On the structure of residuated lattices

A residuated lattice is an ordered algebraic structure L = 〈L,∧,∨, · , e, \ , / 〉 such that 〈L,∧,∨ 〉 is a lattice, 〈L, ·, e 〉 is a monoid, and \ and / are binary operations for which the equivalences a · b ≤ c ⇐ ⇒ a ≤ c/b ⇐ ⇒ b ≤ a\c hold for all a, b, c ∈ L. It is helpful to think of the last two operations as left and right division and thus the equivalences can be seen as “di-viding ” on the right by b and “dividing ” on the left by a. The class of all residuated lattices is denoted by RL. The study of such objects originated in the context of the theory of ring ideals in the 1930’s. The collection of all two-sided ideals of a ring forms a lattice upon which one can impose a natural monoid structure making this object into a residuated lattice. Such ideas were investi-gated by Morgan Ward and R. P. Dilworth in a series of importan