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 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.

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.

scholarly journals Residuation in non-associative MV-algebras

2018 ◽  
Vol 68 (6) ◽  
pp. 1313-1320
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Binary Operation ◽  
Residuated Lattice ◽  
Natural Question ◽  
Natural Conditions ◽  
Double Negation ◽  
Mv Algebra ◽  
Residuated Poset ◽  
Double Negation Law ◽  
Mv Algebras

Abstract It is well known that every MV-algebra can be converted into a residuated lattice satisfying divisibility and the double negation law. In a previous paper the first author and J. Kühr introduced the concept of an NMV-algebra which is a non-associative modification of an MV-algebra. The natural question arises if an NMV-algebra can be converted into a residuated structure, too. Contrary to MV-algebras, NMV-algebras are not based on lattices but only on directed posets and the binary operation need not be associative and hence we cannot expect to obtain a residuated lattice but only an essentially weaker structure called a conditionally residuated poset. Considering several additional natural conditions we show that every NMV-algebra can be converted in such a structure. Also conversely, every such structure can be organized into an NMV-algebra. Further, we study an a bit more stronger version of an algebra where the binary operation is even monotone. We show that such an algebra can be organized into a residuated poset and, conversely, every residuated poset can be converted in this structure.

scholarly journals Isometries and direct decompositions of pseudo MV-algebras

2007 ◽  
Vol 57 (2) ◽  
Author(s):  
Milan Jasem
Keyword(s):  
Direct Decomposition ◽  
The Other ◽  
Other Hand ◽  
Mv Algebra ◽  
Direct Decompositions ◽  
Mv Algebras

AbstractIn the paper isometries in pseudo MV-algebras are investigated. It is shown that for every isometry f in a pseudo MV-algebra $$\mathcal{A}$$ = (A, ⊕, −, ∼, 0, 1) there exists an internal direct decomposition $$\mathcal{A} = \mathcal{B}^0 \times \mathcal{C}^0 $$ of $$\mathcal{A}$$ with $$\mathcal{C}^0 $$ commutative such that $$f(0) = 1_{C^0 } $$ and $$f(x) = x_{B^0 } \oplus (1_{C^0 } \odot (x_{C^0 } )^ - ) = x_{B^0 } \oplus (1_{C^0 } - x_{C^0 } )$$ for each x ∈ A.On the other hand, if $$\mathcal{A} = \mathcal{P}^0 \times \mathcal{Q}^0 $$ is an internal direct decomposition of a pseudo MV-algebra $$\mathcal{A}$$ = (A, ⊕, −, ∼, 0, 1) with $$\mathcal{Q}^0 $$ commutative, then the mapping g: A → A defined by $$g(x) = x_{P^0 } \oplus (1_{Q^0 } - x_{Q^0 } )$$ is an isometry in $$\mathcal{A}$$ and $$g(0) = 1_{Q^0 } $$ .

Higher degrees of distributivity in complete generalized MV-algebras

2011 ◽  
Vol 61 (3) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Mv Algebra ◽  
Mv Algebras

AbstractWe apply the notion of generalized MV-algebra (GMV-algebra, in short) in the sense of Galatos and Tsinakis. Let M be a complete GMV-algebra and let α be a cardinal. We prove that M is α-distributive if and only if it is (α, 2)-distributive. We deal with direct summands of M which are homogeneous with respect to higher degrees of distributivity.

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.

scholarly journals Study of MV-algebras via derivations

2019 ◽  
Vol 27 (3) ◽  
pp. 259-278
Author(s):  
Jun Tao Wang ◽  
Yan Hong She ◽  
Ting Qian
Keyword(s):  
Boolean Algebra ◽  
Direct Product ◽  
Galois Connection ◽  
Product Representation ◽  
Mv Algebra ◽  
One To One ◽  
The Relationship ◽  
Mv Algebras ◽  
Derivation Pair

AbstractThe main goal of this paper is to give some representations of MV-algebras in terms of derivations. In this paper, we investigate some properties of implicative and difference derivations and give their characterizations in MV-algebras. Then, we show that every Boolean algebra (idempotent MV-algebra) is isomorphic to the algebra of all implicative derivations and obtain that a direct product representation of MV-algebra by implicative derivations. Moreover, we prove that regular implicative and difference derivations on MV-algebras are in one to one correspondence and show that the relationship between the regular derivation pair (d, g) and the Galois connection, where d and g are regular difference and implicative derivation on L, respectively. Finally, we obtain that regular difference derivations coincide with direct product decompositions of MV-algebras.

On idempotent modifications of generalized MV-algebras

2010 ◽  
Vol 60 (2) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Present Note ◽  
Subdirectly Irreducible ◽  
Mv Algebra ◽  
Mv Algebras

AbstractThe notion of idempotent modification of an algebra was introduced by Ježek; he proved that the idempotent modification of a group is always subdirectly irreducible. In the present note we show that the idempotent modification of a generalized MV -algebra having more than two elements is directly irreducible if and only if there exists an element in A which fails to be boolean. Some further results on idempotent modifications are also proved.

From Cdc2 to Cdk1: when did the cell cycle kinase join its cyclin partner?

2002 ◽  
Vol 115 (12) ◽  
pp. 2461-2464 ◽  
Author(s):  
Marcel Dorée ◽  
Tim Hunt
Keyword(s):  
Cell Cycle ◽  
Protein Kinase ◽  
Marine Invertebrate ◽  
Direct Proof ◽  
Protein Kinase Activity ◽  
Cell Cycles ◽  
M Phase ◽  
Long Time ◽  
Dependent Protein ◽  
Cyclin Partner

The idea that Cdc2 and cyclins play a key role in the control of the G2/M transition of the cell cycle came largely from genetic analysis of fission yeast and physiological studies of clam, frog, sea urchin and starfish eggs and oocytes. However, it took a long time to realise that Cdc2 and cyclins form a stoichiometric complex and that a cyclin subunit is necessary for the Cdc2 subunit to gain its protein kinase activity. Cyclins were first recognized as proteins whose abundance oscillates during the early cell cycles of marine invertebrate eggs and their connection with MPF (maturation-promoting factor), the entity defined in frog and starfish oocytes whose activity controls entry into M phase, was far from clear at first. Indeed, it was a long time before MPF was shown to be a protein kinase,and direct proof that MPF is a heterodimer comprising one molecule of cyclin and one molecule of Cdc2 was finally obtained only when the Cdc2-associated component of purified starfish MPF was sequenced and found to be cyclin B. When this fundamental discovery was confirmed in vertebrates and mammalian members of the Cdc2 family were also shown to bind cyclins, Cdc2 became Cdk1,the first cyclin-dependent protein kinase.

Sign in / Sign up

Export Citation Format

Share Document