Abstract

Pseudoeffect algebras are non-commutative generalizations of effect algebras, which can serve as models of both quantum structures and non-commutative logics. The main contribution of this study is twofold. Firstly, we initiate an order-theoretic extension of pseudoeffect algebras, called partially ordered pseudoeffect algebras (abbreviated po-PEAs). Secondly, we investigate the fuzzy ideal theory of po-PEAs. In particular, we show that a fuzzy ideal in a po-PEA is finitely generated if and only if it is finitely valued, and every fuzzy ideal in a Noetherian po-PEA is finitely generated.

Key words: Pseudoeffect algebra, fuzzy set, fuzzy ideal, fuzzy logic.