Projectivity in (bounded) integral residuated lattices

In this paper we study projective algebras in varieties of (bounded) commutative integral residuated lattices from an algebraic (as opposed to categorical) point of view. In particular we use a well-established construction in residuated lattices: the ordinal sum. Its interaction with divisibility makes our results have a better scope in varieties of divisibile commutative integral residuated lattices, and it allows us to show that many such varieties have the property that every finitely presented algebra is projective. In particular, we obtain results on (Stonean) Heyting algebras, certain varieties of hoops, and product algebras. Moreover, we study varieties with a Boolean retraction term, showing for instance that in a variety with a Boolean retraction term all finite Boolean algebras are projective. Finally, we connect our results with the theory of Unification

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

Free-decomposability in Varieties of Pseudocomplemented Residuated Lattices

In this paper we prove that the free pseudocomplemented residuated lattices are decomposable if and only if they are Stone, i.e., if and only if they satisfy the identity ¬x ∨ ¬¬x = 1. Some applications are given.Fil: Castaño, Diego Nicolás. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Bahía Blanca. Instituto de Matemática Bahía Blanca (i); Argentina. Universidad Nacional del Sur; ArgentinaFil: Díaz Varela, José Patricio. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Bahía Blanca. Instituto de Matemática Bahía Blanca (i); Argentina. Universidad Nacional del Sur; ArgentinaFil: Torrens, Antoni. Universidad de Barcelona; Españ