We define α-convergence in an
abelian l-group as follows: The net (xi)i∈Iα-converges to x if x is the only element
such that x =∨i≧i0(xi∧ x) =∧i≧i0(xi∨ x) for every i0∈ I. In an Archimedean
l-group (xi) α-converges to x if and only if for every a and b the net (a ∨ xi) ∧ b
order-converges (in the ordinary sense) to (a ∨ x) ∧ b. In general α-convergence is
weaker than this latter condition and is considerably more natural in the
non-Archimedean case. The algebraic operations of an arbitrary abelian l-group G
are continuous relative to α-convergence. If G is completely distributive its
α-convergence derives from a Hausdorff group-topology. Three sufficient conditions
are given for the preservation of the α-convergence of an l-group G when it is
embedded in another l-group E. In an appendix, we formulate a necessary and
sufficient condition in order that an abstract sequential convergence derive from a
topology.
