Lattice-ordered effect algebras and L-algebras

Abstract

It is known that many logical algebras can be considered as L-algebras. In this paper, we first construct an effect algebra (E,⊕,0,1) that cannot be converted into an L-algebra (E,→,0,1) with negation such that x→0 is the orthosupplement. Then we prove that every lattice-ordered effect algebra gives rise to an L-algebra with the same orthosupplement. Finally, we characterize LE-L-algebras as lattice-ordered effect algebras. As an application, we extend those results obtained in some recent references for orthomodular lattices and MV-algebras.