Unordered analog of chain in lattice theory

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In lattice theory, a chain is a fully ordered subset of a lattice, corresponding to a vertical path in a Hasse diagram. What is a fully unordered subset called, corresponding to a horizontal row in a Hasse diagram? Or is it just called an equivalence class?

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A set of pairwise incomparable elements of a partially ordered set is called an antichain or a Sperner family or a clutter; see the Wikipedia page "Antichain" or "Sperner family". (Note, however, that the term "antichain" is also used by some to mean "strong antichain.)