Cardinality of the system of all sequential convergences on an Abelian lattice ordered group. (English) Zbl 0645.06006

The author investigates the number of convergence structures which can be defined on a given abelian lattice-ordered group G. Let \(L_ G\) denote the set of all “compatible” convergence structures which can be introduced on G (a list of 9 axioms defines a convergence on G). One defines \(b(G)=\sup \{card(A):\) A is an orthogonal subset of \(G\}\). The main results of the paper are: (1) If \(b(G)\geq \aleph_ 0\), then \(card(L_ G)\geq 2^{2^{\aleph_ 0}}\), and this estimate is sharp. (2) If b(G) is the integer n, then there is an integer k with \(0\leq k\leq n\) such that \(card(L_ G)=2^ k\). (3) Suppose integers n and k satisfy \(0\leq k\leq n\). Then there exists a proper class \(\{\) G(i): i in \(I\}\) of nonisomorphic abelian lattice-ordered groups G(i) such that for each i in I the relations \(b(G(i))=n\) and card \(L_{G(i)}=2^ k\) are valid.
Reviewer: S.P.Hurd


06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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