A representation of weak effect algebras

Abstract

ABSTRACT. Weak effect algebras were introduced by the author as a generalization of effect algebras and pseudoeffect algebras. It was shown that having a basic algebra, we can restrict its binary operation to orthogonal elements only and what we get is just a weak effect algebra. However, the converse construction is impossible due to the fact that the underlying poset of a basic algebra is a lattice which need not be true for weak effect algebras. Hence, we found a weaker structure than a basic algebra which can serve as a representation of a weak effect algebra. The concept of effect algebra was introduced by D. J. Foulis and M. K. Bennett [9] for the algebraic axiomatization of unsharp reasoning in quantum logics. A. Dvurečenskij and T. Vetterlein generalized this concept by dropping commutativity of the partial binary operation. Their structure is called a pseudoeffect algebra and was introduced in [6]. It was shown in [2] that the partial binary operation of an effect algebra can be extended into a total operation. The resulting structure is a so-called weak basic algebra. This weak basic algebra represents an effect algebra if and only if it satisfies a rather complex identity which in fact describes the condition saying that for orthogonal elements, the partial operation addition is associative and commutative (which need not be true for the total operation). Dropping this assumption, we can derive more general structures, so-called weak effect algebras (see [1]). Hence weak effect algebras form a generalization of effect algebras where also associativity is deleted.2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 06A06, 06A12, 06D35, 03G10, 03G12. K e y w o r d s: weak effect algebra, weak basic algebra, directoid, antitone involution.