Cauchy completions of MV-algebras

Abstract

An MV-convergence is a convergence on an MV-algebra which renders the operations continuous. We show that such convergences on a given MV-algebra A are exactly the restrictions of the bounded -convergences on the abelian -group in which A appears as the unit interval. Thus the theory of -convergence and Cauchy structures transfers to MV-algebras.We outline the general theory, and then apply it to three particular MV-convergences and their corresponding Cauchy completions. The Cauchy completion arising from order convergence coincides with the Dedekind-MacNeille completion of an MV-algebra. The Cauchy completion arising from polar convergence allows a tidy proof of the existence and uniqueness of the lateral completion of an MV-algebra. And the Cauchy completion arising from α-convergence gives rise to the cut completion of an MV-algebra.

On non-associative generalizations of MV-algebras and lattice-ordered commutative loops

“…[15,28]), but the disadvantage of this approach is that it only leads to the positive cone of L, whereas the -loop L itself must be constructed from the positive cone using the method described in [2]. Instead, we present another construction, similar to [1], which directly leads to the -loop L. First, a technical lemma: Lemma 5.1. In any basic algebra A, we have:…”

“…We find a condition under which (L, u) is commutative. In Section 4, generalizing Chang's construction (see [14]), we prove that every linearly ordered commutative basic algebra is of the form (L, u) for a suitable linearly ordered commutative inverse loop and a strong order-unit u in L. In Section 5 we prove that every commutative basic algebra which is a subdirect product of linearly ordered factors (we call such algebras "semilinear") is isomorphic to some (L, u) where again u is a strong order-unit for L. Our method is based on Mundici's good sequences (see [15,22,28]), except that similarly to [1] we actually work with "good functions", which are certain mapping from Z to the algebra. This directly gives L, while the classical construction leads to the positive cone of L from which L must be constructed subsequently.…”

Weak relatively uniform convergences on MV-algebras

“…Let us return to the results of the paper [3] in Theorem 3.3. Essential part of Theorem 3.3 is the following assertion:…”

“…In [3], there is introduced the notion of an M V -convergence as a convergence on an M V -algebra which makes the M V -operations continuous. In an analogical way, a convergence on a unital lattice ordered group (G, u), called lu-convergence, is defined.…”

On a relative uniform completion of an archimedean lattice ordered group

“…G e o r g e s c u , I . L e u s t e a n [2] transfered the theory of l-convergence and the Cauchy completion into MV-algebras. By the restriction of an l-convergence in G to A we obtain a convergence on A making the MV-operations continuous.…”