Subreducts of MV-algebras with product and product residuation

Abstract

Recently, MV -algebras with product have been investigated from different points of view. In particular, in [EGM01], a variety resulting from the combination of MV -algebras and product algebras (see [H98]) has been introduced. The elements of this variety are called LΠ-algebras. In this paper we treat subreducts of LΠ-algebras, with emphasis on quasivarieties of subreducts whose basic operations are continuous in the order topology. We give axiomatizations of the most interesting classes of subreducts, and we connect them with other algebraic classes of algebras, like f -rings and Wajsberg hoops, as well as to structures of co-infinitesimals of LΠ-algebras. In some cases, connections are given by means of equivalences of categories.

Łukasiewicz logic and Riesz spaces

“…A PMV-algebra is a structure (P, ·), where P is an MV-algebra and · : P × P → P satisfies the equations of an internal product. PMV-algebras are an equational class, but the standard model [0, 1] generates only a quasi-variety which is a proper subclass of PMV-algebras [26]. In this context, it was natural to replace the internal product with an external one: a Riesz MV-algebra is a structure (R, ·), where R is an MV-algebra and · : [0, 1]×R → R. Since we prove that the variety of Riesz MV-algebras is generated by [0,1], the propositional calculus RL, that has Riesz MV-algebras as models, is complete with respect to evaluations in [0,1].…”

A General View on Normal Form Theorems for Łukasiewicz Logic with Product

“…Any MV-algebra can be endowed with a lattice order and the standard model is the unit interval [0, 1] with x ⊕ y = min(x + y, 1) and x * = 1 − x. The variety of MV-algebras is generated by the standard model and since [0,1] is closed with respect to the real product, a fruitful research direction is the study of MV-algebras enriched with a product operation [6,24,25,13,9], which can be a binary operation, a scalar multiplication or a combination of both.…”

Stable Non-standard Imprecise Probabilities

“…Remark 1. In [7] it is shown that every c-s-u-f integral domain embeds into an algebra of the form (R * f in ) H , where R * is an ultrapower of the real field, R f in is the c-s-u-f domain consisting of all finite elements of R * , and H is an index set 5 . In particular, every c-s-u-f integral domain embeds into the product of totally ordered integral domains, and this fact justifies the name of these structures.…”

“…(2) (see [7]). Γ R is a functor from the category of c-s-u-f integral domains into the category of PMV + -algebras.…”