Generalized pseudo-effect algebras (Q2784593)

Generalized pseudoeffect algebras (GPE-algebras, for short) are a common generalization of pseudoeffect algebras [the authors, Int. J. Theor. Phys. 40, 685--701 (2001; Zbl 0994.81008)] and generalized effect algebras [\textit{A. Dvurečenskij} and \textit{S. Pulmannová}, New trends in quantum structures. Dordrecht: Kluwer. Bratislava: Ister Science (2000; Zbl 0987.81005)]. Formally, a GPE-algebra is a partial algebra \((E,+,0)\), where \(+\) is a partial binary operation, \(0\) is a fixed element of \(E\), and the following axioms are fulfilled for all \(a,b,c \in E\): (GPE1) if either of the sums \((a+b)+c\) and \(a +(b+c)\) exists, then so does the other one, and both are equal, (GPE2) if \(a + b\) exists, then \(d + a = a + b = b + e\) for some \(d\) and \(e\), (GPE3) if \(a + b = a + c\) or \(b + a = c + a\), then \(b = c\), (GPE4) if \(a + b = 0\), then \(a = 0 = b\), (GPE5) \(a + 0 = a = 0 + a\) (in (GPE3)--(GPE5), an equation \(t_1 = t_2\) is to be read as ``both \(t_1\) and \(t_2\) exist and are equal''). Such an algebra arises from the positive cone \(G^+\) of a po-group \((G,+,0,\leq)\) as follows. Let \(G_0\) be a nonempty subset of \(G^+\) such that, for all \(a,b \in G_0\) with \(a \leq b\), \(a - b \in G_0\) and \(-b + a \in G_0\). If \(+_0\) is the restriction of \(+\) to \(G_0\), then \((G_0,+_0,0)\) is a GPE-algebra. Two conditions that are sufficient for a GPE-algebra \(E\) to be obtainable this way are presented: (A) \(E\) is a meet-semilattice and fulfills the weak Riesz decomposition property, and (B) \(E\) is directed and fulfills the commutational Riesz decomposition property. (These versions of the Riesz decomposition property are defined for GPE-algebras just in the same way as in pseudoeffect algebras.) Furthermore, some properties of the representing groups of GPE-algebras are studied. In particular, it is shown that the category of GPE-algebras satisfying (A) is equivalent to a certain category whose objects are the pairs of the kind \((G,G_0)\), where \(G\) is an \(\ell\)-group and \(G_0\) is a subset of \(G\) as described above.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00009].