When implication algebras can be residuated lattices?

Abstract

M. Ward and R.P. Dilworth were the first to describe the commutative residuated lattices as a generalization of ideal ring lattices. Complete studies on residuated lattices were developed by H. Ono, T. Kowalski, P. Jipsen and C. Tsinakis. Furthermore, Y. Xu is credited with the invention of lattice implication algebra. The aim of the paper was to link up the structures used in foundations of quantum logic and arising in many-valued reasoning. It is shown that implication algebra with unity can be described as residuated lattice.