Residuated lattices and lattice effect algebras

Residuated lattices and lattice effect algebras arose in two rather different fields. In this paper, by introducing two partial operations in effect algebras, we investigate the mutual relationship between involutive residuated lattices and lattice effect algebras. We prove that a lattice effect algebra under certain conditions can be extended to an involutive residuated lattice and the latter with certain properties can be restricted to the former. Especially, an sufficient and necessary condition for an involutive residuated lattice to be a lattice effect algebra with the Riesz decomposition property is obtained.