The following propositions are true: (1) For all natural numbers n, m such that n 6= 0 and m 6= 0 holds (n ·m− n − m) + 1 ­ 0. (2) For all real numbers x, y such that y > 0 holds min(x,y) max(x,y) ¬ 1. (3) For all real numbers x, y such that for every real number c such that c > 0 and c < 1 holds c · x ­ y holds y ¬ 0. (4) Let p be a finite sequence of elements of R. Suppose that for every natu...

In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals (or some other equivalent formulation of completeness), which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time in which algebra was mainly about solving polynomial equations with real or complex coefficie...

States of unital Abelian lattice-groups (normalised positive group homomorphisms to R) provide an abstraction expected-value operators. A well-known theorem due Mundici asserts that the category (with unit-preserving lattice-group as morphisms) is equivalent algebraic MV-algebras, and their homomorphisms. Through this equivalence, states correspond certain [0,1]-valued functionals on which are ...

We study unbounded versions of effect algebras. We show a necessary and sufficient condition, when lattice operations of a such generalized effect algebra P are inherited under its embeding as a proper ideal with a special property and closed under the effect sum into an effect algebra. Further we introduce conditions for a generalized homogeneous, prelattice or MV-effect effect algebras. We pr...

In the present paper, by considering the notion of MV-modules which is the structure that naturally correspond to lu-modules over lu-rings, we prove some results on prime A-ideals and state some conditions to obtain a prime A-ideal in MV-modules. Also, we state some conditions that an A-ideal is not prime and investigate conditions that $Ksubseteq bigcup _{i=1}^{n}K_{i}$ implies $Ksubseteq K_{j...

In this paper we make an algebraic study of the variety of M V * –algebras introduced by C. C.Chang as an algebraic counterpart for a logic with positive and negative truth values. We build the algebraic theory of M V * –algebras within its own limits using a concept of ideal and of prime ideal that are very naturally related to the corresponding concepts in –groups. The main results are a subd...

A square root is a unary operation with some special properties. In the paper, we introduce and study roots on EMV-algebras. First, known properties of defined MV-algebras will be generalized for EMV-algebras, also find new ones MV-algebras. We use to characterize Then, relation between an EMV-algebra its representing top element. show that each strict has element, investigate divisible EMV-alg...

The main aim of this talk is twofold. Firstly, to present an elementary method based on Farkas’ lemma for rationals how to embed any finite partial subalgebra of a linearly ordered MV-algebra into Q [0; 1] and then to establish a new elementary proof of the completeness of the Lukasiewicz axioms for which the MV-algebras community has been looking for a long time. Secondly, to present a direct ...

This paper is an attempt at developing a theory of algebraic systems that would correspond in a natural fashion to the N0-valued prepositional calculus^). For want of a better name, we shall call these algebraic systems MV-algebras where MV is supposed to suggest many-valued logics. It is known that the classical two-valued logic gives rise to the study of Boolean algebras and, as can be expect...