Ideals and quotients in lattice ordered effect algebras

Abstract

 We show that a quotient of a lattice ordered effect algebra L with respect to a Riesz ideal I is linearly ordered if and only if I is a prime ideal, and the quotient is an MV-algebra if and only if I is an intersection of prime ideals. A generalization of the commutators in OMLs is defined in the frame of lattice ordered effect algebras, such that the quotient with respect to a Riesz ideal I is an MV-algebra if and only if I contains all generalized commutators. If L is an OML, generalized commutators coincide with the usual Marsden commutators.