  Effect algebras and pseudoeffect algebras were introduced by Foulis, Bennett,
Dvurecenskij and Vetterlein as so-called quantum structures which serve as an
algebraic axiomatization of the logic of quantum mechanics. A natural question
concerns their connections to substructural logics which are described by means
of residuated lattices or posets. In a previous paper it was shown that an
effect algebra can be organized into a so-called conditionally residuated
structure where the adjointness condition holds only for those elements for
which multiplication and implication are defined. Because this is a very strong
restriction, we try to find another kind of residuation where the terms
occurring in the adjointness condition are everywhere defined though the binary
operation of a given effect algebra is only partial. Moreover, we work with
effect algebras which need not be lattice-ordered and hence the lattice
operations join and meet are replaced by means of upper and lower cones which,
however, are not elements but subsets. Hence, the resulting concept, the
so-called unsharp residuated poset, is equipped with LU-terms which substitute
operations, but are everywhere defined. Although this concept seems rather
complicated at the first glance, we prove that such an unsharp residuated poset
can be conversely organized into an effect algebra or a pseudoeffect algebra
depending on commutativity of the multiplication.
