MV-Algebras and Abelian l-Groups: a Fruitful Interaction

This paper introduces the notion of a functorial torsion class (FTC): in a concrete category $\mathfrak{C}$ which has image factorization, one considers monocoreflective subcategories which are closed under formation of subobjects. Here the interest is in FTCs in the category of abelian lattice-ordered groups with designated strong order unit. The FTCs $\mathfrak{T}$ consisting of archimedean latticeordered groups are characterized: for each subgroup A of the rationals with the identity 1, either $\mathfrak{T} = \mathfrak{S}\left( A \right)$, the class of all lattice-ordered groups of functions on a set X which have finite range in A, or $$\mathfrak{T} = \mathbb{T}\left( A \right)$$, the class of all subgroups of A with 1. As for FTCs possessing non-archimedean groups, it is shown that if $\mathfrak{T}$ is an FTC containing a subgroup A of the reals with 1, of rank two or greater, then $\mathfrak{T}$ contains all ℓ-groups of the form $A\vec \times G$, for all abelian lattice-ordered groups G. Finally, the least FTC that contains a non-archimedean group is the class of all $\mathbb{Z}\vec \times G$, for all abelian lattice-ordered groups G.

In this chapter we shall prove Chang’s completeness theorem stating that if an equation holds in the unit real interval [0,1], then the equation holds in every MV-algebra. Thus, intuitively, the two element structure {0,1} stands to boolean algebras as the interval [0,1] stands to MV-algebras. Our proof is elementary, and makes use of tools (such as “good sequences”) that shall also find applications in a subsequent chapter to show the equivalence between MV-algebras and lattice-ordered abelian groups with strong unit.

Every semifield of non-zero characteristic is either a field of prime characteristic or a semifield of characteristic 1. Semifields of characteristic 1 are equivalent to abelian lattice-ordered groups. It is proved that such a semifield A is algebraically closed if and only if the pure equations xn = a and certain quadratic equations are solvable in A. Using a sheaf representation for z-projectable abelian l-groups on the co-Zariski space of minimal primes, a sheaf-theoretic characterization of algebraic closedness in characteristic 1 is obtained. Concerning the solvability of quadratic equations, the criterion consists in a topological condition for the base space. The results are built upon an analysis of rational functions in characteristic 1. While polynomials satisfy the “fundamental theorem of algebra”, the multiplicative structure of rational functions is determined by means of “divisors”, extracted from the additive structure of A modulo parallelogram identities.

In 1974, J. Martinez introduced the variety ${\mathcal W}$ of weakly Abelian lattice-ordered groups; it is defined by the identity \[ x^{-1}(y\vee 1)x\vee (y\vee 1)^2=(y\vee 1)^2.\]

MV-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.

(1978). Orthomodular Lattices and Quantum Physics. Mathematics Magazine: Vol. 51, No. 5, pp. 299-303.

Free lattice-ordered Abelian groups

A κ-denotational semantics for map theory in ZFC + SI

In this chapter, we consider particular classes of infinite-valued propositional logics which are strongly related to t-norms as conjunction connectives and to the real unit interval as a set of their truth degrees, and which have their implication connectives determined via an adjointness condition.

Algebraic proof of the separation theorem on classical propositional calculus

The first part of the paper_ outlines the development of causality in Greek philosophy. Some remarks are made on how some medieval philosophers approached to the problem. The paper shows also how modern philosophy understood causation. The paper inquires into the characteristics of causal relation as it is accepted in the domain of modern and recent physics.In the second part of the paper one finds some remarks concerning the programme of construing a new system CI of logic of causal propositions. This system is adequate to the manner in which causality is presented in physics. The system CI is being construed to characterize the connective of relativistic conditionals ‘$\rightsquigarrow$’ by means of the methods of recent logic. This connective should be read as follows: ‘if …, then for that reason …’. The arguments of the connective ‘$\rightsquigarrow$’ may be propositional formulas describing a particular event. The new system of non-classical logic is based on the classical propositional calculus, on the one of systems of temporal logic, and on a system ZI of logic of change. The findings concerning causation, as outlined in the first part of the paper, constitute descriptive semantics of the system CI presented in the second part of the paper. This system may play a positive role in the indirect justification of theorems in philosophy in a broad sense.

