×

Pseudoeffect algebras. I: Basic properties. (English) Zbl 0994.81008

Pseudoeffect algebras are a non-commutative generalization of effect algebras. More exactly, a structure \((E,+,0,1)\) with \(+\) a partial binary operation on \(E\) and \(0,1\) constants is said to be a pseudoeffect algebra (pe.a., for short) if it satisfies four axioms: (E1) if one of the sums \((a+b)+c\) and \(a+(b+c)\) is defined, then so is the other, and both are equal, (E2) to every \(a\), there is exactly one \(d\) and exactly one \(e\) such that \(a+d=1=e+a\), (E3) if \(a+b\) is defined, then \(a+b=d+a=b+e\) for some \(d\) and \(e\), (E4) if either \(1+a\) or \(a+1\) exists, then \(a=0\). Such a structure is an effect algebra iff \(a+b\) and \(b+a\) are either both undefined or both defined and equal. Just as in effect algebras, the relation \(\leq\) on \(E\) is defined by \(a \leq b\) iff \(a+c=b\) for some \(c\) iff \(d+a=b\) for some \(d\) is a partial order.
A characteristic example of a pe.a.is provided by the unit interval \(\Gamma(G,u) := \{g \in G^+: g \leq u\}\) of an additive po-group \(G\) with a strong unit \(u\) (i.e., an element \(u\) such that for all \(g \in G\) \(-nu \leq g \leq nu\) for some natural \(n\); such a po-group is said to be unital). The algebra \((\Gamma(G,u),+,0,u)\), where \(+\) is the restriction of the \(G\)-addition to \(\Gamma(G,u)\), is a pe.a., and every algebra isomorphic to an algebra of this kind is said to be an interval pe.a.
In this first part of the paper, several kinds of Riesz type properties for pseudoeffect algebras are introduced. For example, \(E\) is said to have the Commutational Riesz Decomposition Property (RDP\(_1\)) if, for any \(a_1,a_2,b_1,b_2\) such that \(a_1+a_2=b_1+b_2\), there are \(d_1,d_2,d_3,d_4\) such that (i) \(d_1+d_2=a_1\), \(d_3+d_4=a_2\), \(d_1+d_3=b_1\), \(d_2+d_4=b_2\), (ii) \(d'\) and \(d''\) commute in \(E\) forall \(d' \leq d_2\) and \(d'' \leq d_3\). Interconnections between these properties are studied; in particular, they are shown to be lineraly ordered by strength. Similar properties are considered for additive po-groups. Thus, a directed po-group \(G\) is said to have RDP\(_1\) if it satisfies the above condition with the restriction that all \(a_i, b_j, d_k\) are non-negative.
For Part II see [ibid. 40, No. 3, 703–726 (2001; Zbl 0994.81009)].

MSC:

81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
06F15 Ordered groups

Citations:

Zbl 0994.81009
PDFBibTeX XMLCite
Full Text: DOI