3 research outputs found
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 of \brl s has GAP iff there is an unary term such
that satisfies the equations and
, for some }. Using this
characterization, we show that for any variety of bounded
residuated lattice satisfying GAP there is such that the equation
holds in , that is, . 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 such that it satisfies the
equation .} 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