Assuming OCA and Martin's axiom for sigma-linked posets, all isomorphisms between metric reduced products of symmetric groups are trivial, and the outer automorphism group is described explicitly.
Dubuc and Daniele Mundici , Extending Stone duality to multisets and locally finite MV-algebras , Journal of Pure and Applied Algebra 189 (2004), no
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On automorphism groups of metric reduced products of symmetric groups
Assuming OCA and Martin's axiom for sigma-linked posets, all isomorphisms between metric reduced products of symmetric groups are trivial, and the outer automorphism group is described explicitly.