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 $b(x)\vee \mathrm{\neg }b(x)\approx \mathrm{\top }$ and $(x^k\to b(x))\cdot (b(x)\to k\mathrm{.}x)\approx \mathrm{\top }$, 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 $k\mathrm{.}x\vee k\mathrm{.}\mathrm{\neg }x\approx \mathrm{\top }$ holds in $\mathsf{V}$, that is, $\mathsf{V}\subseteq \mathsf{W}{\mathsf{L}}_{\mathsf{k}}$. 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 $k\mathrm{.}{x}^k\vee k\mathrm{.}(\mathrm{\neg }x{)}^k\approx \mathrm{\top }$.} 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.