In this paper we indicate how compact languages containing the Magidor-Malitz quantifiers Qκn in different cardinalities can be amalgamated to yield more expressive, compact languages.
The language Lκ<ω, originally introduced by Magidor and Malitz , is a natural extension of the language L(Q) introduced by Mostowski and investigated by Fuhrken , , Keisler  and Vaught . Intuitively, Lκ<ω is first-order logic together with quantifiers Qκn (n ∈ ω) binding n free variables which express “there is a set X of cardinality κ such than any n distinct elements of X satisfy …”, or in other words, iff the relation on determined by φ contains an n-cube of cardinality κ. With these languages one can express a variety of combinatorial statements of the type considered by Erdös and his colleagues, as well as concepts in universal algebra which are beyond the scope of first-order logic. The model theory of Lκ<ω has been further developed by Badger , Magidor and Malitz  and Shelah .
We refer to a language as being < κ compact if, given any set of sentences Σ of the language, if Σ is finitely satisfiable and ∣Σ∣ < κ, then Σ has a model. The phrase countably compact is used in place of <ℵ1 compact.