Elementary equivalent pairs of algebras associated with sets

The elementary equivalence of two full relation algebras, partition lattices or function monoids are shown to be equivalent to the second order equivalence of the cardinalities of the corresponding sets. This is shown to be related to elementary equivalence of permutation groups and ordinals. Infinite function monoids are shown to be ultrauniversal.Presented by Walter Taylor.The work of the second author was supported by a grant from the University of Cape Town Research Committee, and by the Topology Research Group from the University of Cape Town and the South African Council for Scientific and Industrial Research.