  We consider orthomodular posets endowed with a symmetric difference. We call
them ODPs. Expressed in the quantum logic language, we consider quantum logics
with an XOR-type connective. We study three classes of "almost Boolean" ODPs,
two of them defined by requiring rather specific behaviour of infima and the
third by a Boolean-like behaviour of Frink ideals. We establish a (rather
surprising) inclusion between the three classes, shadding thus light on their
intrinsic properties. (More details can be found in the Introduction that
follows.) Let us only note that the orthomodular posets pursued here, though
close to Boolean algebras (i.e., close to standard quantum logics), still have
a potential for an arbitrarily high degree of non-compatibility and hence they
may enrich the studies of mathematical foundations of quantum mechanics.
