The lattices of 𝔏-fuzzy state filters in state residuated lattices

2020 ◽  
Vol 70 (6) ◽  
pp. 1289-1306
Author(s):  
Pengfei He ◽  
Juntao Wang ◽  
Jiang Yang
Keyword(s):  
Boolean Algebra ◽  
Fuzzy Set ◽  
Heyting Algebra ◽  
Stable State ◽  
Residuated Lattices ◽  
Complete Boolean Algebra ◽  
Lattice Structures ◽  
Algebra 2 ◽  
Fuzzy State

AbstractIn the paper, we introduce 𝔏-fuzzy state filters in state residuated lattices and investigate their related properties, where 𝔏 is a complete Heyting algebra. Moreover, we study the 𝔏-fuzzy state co-annihilator of an 𝔏-fuzzy set with respect to an 𝔏-fuzzy state filter. Finally, using the 𝔏-fuzzy state co-annihilator, we investigate lattice structures of the set of some types of 𝔏-fuzzy state filters in state residuated lattices. In particular, we prove that: (1) the set FSF[L] of all 𝔏-fuzzy state filters is a complete Heyting algebra; (2) the set SνFSF[L] of all stable state filters relative to an 𝔏-fuzzy set ν is also a complete Heyting algebra; (3) the set IμFSF[L] of all involutory 𝔏-fuzzy state filters relative to an 𝔏-fuzzy state filter μ is a complete Boolean algebra.

Fuzzy Quantum Space

2017 ◽  
pp. 84-114
Author(s):  
Renáta Bartková ◽  
Beloslav Riečan ◽  
Anna Tirpáková
Keyword(s):  
Probability Theory ◽  
Boolean Algebra ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Probability Space ◽  
Quantum Space ◽  
Fuzzy State ◽  
Fuzzy Observable

In this chapter we study the existence of a sum of fuzzy observables in a fuzzy quantum space which generalizes the Kolmogorov probability space using the ideas of fuzzy set theory. We also study some properties of the sum of fuzzy observables. To study the above mentioned, we also include the basic notions from the probability theory on fuzzy quantum space in this chapter, i.e. the notion of fuzzy quantum space, a fuzzy observable, an indicator of a fuzzy set, a null fuzzy observable, a Boolean algebra on fuzzy quantum space, fuzzy state etc.

M-structure in Banach spaces

1967 ◽  
Vol 63 (3) ◽  
pp. 613-629 ◽  
Author(s):  
F. Cunningham
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Banach Spaces ◽  
Measure Space ◽  
Complete Boolean Algebra ◽  
Ordinary Sense ◽  
One Dimensional ◽  
Algebra 2 ◽  
Image Position ◽  
Vector Valued

L-structure in a Banach space X was defined in (3) by L-projections, that is projections P satisfyingfor all x ∈ X. The significance of L-structure is shown by the following facts: (1) All L-projections on X commute and together form a complete Boolean algebra. (2) X can be isometrically represented as a vector-valued L1 on a measure space constructed from the Boolean algebra of its L-projections (2). (3) L1-spaces in the ordinary sense are characterized among Banach spaces by properties equivalent to having so many L-projections that the representation in (2) is everywhere one-dimensional.

scholarly journals When do L-fuzzy ideals of a ring generate a distributive lattice?

2016 ◽  
Vol 14 (1) ◽  
pp. 531-542
Author(s):  
Ninghua Gao ◽  
Qingguo Li ◽  
Zhaowen Li
Keyword(s):  
Distributive Lattice ◽  
Heyting Algebra ◽  
Lattice Structures ◽  
Boolean Ring ◽  
Fuzzy Subset ◽  
The Family ◽  
Essential Properties ◽  
Fuzzy Ideal

