scholarly journals Galois connection of stabilizers in residuated lattices

Filomat ◽  
2020 ◽  
Vol 34 (4) ◽  
pp. 1223-1239
Author(s):  
Saeed Rasouli
Keyword(s):  
Complete Lattice ◽  
Residuated Lattice ◽  
Galois Connection ◽  
Residuated Lattices ◽  
Pseudocomplemented Lattices

The paper is devoted to introduce the notions of some types of stabilizers in non-commutative residuated lattices and to investigate their properties. We establish a connection between (contravariant) Galois connection and stabilizers of a residuated lattices. If A is a residuated lattice and F be a filter of A, we show that the set of all stabilizers relative to F of a same type forms a complete lattice. Furthermore, we prove that ST - F?l, ST - Fl and ST - Fs are pseudocomplemented lattices.

INTUITIONISTIC FUZZY FILTERS OF RESIDUATED LATTICES

2011 ◽  
Vol 07 (03) ◽  
pp. 499-513 ◽  
Author(s):  
SHOKOOFEH GHORBANI
Keyword(s):  
Fuzzy Sets ◽  
Complete Lattice ◽  
Residuated Lattice ◽  
Intuitionistic Fuzzy Sets ◽  
Residuated Lattices ◽  
Intuitionistic Fuzzy ◽  
Correspondence Theorem ◽  
Fuzzy Filters

In this paper, the concept of intuitionistic fuzzy sets is applied to residuated lattices. The notion of intuitionistic fuzzy filters of a residuated lattice is introduced and some related properties are investigated. The characterizations of intuitionistic fuzzy filters are obtained. We show that the set of all the intuitionistic fuzzy filters of a residuated lattice forms a complete lattice and we find the distributive sublattices of it. Finally, the correspondence theorem for intuitionistic fuzzy filters is established.

scholarly journals Sectionally Pseudocomplemented Residual Lattice

1970 ◽  
Vol 6 (2) ◽  
pp. 53-54
Author(s):  
MD Zaidur Rahman ◽  
Md Abdul Kalam Azad ◽  
Md Nazmul Hasan
Keyword(s):  
Key Words ◽  
Basic Concept ◽  
Residuated Lattice ◽  
Residuated Lattices ◽  
Commutative Monoid ◽  
Bounded Lattice ◽  
Complemented Lattices ◽  
Pseudocomplemented Lattices

At first, we recall the basic concept, By a residual lattice is meant an algebra L = (L,∨,∧,∗,o,0,1) such that (i) L = (L,∨,∧,0,1) is a bounded lattice, (ii) L = (L,∗,1) is a commutative monoid, (iii) it satisfies the so-called adjoin ness property: (x ∨ y) ∗ z = y if and only if y ≤ z ≤ x o y Let us note [7] that x ∨ y is the greatest element of the set (x ∨ y) ∗ z = y Moreover, if we consider x ∗ y = x ∧ y , then x o y is the relative pseudo-complement of x with respect to y, i. e., for ∗ = ∧ residuated lattices are just relatively pseudo-complemented lattices. The identities characterizing sectionally pseudocomplemented lattices are presented in [3] i.e. the class of these lattices is a variety in the signature {∨,∧,o,1}. We are going to apply a similar approach for the adjointness property: Key words: Residuated lattice; non Distributive; Residuated Abeliean; commutative monoid: DOI: http://dx.doi.org/10.3329/diujst.v6i2.9345 DIUJST 2011; 6(2): 53-54

scholarly journals Relatively residuated lattices and posets

2020 ◽  
Vol 70 (2) ◽  
pp. 239-250
Author(s):  
Ivan Chajda ◽  
Jan Kühr ◽  
Helmut Länger
Keyword(s):  
Residuated Lattice ◽  
Unary Operation ◽  
General Concept ◽  
Residuated Lattices ◽  
Distributive Lattices ◽  
Pseudocomplemented Lattice ◽  
Pseudocomplemented Lattices

Abstract It is known that every relatively pseudocomplemented lattice is residuated and, moreover, it is distributive. Unfortunately, non-distributive lattices with a unary operation satisfying properties similar to relative pseudocomplementation cannot be converted in residuated ones. The aim of our paper is to introduce a more general concept of a relatively residuated lattice in such a way that also non-modular sectionally pseudocomplemented lattices are included. We derive several properties of relatively residuated lattices which are similar to those known for residuated ones and extend our results to posets.

scholarly journals Vague Filters of Residuated Lattices

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Shokoofeh Ghorbani
Keyword(s):  
Complete Lattice ◽  
Residuated Lattice ◽  
Residuated Lattices

Notions of vague filters, subpositive implicative vague filters, and Boolean vague filters of a residuated lattice are introduced and some related properties are investigated. The characterizations of (subpositive implicative, Boolean) vague filters is obtained. We prove that the set of all vague filters of a residuated lattice forms a complete lattice and we find its distributive sublattices. The relation among subpositive implicative vague filters and Boolean vague filters are obtained and it is proved that subpositive implicative vague filters are equivalent to Boolean vague filters.

