Complemented MacNeille completions and algebras of fractions

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