Implicative algebras and Heyting algebras can be residuated lattices

Abstract

The commutative residuated lattices were first introduced by M. Ward and R.P. Dilworth as generalization of ideal lattices of rings. Complete studies on residuated lattices were developed by H. Ono, T. Kowalski, P. Jipsen and C. Tsinakis. Also, the concept of lattice implication algebra is due to Y. Xu. And Luitzen Brouwer founded the mathematical philosophy of intuitionism, which believed that a statement could only be demonstrated by direct proof. Arend Heyting, a student of Brouwer’s, formalized this thinking into his namesake algebras. In this paper, we investigate the relationship between implicative algebras, Heyting algebras and residuated lattices. In fact, we show that implicative algebras and Heyting algebras can be described as residuated lattices.