  Pseudo-effect algebras are partial algebraic structures, that were introduced
as a non-commutative generalization of effect algebras. In the present paper,
lattice ordered pseudo-effect algebras are considered as possible algebraic
non-commutative analogs of non-commutative non-standard reasoning. To this aim,
the interplay among conjunction, implication and negation connectives is
studied. It turns out that in the non-commutative reasoning, all these
connectives are doubled. In particular, there are two negations and two pairs
consisting of conjunction and implication, related by residuation laws. The
main result of the paper is a characterization of lattice pseudo-effect
algebras in terms of so-called pseudo Sasaki algebras. We also show that all
pseudo-effect algebras can be characterized in terms of certain partially
defined double residuated structures.
