  We present a family of local involutive integral bounded residuated
lattice-ordered commutative monoids (involutive residuated lattices, for short)
having Boolean term, radical term (see \cite{CT12} and \cite{T23}), and
satisfying GAP (Generalized Appel property) (see \cite{T23}).The construction
of this family is based in the examples given in \cite[Subsection 5.1]{CT12},
by taking $\bfm{\LLn}$ in place of $\bfm{L_{n+1}}$ and $\bfm{L_{p+1}}$ in place
of the two element Boolean algebra. In some sense this paper is a companion of
the paper \cite{T23}, in which the General Apple Property (GAP) and Boolean
terms are studied.
