Congruences and ideals in pseudoeffect algebras

Abstract

This paper is devoted to congruences and ideals in pseudoeffect algebras. Let I be a normal ideal in a pseudoeffect algebra E. We show that: (1) the relation ~ I induced by I is a congruence if and only if for every a?E, I? [0,a] is upper directed; (2) the relation ~ I induced by I is a strong congruence if and only if I is a normal weak Riesz ideal in a pseudoeffect algebra E. Moreover, we introduce a stronger concept of congruence--namely Riesz strong congruence--and we prove that, if I is a normal weak Riesz ideal in a pseudoeffect algebra E, then ~ I is a Riesz strong congruence and, conversely, if ~ is a Riesz strong congruence, then I = [0]~ is a normal weak Riesz ideal, and ~ I = ~.