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It is clear that the model M (the family M) includes absolutely denumerable sets.
It was proved in [22] that for a relational structure with a finite signature, this happens just in case E is at most denumerable and can be divided into finitely many blocks such that every permutation of E which preserves each block is an automorphism of R.
Let R be the poset product of a k-element chain by a denumerable antichain.