scholarly journals Representation of perfect and local MV-algebras

2011 ◽  
Vol 61 (3) ◽  
Author(s):  
Brunella Gerla ◽  
Ciro Russo ◽  
Luca Spada
Keyword(s):  
Unit Interval ◽  
Representation Theorems ◽  
Strong Unit ◽  
Mv Algebras

AbstractWe describe representation theorems for local and perfect MV-algebras in terms of ultraproducts involving the unit interval [0, 1]. Furthermore, we give a representation of local Abelian ℓ-groups with strong unit as quasi-constant functions on an ultraproduct of the reals. All the above theorems are proved to have a uniform version, depending only on the cardinality of the algebra to be embedded, as well as a definable construction in ZFC.The paper contains both known and new results and provides a complete overview of representation theorems for such classes.

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 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 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.

Ulam Games, Łukasiewicz Logic, and AF C*-Algebras

1993 ◽  
Vol 18 (2-4) ◽  
pp. 151-161
Author(s):  
Daniele Mundici
Keyword(s):  
Search Space ◽  
Error Correcting Codes ◽  
Strong Unit ◽  
C Algebras ◽  
Finite Dimensional ◽  
Minimum Number ◽  
Boolean Search ◽  
Game Theoretic ◽  
The Relationship ◽  
Mv Algebras

Ulam asked what is the minimum number of yes-no questions necessary to find an unknown number in the search space (1, …, 2n), if up to l of the answers may be erroneous. The solutions to this problem provide optimal adaptive l error correcting codes. Traditional, nonadaptive l error correcting codes correspond to the particular case when all questions are formulated before all answers. We show that answers in Ulam’s game obey the (l+2)-valued logic of Łukasiewicz. Since approximately finite-dimensional (AF) C*-algebras can be interpreted in the infinite-valued sentential calculus, we discuss the relationship between game-theoretic notions and their C*-algebraic counterparts. We describe the correspondence between continuous trace AF C*-algebras, and Ulam games with separable Boolean search space S. whose questions are the clopen subspaces of S. We also show that these games correspond to finite products of countable Post MV algebras, as well as to countable lattice-ordered Specker groups with strong unit.

scholarly journals On the Equivalence Between MV-Algebras and l-Groups with Strong Unit

Studia Logica ◽  
2014 ◽  
Vol 103 (4) ◽  
pp. 807-814 ◽  
Author(s):  
Eduardo J. Dubuc ◽  
Y. A. Poveda
Keyword(s):  
Strong Unit ◽  
Mv Algebras

Measures Induced by Units

2013 ◽  
Vol 78 (3) ◽  
pp. 886-910
Author(s):  
Giovanni Panti ◽  
Davide Ravotti
Keyword(s):  
Propositional Logic ◽  
Continuous Functions ◽  
Unit Interval ◽  
Algebraic Semantics ◽  
Absolutely Continuous ◽  
Strong Unit ◽  
Borel Measures ◽  
Maximal Spectrum ◽  
Dense Set ◽  
Finitely Presented

AbstractThe half-open real unit interval (0,1] is closed under the ordinary multiplication and its residuum. The corresponding infinite-valued propositional logic has as its equivalent algebraic semantics the equational class of cancellative hoops. Fixing a strong unit in a cancellative hoop—equivalently, in the enveloping lattice-ordered abelian group—amounts to fixing a gauge scale for falsity. In this paper we show that any strong unit in a finitely presented cancellative hoopHinduces naturally (i.e., in a representation-independent way) an automorphism-invariant positive normalized linear functional onH. SinceHis representable as a uniformly dense set of continuous functions on its maximal spectrum, such functionals—in this context usually called states—amount to automorphism-invariant finite Borel measures on the spectrum. Different choices for the unit may be algebraically unrelated (e.g., they may lie in different orbits under the automorphism group ofH), but our second main result shows that the corresponding measures are always absolutely continuous w.r.t. each other, and provides an explicit expression for the reciprocal density.

Monadic MV-algebras are Equivalent to Monadic ℓ-groups with Strong Unit

Studia Logica ◽  
2011 ◽  
Vol 98 (1-2) ◽  
pp. 175-201 ◽  
Author(s):  
C. Cimadamore ◽  
J. P. Díaz Varela
Keyword(s):  
Strong Unit ◽  
Mv Algebras

On the Riesz structures of a lattice ordered abelian group

2019 ◽  
Vol 69 (6) ◽  
pp. 1237-1244
Author(s):  
Giacomo Lenzi
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Positive Element ◽  
Space Structure ◽  
Real Vector ◽  
Ordered Group ◽  
Positive Real ◽  
Strong Unit ◽  
Totally Ordered Group ◽  
Mv Algebras

Abstract A Riesz structure on a lattice ordered abelian group G is a real vector space structure where the product of a positive element of G and a positive real is positive. In this paper we show that for every cardinal k there is a totally ordered abelian group with at least k Riesz structures, all of them isomorphic. Moreover two Riesz structures on the same totally ordered group are partially isomorphic in the sense of model theory. Further, as a main result, we build two nonisomorphic Riesz structures on the same l-group with strong unit. This gives a solution to a problem posed by Conrad in 1975. Finally we apply the main result to MV-algebras and Riesz MV-algebras.

scholarly journals Some classes of pseudo-BL algebras

2002 ◽  
Vol 73 (1) ◽  
pp. 127-154 ◽  
Author(s):  
George Georgescu ◽  
Laurenţiu Leuştean
Keyword(s):  
Strong Unit ◽  
Mv Algebras

AbstractPseudo-BL algebras are noncommutative generalizations of BL-algebras and they include pseudo-MV algebras, a class of structures that are categorically equivalent to l-groups with strong unit. In this paper we characterize directly indecomposable pseudo-BL algebras and we define and study different classes of these structures: local, good, perfect, peculiar, and (strongly) bipartite pseudo-BL algebras.

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