Complemented MacNeille completions and algebras of fractions

We introduce (t)bimonoids as ordered algebras consisting of two compatible monoidal structures on a partially ordered (lattice-ordered) set. Bimonoids form an appropriate frame-work for the study of a general notion of complementation, which subsumes both Boolean complements in bounded dis-tributive lattices and multiplicative inverses in monoids. The central question of the paper is whether and how bimonoids can be embedded into complemented bimonoids, generaliz-ing the embedding of cancellative commutative monoids into their groups of fractions and of bounded distributive lattices into their free Boolean extensions. We prove that each com-mutative (8-)bimonoid embeds into a complete complemented commutative .@-bimonoid in a doubly dense way reminiscent of the Dedekind-MacNeille completion. Moreover, this com-plemented completion, which is term equivalent to a com-mutative involutive residuated lattice, sometimes contains a tighter complemented envelope analogous to the group of frac-tions. In the case of cancellative commutative monoids this algebra of fractions is precisely the familiar group of frac-tions, while in the case of Brouwerian (Heyting) algebras it is a (bounded) idempotent involutive commutative residuated lattice. This construction of the algebra of fractions in fact yields a categorical equivalence between varieties of integral and of involutive residuated structures which subsumes as spe-cial cases the known equivalences between Abelian l-groups and their negative cones, and between Sugihara monoids and their negative cones.(c) 2023 Published by Elsevier Inc.