  In this paper we provide a preliminary investigation of subclasses of bounded
posets with antitone involution which are "pastings" of their maximal Kleene
sub-lattices. Specifically, we introduce super-paraorthomodular lattices,
namely paraothomodular lattices whose order determines, and it is fully
determined by, the order of their maximal Kleene sub-algebras. It will turn out
that the (spectral) paraorthomodular lattice of effects over a separable
Hilbert space can be considered as a prominent example of such. Therefore, it
arguably provides an algebraic/order theoretical rendering of complementarity
phenomena between unsharp observables. A number of examples, properties and
characterization theorems for structures we deal with will be outlined. For
example, we prove a forbidden configuration theorem and we investigate the
notion of commutativity for modular pseudo-Kleene lattices, examples of which
are (spectral) paraorthomodular lattices of effects over finite-dimensional
Hilbert spaces. Finally, we show that structures introduced in this paper yield
paraconsistent partial referential matrices, the latter being generalizations
of J. Czelakowski's partial referential matrices. As a consequence, a link
between some classes of posets with antitone involution and algebras of partial
"unsharp" propositions is established.
