Relational groupoids and residuated lattices

Residuated structures are important lattice-ordered algebras both for mathematics and for logics; in particular, the development of lattice-valued mathematics and related non-classical logics is based on a multitude of lattice-ordered structures that suit for many-valued reasoning under uncertainty and vagueness. Extended-order algebras, introduced in [10] and further developed in [1], give an order-theoretical approach to a general description of the algebras of logics which goes along the line of an implication-based view. In the metamathematical framework of classical and intuitionistic logics the algebraic structure of their semantics depends entirely on an order relation in the set of the truth values. In fact, the algebraic structure of both boolean and Heyting algebras (or, as one should prefer saying, boolean and Heyting lattices) is completely determined by the underlying order relation. Looking at non-classical logics, extended-order algebras have been introduced on the base of the following principle: ”just like an order relation ≤ in a set L determines completely the lattice structure of L, each of its extensions relative to a true value > ∈ L, e.g. any implication →: L × L → L such that for all a, b ∈ L : a ≤ b ⇔ a → b = >, completely determines the richer lattice-ordered algebraic structure on L, to be used either in classical or in non-classical logics”. The implicative structure (L,→,>) so obtained has been called implicative algebra in the monograph of H. Rasiowa [12], where it has been specialized to characterize either algebras of subsets (implication algebras) or algebras of open sets (positive implication algebras also called Hilbert algebras). Instead, the above described motivation has led to call (L,→,>) weak extended-order algebra (w-eo algebra) in [10, 1]; there, additional conditions have been considered and discussed, including a weak, but important requirement that characterizes extended-order algebras (eo algebras), giving characterizations of several classes of residuated structures; in particular, it is seen that every integral residuated lattice is a symmetrical distributive and associative eo algebra. It has to be noted that