D-posets

Chovanec, Ferdinand; Kōpka, František

doi:10.1016/b978-044452870-4/50031-5

Cited by 40 publications

(44 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An MV-effect algebra is a lattice ordered effect algebra M in which, for all a, b ∈ M , (a ∨ b) a = b (a ∧ b). It is proved in [4] that there is a natural, one-to one correspondence between MV-effect algebras and MV-algebras given by the following rules. Let (M, ⊕, 0, 1) be an MV-effect algebra.…”

Section: Definitions and Basic Relationshipsmentioning

confidence: 99%

See 1 more Smart Citation

The category of MV-pairs

Nola

Holčapek

Jenča³

2009

Logic Journal of IGPL

View full text Add to dashboard Cite

Abstract. An MV-pair is a pair (B, G), where B is a Boolean algebra and G is a subgroup of the automorphism group of B satisfying certain condition. Recently it was proved by one of the authors that for an MV-pair (B, G), ∼ G is an effect-algebraic congruence and B/ ∼ G is an MV-algebra. Moreover, every MV-algebra M can be represented by an MV-pair in this way.In this paper we show that one can define a suitable category of MV-pairs in such a way that there exist a faithful functor from the category of MV-algebras to the aforementioned category and a functor in the reversed direction.

show abstract

Section: Definitions and Basic Relationshipsmentioning

confidence: 99%

“…Three of them are given in the following proposition. [4] Let E be a lattice ordered effect algebra. The following are equivalent (a) E is an MV-effect algebra.…”

Section: Definitions and Basic Relationshipsmentioning

confidence: 99%

The category of MV-pairs

Nola

Holčapek

Jenča³

2009

Logic Journal of IGPL

View full text Add to dashboard Cite

show abstract

“…Following J. N o vá k [22]- [24], using some recent results on generalized random events and states [2], [3], [20], [21], [30] and basic categorical methods [1], [15], in [19] we have studied the process of extending generalized probability measures.…”

Section: Introductionmentioning

confidence: 99%

“…D-posets have been introduced in [20] in order to model events in quantum probability. They generalize Boolean algebras, MV-algebras and other probability domains, and provide a category in which observables and states become morphisms [2], [11]. Recall that a D-poset is a partially ordered set with the greatest element 1 X , the least element 0 X , and a partial binary operation called difference, such that a b is defined iff b ≤ a, and the following axioms are assumed: [16], [17], i.e., systems X ⊆ I X carrying the coordinatewise partial order, coordinatewise convergence of sequences, containing the top and bottom elements of I X , and closed with respect to the partial operation difference defined coordinatewise.…”

Section: Introductionmentioning

confidence: 99%

Real functions and the extension of generalized probability measures II

Havlíčková

2015

Tatra Mountains Mathematical Publications

View full text Add to dashboard Cite

ABSTRACT. We continue our study of the extensions of generalized probability measures. First, we describe some extensions of generalized random events (represented by classes of functions with values in [0,1]) to which generalized probability measures can be extended. Second, we study products of domains of probability and describe states on such products. Third, we show that the events in IF-probability, introduced by B. Riečan, form a suitable category isomorphic to a subcategory of the category of fuzzy random events. Consequently, IF-probability can be interpreted within fuzzy probability theory. We put forward some problems related to the extensions of probability domains and hint some applications.

show abstract

“…In this context it should be noted that there exist much more general, categorical and fuzzy (M V -algebras) approaches, in which the notion of D-posets, introduced by F. Kôpka and F. Chovanec (see [30], [32], [8], [31]) on the base of the partial operation of difference, plays an important role as a model in quantum probability theory. The notion was intensively investigated by R. Frič and M. Papčo (see [16]- [20], [33]- [35]).…”

Section: Introductionmentioning

confidence: 99%

On duality between order and algebraic structures in Boolean systems

Dadej¹,

Halik²

2013

JMA

View full text Add to dashboard Cite

Abstract:We present an extension of the known one-to-one correspondence between Boolean algebras and Boolean rings with unit being two types of Boolean systems endowed with order and algebraic structures, respectively. Two equivalent generalizations of Boolean algebras are discussed. We show that there is a one-to-one correspondence between any of the two mentioned generalized Boolean algebras and Boolean rings without unit.

show abstract

D-posets

Cited by 40 publications

References 39 publications

The category of MV-pairs

The category of MV-pairs

Real functions and the extension of generalized probability measures II

On duality between order and algebraic structures in Boolean systems

Contact Info

Product

Resources

About