Perfect residuated lattice ordered monoids

2010 ◽  
Vol 60 (6) ◽  
Author(s):  
Jiří Rachůnek ◽  
Dana Šalounová
Keyword(s):  
Probability Measure ◽  
Residuated Lattice ◽  
Residuated Lattices ◽  
Heyting Algebras ◽  
Effect Algebras ◽  
Quantum Logics ◽  
Mv Algebras ◽  
Intuitionistic Logics

AbstractBounded Rℓ-monoids form a large subclass of the class of residuated lattices which contains certain of algebras of fuzzy and intuitionistic logics, such as GMV-algebras (= pseudo-MV-algebras), pseudo-BL-algebras and Heyting algebras. Moreover, GMV-algebras and pseudo-BL-algebras can be recognized as special kinds of pseudo-MV-effect algebras and pseudo-weak MV-effect algebras, i.e., as algebras of some quantum logics. In the paper, bipartite, local and perfect Rℓ-monoids are investigated and it is shown that every good perfect Rℓ-monoid has a state (= an analogue of probability measure).

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 Left residuated lattices induced by lattices with a unary operation

2019 ◽  
Vol 24 (2) ◽  
pp. 723-729
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Orthomodular Lattice ◽  
Residuated Lattice ◽  
Unary Operation ◽  
Residuated Lattices ◽  
Boolean Algebras ◽  
Natural Question ◽  
Similar Technique ◽  
Orthomodular Lattices

Abstract In a previous paper, the authors defined two binary term operations in orthomodular lattices such that an orthomodular lattice can be organized by means of them into a left residuated lattice. It is a natural question if these operations serve in this way also for more general lattices than the orthomodular ones. In our present paper, we involve two conditions formulated as simple identities in two variables under which this is really the case. Hence, we obtain a variety of lattices with a unary operation which contains exactly those lattices with a unary operation which can be converted into a left residuated lattice by use of the above mentioned operations. It turns out that every lattice in this variety is in fact a bounded one and the unary operation is a complementation. Finally, we use a similar technique by using simpler terms and identities motivated by Boolean algebras.

State hyperstructures of tree automata based on lattice-valued logic

2018 ◽  
Vol 52 (1) ◽  
pp. 23-42 ◽  
Author(s):  
Maryam Ghorani
Keyword(s):  
Residuated Lattice ◽  
Residuated Lattices ◽  
The Other ◽  
Equivalence Relations ◽  
Tree Automaton ◽  
Tree Automata ◽  
Other Hand ◽  
Complete Residuated Lattice ◽  
The Creation ◽  
The Given

In this paper, an association is organized between the theory of tree automata on one hand and the hyperstructures on the other hand, over complete residuated lattices. To this end, the concept of order of the states of a complete residuated lattice-valued tree automaton (simply L-valued tree automaton) is introduced along with several equivalence relations in the set of the states of an L-valued tree automaton. We obtain two main results from this study: one of the relations can lead to the creation of Kleene’s theorem for L-valued tree automata, and the other one leads to the creation of a minimal v-valued tree automaton that accepts the same language as the given one.

On ideals of residuated lattices

2021 ◽  
pp. 1-11
Author(s):  
Yan Yan Dong ◽  
Jun Tao Wang
Keyword(s):  
Residuated Lattice ◽  
Lattice Structure ◽  
Residuated Lattices ◽  
The Ideal

In this paper, we first point out some mistakes in [12]. Especially the Theorem 3.9 [12] showed that: Let A be residuated lattice and ∅ ≠ X ⊆ A, then the least ideal containing X can be expressed as: 〈X〉 = {a ∈ A|a ≤ (·· · ((x1 ⊕ x2) ⊕ x3) ⊕ ·· ·) ⊕ xn, xi ∈ X, i = 1, 2 ·· · , n}. But we present an example to illustrate the ideal generation formula may not hold on residuated lattices. Further we give the correct ideal generation formula on residuated lattices. Moreover, we extend the concepts of annihilators and α-ideals to MTL-algebras and focus on studying the relations between them. Furthermore, we show that the set Iα (M) of all α-ideals on a linear MTL-algebra M only contains two trivial α-ideals {0} and M. However, the authors [24] studied the structure of Iα (M) in a linear BL-algebra M, which means some results with respect to Iα (M) given in [24] are trivial. Unlike that, we investigate the lattice structure of Iα (M) on general MTL-algebras.

scholarly journals The Relations between Residuated Frames and Residuated Connections

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 295
Author(s):  
Yong Chan Kim ◽  
Ju-Mok Oh
Keyword(s):  
Fuzzy Logic ◽  
Complete Lattice ◽  
Rough Sets ◽  
Residuated Lattice ◽  
Macneille Completion ◽  
Relational Semantics ◽  
Fuzzy Rough Sets ◽  
Complete Residuated Lattice

We introduce the notion of (dual) residuated frames as a viewpoint of relational semantics for a fuzzy logic. We investigate the relations between (dual) residuated frames and (dual) residuated connections as a topological viewpoint of fuzzy rough sets in a complete residuated lattice. As a result, we show that the Alexandrov topology induced by fuzzy posets is a fuzzy complete lattice with residuated connections. From this result, we obtain fuzzy rough sets on the Alexandrov topology. Moreover, as a generalization of the Dedekind–MacNeille completion, we introduce R-R (resp. D R - D R ) embedding maps and R-R (resp. D R - D R ) frame embedding maps.

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