scholarly journals Loomis-sikorski theorem for σ-complete MV-algebras and ℓ-groups

2000 ◽  
Vol 68 (2) ◽  
pp. 261-277 ◽  
Author(s):  
Anatolij Dvurečenskij
Keyword(s):  
Fuzzy Sets ◽  
Homomorphic Image ◽  
Representation Theorem ◽  
Strong Unit ◽  
Direct Generalization ◽  
Mv Algebra ◽  
Mv Algebras

AbstractWe show that every σ-complete MV-algebra is an MV-σ-homomorphic image of some σ-complete MV- algebra of fuzzy sets, called a tribe, which is a system of fuzzy sets of a crisp set Ω containing 1Ω and closed under fuzzy complementation and formation of min {∑nfn, 1}. Since a tribe is a direct generalization of a σ-algebra of crisp subsets, the representation theorem is an analogue of the Loomis-Sikorski theorem for MV-algebras. In addition, this result will be extended also for Dedekind σ-complete ℓ-groups with strong unit.

scholarly journals Pseudo MV-algebras are intervals in ℓ-groups

2002 ◽  
Vol 72 (3) ◽  
pp. 427-446 ◽  
Author(s):  
Anatolij Dvurečenskij
Keyword(s):  
Strong Unit ◽  
Famous Result ◽  
Mv Algebra ◽  
Mv Algebras

AbstractWe show that any pseudo MV-algebra is isomorphic with an interval Γ(G, u), where G is an ℓ-group not necessarily Abelian with a strong unit u. In addition, we prove that the category of unital ℓ-groups is categorically equivalent with the category of pseudo MV-algebras. Since pseudo MV-algebras are a non-commutative generalization of MV-algebras, our assertions generalize a famous result of Mundici for a representation of MV-algebras by Abelian unital ℓ-groups. Our methods are completely different from those of Mundici. In addition, we show that any Archimedean pseudo MV-algebra is an MV-algebra.

THE LOOMIS–SIKORSKI THEOREM FOR -ALGEBRAS

2018 ◽  
Vol 106 (2) ◽  
pp. 200-234 ◽  
Author(s):  
ANATOLIJ DVUREČENSKIJ ◽  
OMID ZAHIRI
Keyword(s):  
Fuzzy Sets ◽  
Homomorphic Image ◽  
The State ◽  
Topological Properties ◽  
Maximal Ideals ◽  
Mv Algebra ◽  
Algebraic Operations

An EMV-algebra resembles an MV-algebra in which a top element is not guaranteed. For$\unicode[STIX]{x1D70E}$-complete$EMV$-algebras, we prove an analogue of the Loomis–Sikorski theorem showing that every$\unicode[STIX]{x1D70E}$-complete$EMV$-algebra is a$\unicode[STIX]{x1D70E}$-homomorphic image of an$EMV$-tribe of fuzzy sets where all algebraic operations are defined by points. To prove it, some topological properties of the state-morphism space and the space of maximal ideals are established.

scholarly journals The Riesz Hull of a Semisimple MV-Algebra

2015 ◽  
Vol 65 (4) ◽  
Author(s):  
D. Diaconescu ◽  
I. Leuștean
Keyword(s):  
Vector Lattice ◽  
Strong Unit ◽  
Riesz Spaces ◽  
Vector Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Mv Algebra ◽  
Standard Construction ◽  
Abelian Lattice ◽  
Mv Algebras

AbstractMV-algebras and Riesz MV-algebras are categorically equivalent to abelian lattice-ordered groups with strong unit and, respectively, with Riesz spaces (vector-lattices) with strong unit. A standard construction in the literature of lattice-ordered groups is the vector-lattice hull of an archimedean latticeordered group. Following a similar approach, in this paper we define the Riesz hull of a semisimple MV-algebra.

scholarly journals An elementary proof of the completeness of the Lukasiewicz axioms

10.29007/s5h9 ◽  
2018 ◽  
Author(s):  
Michal Botur ◽  
Jan Paseka
Keyword(s):  
Representation Theorem ◽  
Elementary Proof ◽  
Direct Proof ◽  
Rational Numbers ◽  
Farkas Lemma ◽  
Mv Algebra ◽  
Long Time ◽  
Mv Algebras

