized BL- or MV-algebras, partially ordered groups, pseudo-BCK algebras, residuated posets,...
Relationship of algebraic theories to powerset theories and fuzzy topological theories for lattice-valued mathematics.
Sheaves for an involutive quantaloid
Symmetry and Cauchy completion of quantaloid-enriched categories.