Quotients and weakly algebraic sets in pseudoeffect algebras

In the paper, we show that the quotient E]_IE]_I of a lattice-ordered pseudoeffect algebra EE with respect to a normal weak Riesz ideal II is linearly ordered if and only if II is a prime normal weak Riesz ideal, and E]_IE]_I is a representable pseudo MV-algebra if and only if II is an intersection of prime normal weak Riesz ideals. Moreover, we introduce the concept of weakly algebraic sets in pseudoeffect algebras, discuss the characterizations of weakly algebraic sets and show that weakly algebraic sets in pseudoeffect algebra EE are in a one-to-one correspondence with normal weak Riesz ideals in pseudoeffect algebra E.E.