Integrally Closed Residuated Lattices

Abstract

A residuated lattice is said to be integrally closed if it satisfies the quasiequations $$xy \le x \implies y \le {\mathrm {e}}$$ and $$yx \le ~x \implies y \le {\mathrm {e}}$$ , or equivalently, the equations $$x \backslash x \approx {\mathrm {e}}$$ and $$x /x \approx {\mathrm {e}}$$ . Every integral, cancellative, or divisible residuated lattice is integrally closed, and, conversely, every bounded integrally closed residuated lattice is integral. It is proved that the mapping $$a \mapsto (a \backslash {\mathrm {e}})\backslash {\mathrm {e}}$$ on any integrally closed residuated lattice is a homomorphism onto a lattice-ordered group. A Glivenko-style property is then established for varieties of integrally closed residuated lattices with respect to varieties of lattice-ordered groups, showing in particular that integrally closed residuated lattices form the largest variety of residuated lattices admitting this property with respect to lattice-ordered groups. The Glivenko property is used to obtain a sequent calculus admitting cut-elimination for the variety of integrally closed residuated lattices and to establish the decidability, indeed PSPACE-completenes, of its equational theory. Finally, these results are related to previous work on (pseudo) BCI-algebras, semi-integral residuated pomonoids, and Casari’s comparative logic.