## 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

### MSC:

 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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### References:

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