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 similar to (I) induced by I is a congruence if and only if for every a is an element of E, I boolean AND [0,a] is upper directed; (2) the relation similar to (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 similar to (I) is a Riesz strong congruence and, conversely, if similar to is a Riesz strong congruence, then I = [0](similar to) is a normal weak Riesz ideal, and similar to(I) = similar to.
congruences ideals pseudoeffect algebras strong congruences normal weak riesz ideals riesz ideals riesz strong congruences