AbstractThe notion of L-fuzzy extended ideals is introduced in a Boolean ring, and their essential properties are investigated. We also build the relation between an L-fuzzy ideal and the class of its L-fuzzy extended ideals. By defining an operator “⇝” between two arbitrary L-fuzzy ideals in terms of L-fuzzy extended ideals, the result that “the family of all L-fuzzy ideals in a Boolean ring is a complete Heyting algebra” is immediately obtained. Furthermore, the lattice structures of L-fuzzy extended ideals of an L-fuzzy ideal, L-fuzzy extended ideals relative to an L-fuzzy subset, L-fuzzy stable ideals relative to an L-fuzzy subset and their connections are studied in this paper.

scholarly journals Essential extensions of filters in residuated lattices

2021 ◽  
Author(s):  
Masoud Haveshki
Keyword(s):  
Boolean Algebra ◽  
Residuated Lattice ◽  
Residuated Lattices ◽  
Essential Extension ◽  
Mv Algebra ◽  
Essential Extensions

Abstract We define the essential extension of a filter in the residuated lattice A associated to an ideal of L(A) and investigate its related properties. We prove the residuated lattice A is a Boolean algebra, G(RL)-algebra or MV -algebra if and only if the essential extension of {1} associated to A \ P is a Boolean filter, G-filter or MV -filter (for all P ∈ SpecA), respectively. Also, some properties of lattice of essential extensions are studied.

scholarly journals Refined Neutrosophy and Lattices vs. Pair Structures and YinYang Bipolar Fuzzy Set

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 353 ◽  
Author(s):  
Florentin Smarandache
Keyword(s):  
Fuzzy Set ◽  
Infinite Number ◽  
Neutrosophic Set ◽  
Lattice Structures

In this paper, we present the lattice structures of neutrosophic theories. We prove that Zhang-Zhang’s YinYang bipolar fuzzy set is a subclass of the Single-Valued bipolar neutrosophic set. Then we show that the pair structure is a particular case of refined neutrosophy, and the number of types of neutralities (sub-indeterminacies) may be any finite or infinite number.

Power-collapsing games

2008 ◽  
Vol 73 (4) ◽  
pp. 1433-1457 ◽  
Author(s):  
Miloš S. Kurilić ◽  
Boris Šobot
Keyword(s):  
Boolean Algebra ◽  
Dense Subset ◽  
Winning Strategy ◽  
Zero Element ◽  
The Other ◽  
Complete Boolean Algebra ◽  
Infinite Cardinal ◽  
Other Hand ◽  
Game Theoretic

AbstractThe game is played on a complete Boolean algebra , by two players. White and Black, in κ-many moves (where κ is an infinite cardinal). At the beginning White chooses a non-zero element p ∈ . In the α-th move White chooses pα ∈ (0, p) and Black responds choosing iα ∈{0, 1}. White winsthe play iff . where and .The corresponding game theoretic properties of c.B.a.'s are investigated. So, Black has a winning strategy (w.s.) if κ ≥ π() or if contains a κ-closed dense subset. On the other hand, if White has a w.s., then κ ∈ . The existence of w.s. is characterized in a combinatorial way and in terms of forcing. In particular, if 2<κ = κ ∈ Reg and forcing by preserves the regularity of κ, then White has a w.s. iff the power 2κ is collapsed to κ in some extension. It is shown that, under the GCH, for each set S ⊆ Reg there is a c.B.a. such that White (respectively. Black) has a w.s. for each infinite cardinal κ ∈ S (resp. κ ∉ S). Also it is shown consistent that for each κ ∈ Reg there is a c.B.a. on which the game is undetermined.

scholarly journals A Complete Boolean Algebra That Has No Proper Atomless Complete Subalgebra

1996 ◽  
Vol 182 (3) ◽  
pp. 748-755 ◽  
Author(s):  
Thomas Jech ◽  
Saharon Shelah
Keyword(s):  
Boolean Algebra ◽  
Complete Boolean Algebra
Sign in / Sign up

Export Citation Format

Share Document