Congruences and ideals in pseudoeffect algebras

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 = ~.