L-algebras, self-similarity, and l-groups

Every set X with a binary operation satisfying (x y) (x z)=(y x) (y z) corresponds to a solution of the quantum Yang–Baxter equation if the left multiplication is bijective [W. Rump, A decomposition theorem for square-free unitary solutions of the quantum Yang–Baxter equation, Adv. Math. 193 (2005) 40–55]. The same equation becomes a true statement of propositional logic if the binary operation is interpreted as implication. In the present paper, L-algebras, essentially defined by the mentioned equation, are introduced and studied. For example, Hilbert algebras, locales, (left) hoops, (pseudo) MV-algebras, and l-group cones, are L-algebras. The main result states that every L-algebra admits a self-similar closure. In a further step, a structure group G(X) is associated to any L-algebra X. One more equation implies that the structure group G(X) is lattice-ordered. As an application, we characterize the L-algebras with a natural embedding into the negative cone of an l-group. In particular, this implies Mundiciʹs equivalence between MV-algebras and unital abelian l-groups, and Dvurečenskijʹs non-commutative generalization.