The main aim of this talk is twofold. Firstly, to present an elementarymethod based on Farkas' lemma for rationals how to embed any finite partial subalgebraof a linearly ordered MV-algebra into Q \ [0; 1] and then to establish a new elementaryproof of the completeness of the Lukasiewicz axioms for which the MV-algebras communityhas been looking for a long time. Secondly, to present a direct proof of Di Nola'srepresentation Theorem for MV-algebras and to extend his results to the restriction ofthe standard MV-algebra on rational numbers.

scholarly journals Ideals in residuated lattices

2021 ◽  
Vol 37 (1) ◽  
pp. 53-63
Author(s):  
DUMITRU BUŞNEG ◽  
DANA PICIU ◽  
ANCA-MARIA DINA
Keyword(s):  
Fuzzy Sets ◽  
Binary Relation ◽  
Congruence Relation ◽  
Natural Generalization ◽  
Residuated Lattices ◽  
Algebraic Analysis ◽  
Deductive Systems ◽  
Mv Algebra ◽  
The Relationship ◽  
Mv Algebras

"The notion of ideal in residuated lattices is introduced in [Kengne, P. C., Koguep, B. B., Akume, D. and Lele, C., L-fuzzy ideals of residuated lattices, Discuss. Math. Gen. Algebra Appl., 39 (2019), No. 2, 181–201] and [Liu, Y., Qin, Y., Qin, X. and Xu, Y., Ideals and fuzzy ideals in residuated lattices, Int. J. Math. Learn & Cyber., 8 (2017), 239–253] as a natural generalization of that of ideal in MV algebras (see [Cignoli, R., D’Ottaviano, I. M. L. and Mundici, D., Algebraic Foundations of many-valued Reasoning, Trends in Logic-Studia Logica Library 7, Dordrecht: Kluwer Academic Publishers, 2000] and [Chang, C. C., Algebraic analysis of many-valued logic, Trans. Amer. Math. Soc., 88 (1958), 467–490]). If A is an MV algebra and I is an ideal on A then the binary relation x ∼I y iff x^{*}Ꙩ y; x Ꙩy^{*} ∈ I , for x; y ∈ A; is a congruence relation on A. In this paper we find classes of residuated lattices for which the relation ∼ I (defined for MV algebras) is a congruence relation and we give new characterizations for i-ideals and prime i-ideals in residuated lattices. As a generalization of the case of BL algebras (see [Lele, C. and Nganou, J. B., MV-algebras derived from ideals in BL-algebras, Fuzzy Sets and Systems, 218 (2013), 103–113]), we investigate the relationship between i-ideals and deductive systems in residuated lattices."

scholarly journals ON THE LOOMIS–SIKORSKI THEOREM FOR MV-ALGEBRAS WITH INTERNAL STATE

2010 ◽  
Vol 89 (3) ◽  
pp. 317-333 ◽  
Author(s):  
A. DI NOLA ◽  
A. DVUREČENSKIJ ◽  
A. LETTIERI
Keyword(s):  
Fuzzy Logic ◽  
Homomorphic Image ◽  
Internal State ◽  
Algebraic Approach ◽  
Mv Algebra ◽  
Complete State ◽  
Internal States ◽  
Mv Algebras

AbstractIn Flaminio and Montagna [‘An algebraic approach to states on MV-algebras’, in: Fuzzy Logic 2, Proc. 5th EUSFLAT Conference, Ostrava, 11–14 September 2007 (ed. V. Novák) (Universitas Ostraviensis, Ostrava, 2007), Vol. II, pp. 201–206; ‘MV-algebras with internal states and probabilistic fuzzy logic’, Internat. J. Approx. Reason.50 (2009), 138–152], the authors introduced MV-algebras with an internal state, called state MV-algebras. (The letters MV stand for multi-valued.) In Di Nola and Dvurečenskij [‘State-morphism MV-algebras’, Ann. Pure Appl. Logic161 (2009), 161–173], a stronger version of state MV-algebras, called state-morphism MV-algebras, was defined. In this paper, we present the Loomis–Sikorski theorem for σ-complete MV-algebras with a σ-complete state-morphism-operator, showing that every such MV-algebra is aσ-homomorphic image of a tribe of functions with an internal state induced by a function where all the MV-operations are defined by points.

