A general Glivenko theorem

Abstract

The equation $${({\rm L}): (x \rightarrow y) \rightarrow (x \rightarrow z) = (y \rightarrow x) \rightarrow (y \rightarrow z)}$$ occurs in algebraic logic and in the theory of the quantum Yang-Baxter equation. KL-algebras are based on this equation and generalize, e.g., Hilbert algebras and locales, (one-sided) hoops, (pseudo) MV-algebras, and l-group cones. Every KL-algebra admits a universal map into its structure group, a map that generalizes classical double negation. Using this map, a Glivenko type theorem for KL-algebras is obtained. In the special case where X has a smallest element and double negation is an endomorphism d,it is shown that δ(X) is an MV-algebra which operates on the kernel δ −1(1), a BCK-algebra satisfying (L). This leads to an embedding of X into a restricted semidirect product, $${\delta(X)\mathop \varpropto \limits^{b}\delta^{-1}(1)}$$ . Conversely, it is shown that any (restricted) semidirect product of an MV-algebra with a BCK-algebra satisfying (L) is bounded such that double negation is an endomorphism.