The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices

Abstract In this paper, by using the ideal theory in residuated lattices, we construct the prime and maximal spectra (Zariski topology), proving that the prime and maximal spectra are compact topological spaces, and in the case of De Morgan residuated lattices they become compact {T}_{0} topological spaces. At the same time, we define and study the reticulation functor between De Morgan residuated lattices and bounded distributive lattices. Moreover, we study the I-topology (I comes from ideal) and the stable topology and we define the concept of pure ideal. We conclude that the I-topology is in fact the restriction of Zariski topology to the lattice of ideals, but we use it for simplicity. Finally, based on pure ideals, we define the normal De Morgan residuated lattice (L is normal iff every proper ideal of L is a pure ideal) and we offer some characterizations.

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On ideals of residuated lattices

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-202417 ◽

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.

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Relatively residuated lattices and posets

Mathematica Slovaca ◽

10.1515/ms-2017-0347 ◽

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.

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Rings which resemble rings of entire functions

Glasgow Mathematical Journal ◽

10.1017/s0017089500005000 ◽

1983 ◽

Vol 24 (1) ◽

pp. 7-16

Author(s):

David E. Rush

Keyword(s):

Riemann Surface ◽

Partially Ordered Set ◽

Riemann Surfaces ◽

Entire Functions ◽

Ordered Set ◽

Ideal Theory ◽

Prime Ideals ◽

Topological Spaces ◽

Valuation Theory ◽

The Ideal

Since Helmer's 1940 paper [9] laid the foundations for the study of the ideal theory of the ring A(ℂ) of entire functions, many interesting results have been obtained for the rings A(X) of analytic functions on non-compact connected Riemann surfaces. For example, the partially ordered set Spec (A(ℂ) of prime ideals of A(ℂ) has been described by Henrikson and others [2], [10], [11]. Also, it has been shown by Ailing [4] that Spec(A(ℂ))sSpec(A(X)) as topological spaces for any non-compact connected Riemann surface X. Many results on the valuation theory of A(X) have also been obtained [1], [2]. In this note we show that a large portion of the results on the rings A(X) extend to the W-rings with complete principal divisor space which were defined by J. Klingen in [15], [16]. Therefore, many properties of A(ℂ) are shared by its non-archimedian counterparts studied by M. Lazard, M. Krasner, and others [8], [17], [18].

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Ideal theory on EQ-algebras

AIMS Mathematics ◽

10.3934/math.2021679 ◽

2021 ◽

Vol 6 (11) ◽

pp. 11686-11707

Author(s):

Jie Qiong Shi ◽

◽

Xiao Long Xin ◽

Keyword(s):

Topological Space ◽

Maximal Ideal ◽

Ideal Theory ◽

Distributive Lattices ◽

Prime Ideals ◽

Topological Properties ◽

Maximal Ideals ◽

Primary Ideals ◽

The Ideal

<abstract><p>In this article, we introduce ideals and other special ideals on EQ-algebras, such as implicative ideals, primary ideals, prime ideals and maximal ideals. At first, we give the notion of ideal and its related properties on EQ-algebras, and give its equivalent characterizations. We discuss the relations between ideals and filters, and study the generating formula of ideals on EQ-algebras. Moreover, we study the properties of implicative ideals, primary ideals, prime ideals and maximal ideals and their relations. For example, we prove that every maximal ideal is prime and if prime ideals are implicative, then they are maximal in the EQ-algebra with the condition $ (DNP) $. Finally, we introduce the topological properties of prime ideals. We get that the set of all prime ideals is a compact $ T_{0} $ topological space. Also, we transferred the spectrum of EQ-algebras to bounded distributive lattices and given the ideal reticulation of EQ-algebras.</p></abstract>

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A Hesitant Intuitionistic Fuzzy Set Approach to Study Ideals of Semirings

International Journal of Fuzzy System Applications ◽

10.4018/ijfsa.2021070101 ◽

2021 ◽

Vol 10 (3) ◽

pp. 1-17

Author(s):

Debabrata Mandal

Keyword(s):

Set Theory ◽

Fuzzy Set ◽

Cartesian Product ◽

Intuitionistic Fuzzy Set ◽

Ideal Theory ◽

Hesitant Fuzzy Set ◽

Neutrosophic Set ◽

Intuitionistic Fuzzy ◽

Interval Valued ◽

The Ideal

The classical set theory was extended by the theory of fuzzy set and its several generalizations, for example, intuitionistic fuzzy set, interval valued fuzzy set, cubic set, hesitant fuzzy set, soft set, neutrosophic set, etc. In this paper, the author has combined the concepts of intuitionistic fuzzy set and hesitant fuzzy set to study the ideal theory of semirings. After the introduction and the priliminary of the paper, in Section 3, the author has defined hesitant intuitionistic fuzzy ideals and studied several properities of it using the basic operations intersection, homomorphism and cartesian product. In Section 4, the author has also defined hesitant intuitionistic fuzzy bi-ideals and hesitant intuitionistic fuzzy quasi-ideals of a semiring and used these to find some characterizations of regular semiring. In that section, the author also has discussed some inter-relations between hesitant intuitionistic fuzzy ideals, hesitant intuitionistic fuzzy bi-ideals and hesitant intuitionistic fuzzy quasi-ideals, and obtained some of their related properties.

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The Foundations of the Ideal Theory of Zolotarev

American Mathematical Monthly ◽

10.2307/2299413 ◽

1930 ◽

Vol 37 (3) ◽

pp. 117 ◽

Cited By ~ 1

Author(s):

N. Tchebotarev

Keyword(s):

Ideal Theory ◽

The Ideal

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Perfect residuated lattice ordered monoids

Mathematica Slovaca ◽

10.2478/s12175-010-0050-6 ◽

2010 ◽

Vol 60 (6) ◽

Cited By ~ 1

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).

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Essential extensions of filters in residuated lattices

10.21203/rs.3.rs-185624/v1 ◽

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.

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Left residuated lattices induced by lattices with a unary operation

Soft Computing ◽

10.1007/s00500-019-04461-x ◽

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.

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