scholarly journals L-fuzzy ideals and L-fuzzy subalgebras of Novikov algebras

2019 ◽  
Vol 17 (1) ◽  
pp. 1538-1546
Author(s):  
Xin Zhou ◽  
Liangyun Chen ◽  
Yuan Chang
Keyword(s):  
Fuzzy Sets ◽  
Homomorphic Image ◽  
Quotient Algebra ◽  
Algebraic Properties ◽  
Novikov Algebras ◽  
Fuzzy Ideal

Abstract In this paper, we apply the concept of fuzzy sets to Novikov algebras, and introduce the concepts of L-fuzzy ideals and L-fuzzy subalgebras. We get a sufficient and neccessary condition such that an L-fuzzy subspace is an L-fuzzy ideal. Moreover, we show that the quotient algebra A/μ of the L-fuzzy ideal μ is isomorphic to the algebra A/Aμ of the non-fuzzy ideal Aμ. Finally, we discuss the algebraic properties of surjective homomorphic image and preimage of an L-fuzzy ideal.

Image development in the framework of ξ –complex fuzzy morphisms

2021 ◽  
pp. 1-13
Author(s):  
Aneeza Imtiaz ◽  
Umer Shuaib ◽  
Abdul Razaq ◽  
Muhammad Gulistan
Keyword(s):  
Decision Making ◽  
Comparative Analysis ◽  
Fuzzy Sets ◽  
Unit Disk ◽  
Homomorphic Image ◽  
Significant Role ◽  
Original Form ◽  
Mathematical Tool ◽  
Fuzzy Subgroups ◽  
Fundamental Theorems

The study of complex fuzzy sets defined over the meet operator (ξ –CFS) is a useful mathematical tool in which range of degrees is extended from [0, 1] to complex plane with unit disk. These particular complex fuzzy sets plays a significant role in solving various decision making problems as these particular sets are powerful extensions of classical fuzzy sets. In this paper, we define ξ –CFS and propose the notion of complex fuzzy subgroups defined over ξ –CFS (ξ –CFSG) along with their various fundamental algebraic characteristics. We extend the study of this idea by defining the concepts of ξ –complex fuzzy homomorphism and ξ –complex fuzzy isomorphism between any two ξ –complex fuzzy subgroups and establish fundamental theorems of ξ –complex fuzzy morphisms. In addition, we effectively apply the idea of ξ –complex fuzzy homomorphism to refine the corrupted homomorphic image by eliminating its distortions in order to obtain its original form. Moreover, to view the true advantage of ξ –complex fuzzy homomorphism, we present a comparative analysis with the existing knowledge of complex fuzzy homomorphism which enables us to choose this particular approach to solve many decision-making problems.

Radical of a-Ideals in Mv -Modules

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
F. Forouzesh ◽  
E. Eslami ◽  
A. Borumand Saeid
Keyword(s):  
Maximal Ideal ◽  
The Other ◽  
Subject Classification ◽  
Mv Algebra ◽  
Mv Algebras

Abstract In this paper, we introduce the notion of the radical of an ideal in MV - algebras. Several characterizations of this radical is given. We define the notion of a semi-maximal ideal in an MV -algebra and prove some theorems which give relations between this semi-maximal ideal and the other types of ideals in MV -algebras. Also we prove that A/I is a semi-simple MV -algebra if and only if I is a semi-maximal ideal of an MV -algebra A. The above notions are used to define the radical of A-ideals in MV -modules and investigate some properties. Mathematics Subject Classification 2010: 03B50, 03G25, 06D35

Effect-like algebras induced by means of basic algebras

2010 ◽  
Vol 60 (1) ◽  
Author(s):  
Ivan Chajda
Keyword(s):  
Distributive Lattice ◽  
Effect Algebra ◽  
Binary Operation ◽  
Basic Algebra ◽  
Natural Question ◽  
Lattice Effect Algebra ◽  
Mv Algebra ◽  
Lattice Effect ◽  
Mv Algebras

AbstractHaving an MV-algebra, we can restrict its binary operation addition only to the pairs of orthogonal elements. The resulting structure is known as an effect algebra, precisely distributive lattice effect algebra. Basic algebras were introduced as a generalization of MV-algebras. Hence, there is a natural question what an effect-like algebra can be reached by the above mentioned construction if an MV-algebra is replaced by a basic algebra. This is answered in the paper and properties of these effect-like algebras are studied.

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