In this paper we study the algebraization of two Gentzen systems, both of them generating the implication-less fragment of the intuitionistic propositional calculus. We prove that they are algebraizable, the variety of pseudocomplemented distributive lattices being an equivalent algebraic semantics for them, in the sense that their Gentzen deduction and the equational deduction over this variety are interpretable in one another, these interpretations being essentially inverse to one another. As a consequence, the consistent deductive systems that satisfy the properties of Conjunction, Disjunction and Pseudo-Reductio ad Absurdum are described by giving apropriate Gentzen systems for them. All these Gentzen systems are algebraizable, the subvarieties of the variety of pseudocomplemented distributive lattices being their equivalent algebraic semantics respectively. Finally we give a Gentzen system for the conjunction and disjunction fragment of the classical propositional calculus, prove that the variety of distributive lattices is an equivalent algebraic semantics for it and give a Gentzen system, weaker than the latter, the variety of lattices being an equivalent algebraic semantics for it.

Context. Such perfect algebraic constructions of many-valued logic as ternary bent-functions and their truth tables which are called as 3-bent-sequences, are used very often in modern cryptographic algorithms, in particular, in pseudorandom sequence generators. However, today there are no methods for synthesizing the ternary bent-functions class for a number of variables greater than two, which significantly limits the ability to scale the number of protection levels of the of pseudorandom sequence generators based on the ternary bent-functions. This circumstance generates the task of developing methods for the synthesis of ternary bent-functions, which is solved in this paper for the case of ternary bent-functions of three variables. The object of this research is the process of efficiency increasing of the cryptographic algorithms based on the functions of many-valued logic.Objective. The purpose of the paper is to construct a method for the synthesis of the set of ternary bent-functions of three variables.Method. The mathematical apparatus of the Reed-Muller transform (algebraic normal form) was used as the basis of the proposed constructive method for the synthesis of ternary bent-functions of three variables. So, on the basis of the established properties of the algebraic normal form of ternary bent-functions and limited enumeration, the search for ternary bent-functions up to affine terms is performed, after which we apply the procedure of reproduction.Results. As a result of using of the proposed method for the synthesis of ternary bent-functions of three variables, 155844 3-bentsequences were found up to an affine term, while the cardinality of the full set of found 3-bent-sequences is 12623364. The research performed made it possible to determine that in this set there are 3-bent-sequences of six different weight structures, on the basis of which 12 different triple sets can be compiled for use in pseudorandom sequence generators. A scheme for a cryptographically stable pseudorandom sequence generator based on the found set of 3-bent-sequences of length N = 27 is proposed. It is shown that the protection levels number of such a generator of pseudorandom sequences is 41 7.041 10    which is comparable with the protection levels number of modern block symmetric cryptographic algorithms, for example, AES-128.Conclusions. The further development of modern cryptographic algorithms, in particular, cryptographically stable pseudorandom sequence generators, is largely based on the use of perfect algebraic constructions of many-valued logic. For the first time, a constructive method for the synthesis of ternary bent-functions of three variables is proposed. For the found set of ternary bent-functions, the distribution of weight structures is found, and the possible triple sets are established. Based on the constructed set of ternary bentfunctions, a pseudorandom sequence generator scheme is proposed that has a protection levels number that is comparable with modern block symmetric cryptographic algorithms. We note that the constructed class of ternary bent-functions can also be used for the synthesis of cryptographically strong S-boxes, codes of constant amplitude, as well as error correction codes. As an actual area of further research, we can note the development of methods for the synthesis of ternary bent-functions of a larger number of variables.

We provide a generalization of Mundici’s equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category of unital commutative lattice-ordered monoids is equivalent to the category of MV-monoidal algebras. Roughly speaking, unital commutative lattice-ordered monoids are unital Abelian lattice-ordered groups without the unary operation x mapsto -x. The primitive operations are +, vee , wedge , 0, 1, -1. A prime example of these structures is mathbb {R}, with the obvious interpretation of the operations. Analogously, MV-monoidal algebras are MV-algebras without the negation x mapsto lnot x. The primitive operations are oplus , odot , vee , wedge , 0, 1. A motivating example of MV-monoidal algebra is the negation-free reduct of the standard MV-algebra [0, 1]subseteq mathbb {R}. We obtain the original Mundici’s equivalence as a corollary of our main result.

We introduce Boolean-like algebras of dimension n (n{mathrm {BA}}s) having n constants {{{mathsf {e}}}}_1,ldots ,{{{mathsf {e}}}}_n, and an (n+1)-ary operation q (a “generalised if-then-else”) that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of n{mathrm {BA}}s share many remarkable properties with the variety of Boolean algebras and with primal varieties. The n{mathrm {BA}}s provide the algebraic framework for generalising the classical propositional calculus to the case of n–perfectly symmetric–truth-values. Every finite-valued tabular logic can be embedded into such a n-valued propositional logic, n{mathrm {CL}}, and this embedding preserves validity. We define a confluent and terminating first-order rewriting system for deciding validity in n{mathrm {CL}}, and, via the embeddings, in all the finite tabular logics.