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QUANTALE

Quantale

Algebraic structure

In mathematics, quantales are certain partially ordered algebraic structures that generalize locales (point free topologies) as well as various multiplicative

Quantale

1000 (number)

59 into prime parts 1851 = sum of the first 32 primes 1852 = number of quantales on 5 elements, up to isomorphism 1853 = sum of primitive roots of 27-th

1000 (number)

1000_(number)

Monoid (category theory)

Mathematical concept in category theory

(with the monoidal structure induced by the Cartesian product) is a unital quantale. A monoid object in (Ab, ⊗Z, Z), the category of abelian groups, is a ring

Monoid (category theory)

Monoid_(category_theory)

Continuous function

Mathematical function with no sudden changes

generalization of metric spaces and posets, which uses the concept of quantales, and that can be used to unify the notions of metric spaces and domains

Continuous function

Continuous_function

Ideal (ring theory)

Submodule of a mathematical ring

join), and product make the set of ideals of a commutative ring into a quantale. If a , b {\displaystyle {\mathfrak {a}},{\mathfrak {b}}} are ideals of

Ideal (ring theory)

Ideal_(ring_theory)

List of order theory topics

(structure) Boolean ring Complete Boolean algebra Orthocomplemented lattice Quantale Partially ordered monoid Ordered group Archimedean property Ordered ring

List of order theory topics

List_of_order_theory_topics

Bisimulation

Relation between transition systems in computer science

associated treatment of bisimilarity, can be interpreted in any involutive quantale. Bisimilarity can also be defined in order-theoretical fashion, in terms

Bisimulation

Converse relation

Reversal of the order of elements of a binary relation

structure (with inclusion of relations as sets), and actually an involutive quantale. Similarly, the category of heterogeneous relations, Rel is also an ordered

Converse relation

Converse_relation

Fuzzy set

Sets whose elements have degrees of membership

1987, pp. 325–332 Gylys, R. P. (1994) Quantal sets and sheaves over quantales, Liet. Matem. Rink., v. 34, No. 1, pp. 9–31. Ulrich Höhle, Stephen Ernest

Fuzzy set

Fuzzy_set

Linear logic

System of resource-aware logic

mathematical objects. The algebraic semantics of linear logic is that of quantales.[citation needed] In linguistics, linear logic models grammatical parsing

Linear logic

Linear_logic

Quantaloid

\mathrm {Hom} (X,X)} of any object X {\displaystyle X} in a quantaloid is a quantale, whence the name. Rosenthal, Kimmo I. (1996), The theory of quantaloids

Quantaloid

Kripke semantics

Formal semantics for non-classical logic systems

Robert (2006b). "A Kripke-Joyal Semantics for Noncommutative Logic in Quantales" (PDF). In Governatori, G.; Hodkinson, I.; Venema, Y. (eds.). Advances

Kripke semantics

Kripke_semantics

Semiring

Algebraic ring that need not have additive negative elements

matrix multiplication corresponds to composition of relations. Any unital quantale is a semiring under join and multiplication. A normal skew lattice in a

Semiring

Order theory

Branch of mathematics

combine orders with even more algebraic operations, as in the case of quantales, that allow for the definition of an addition operation. Many other important

Order theory

Order_theory

Relation algebra

Type of residuated Boolean algebra with extra structure

logic Involution Logic of relatives Logical matrix Predicate functor logic Quantale Relation Relational calculus Relational algebra Residuated Boolean algebra

Relation algebra

Relation_algebra

Residuated lattice

In mathematics, an algebraic structure

that are also Kleene algebras, for which ∧ is ordinarily not required. Quantale Residuated mapping Substructural logic Residuated Boolean algebra Ward

Residuated lattice

Residuated_lattice

André Joyal

Canadian mathematician

Robert Goldblatt, A Kripke-Joyal semantics for noncommutative logic in quantales; Advances in Modal Logic 6, 209—225, Coll. Publ., London, 2006; MR 2396933

André Joyal

André_Joyal

Category of preordered sets

Concept in set theory

Javier; Höhle, Ulrich; Kortelainen, Jari (2018). Semigroups in Complete Lattices: Quantales, Modules and Related Topics. Springer. ISBN 978-3319789484.

Category of preordered sets

Category_of_preordered_sets

Noncommutative logic

Extension of linear logic

2307/2310058. ISSN 0002-9890. JSTOR 2310058. Yetter, David N. (1990). "Quantales and (Noncommutative) Linear Logic". The Journal of Symbolic Logic. 55

Noncommutative logic

Noncommutative_logic

Jiří Rosický (mathematician)

Czech mathematician

categories. He was active in several other topics, ranging from the theory of quantales to many aspects of general category theory: he introduced tangent categories

Jiří Rosický (mathematician)

Jiří_Rosický_(mathematician)