The General Apple Property and Boolean terms in Integral Bounded Residuated Lattice-ordered Commutative Monoids

In this paper we give equational presentations of the varieties of {\em integral bounded residuated lattice-ordered commutative monoids} (bounded residuated lattices for short) satisfying the \emph{General Apple Property} (GAP), that is, varieties in which all of its directly indecomposable members are local. This characterization is given by means of Boolean terms: \emph{A variety $\mathsf{V}$ of \brl s has GAP iff there is an unary term $b(x)$ such that $\mathsf{V}$ satisfies the equations $b(x)\lor\neg b(x)\approx \top$ and $(x^k\to b(x))\cdot(b(x)\to k.x)\approx \top$, for some $k>0$}. Using this characterization, we show that for any variety $\mathsf{V}$ of bounded residuated lattice satisfying GAP there is $k>0$ such that the equation $k.x\lor k.\neg x\approx \top$ holds in $\mathsf{V}$, that is, $\mathsf{V} \subseteq \mathsf{WL_\mathsf{k}}$. As a consequence we improve Theorem 5.7 of \cite{CT12}, showing in theorem that a\emph{ variety of \brls\ has Boolean retraction term if and only if there is $k>0$ such that it satisfies the equation $k.x^k\lor k.(\neg k)^n\approx\top$.} We also see that in Bounded residuated lattices GAP is equivalent to Boolean lifting property (BLP) and so, it is equivalent to quasi-local property (in the sense of \cite{GLM12}). Finally, we prove that a variety of \brl s has GAP and its semisimple members form a variety if and only if there exists an unary term which is simultaneously Boolean and radical for this variety.Comment: 25 pages, 1 figure, 2 table

Cadenes de Sales discretes

The Sales algebras are the substract of some positive implicational calculi . As a first aproximation to their representation we study the Sales algebras that are discrete and linear . In this paper we present a caracteritzation of the finite Sales algebras and we give also an infinite discrete Sales algebra such that we can identify with it every linear infinite simple Sales algebra with a penultimate element . The most interes ting result is the first Theorem In it we show that certain elements of the Sales algebras generate finit linear subalgebras of them

Estudi i algebraització de certes lògiques: Àlgebres d-completes.

En aquesta tesi doctoral s'obtenen i estudien les àlgebres d-completes com les àlgebres implicatives associades a uns determinats càlculs proposicionals implicatius, que satisfan un teorema de la deducció feble i contenen als càlculs proposicionals multi-valorats donats per Lukasiewicz. Per altra banda, també s'estudien els sistemes deductius d'aquestes àlgebres

Elements crisipians en àlgebres d-completes i àlgebres de Sales

El autor estudia los elementos de comportamiento clásico, o crisipianos, en álgebras d-completas (introducidas por él mismo como el sustrato algebraico de las lógicas completas) y en álgebras de Sales (sustrato algebraico de las lógicas multivaloradas). Da caracterizaciones de estos elementos en ambos casos. Estudia la relación de dichos elementos con los espectros irreducible, primo y completamente irreducible. Además obtiene que el conjunto de elementos crisipianos de un álgebra de Sales es una subálgebra y es un álgebra de Abbott (o de implicación)

W-algebras which are boolean products of members of SR[1] and CW-algebras

Preprint enviat per a la seva publicació en una revista científica: Stud Logica 46, 265–274 (1987). [https://doi.org/10.1007/BF00372551]We show that the class of all isomorphic images of Booleans producís of members of SR[1] is the class of all Archimedean W-algebras. And the class of all isomorphic images of CW-lgebras is the class of all W-algebras such that the family of all minimal prime implicative filters is the family of all Stone ultrafilters

Cadenes de Sales discretes

The Sales algebras are the substract of some positive implicational calculi . As a first aproximation to their representation we study the Sales algebras that are discrete and linear . In this paper we present a caracteritzation of the finite Sales algebras and we give also an infinite discrete Sales algebra such that we can identify with it every linear infinite simple Sales algebra with a penultimate element . The most interes ting result is the first Theorem In it we show that certain elements of the Sales algebras generate finit linear subalgebras of them

Elements crisipians en àlgebres d-completes i àlgebres de Sales

El autor estudia los elementos de comportamiento clásico, o crisipianos, en álgebras d-completas (introducidas por él mismo como el sustrato algebraico de las lógicas completas) y en álgebras de Sales (sustrato algebraico de las lógicas multivaloradas). Da caracterizaciones de estos elementos en ambos casos. Estudia la relación de dichos elementos con los espectros irreducible, primo y completamente irreducible. Además obtiene que el conjunto de elementos crisipianos de un álgebra de Sales es una subálgebra y es un álgebra de Abbott (o de implicación)