I. Observe and . Ex, 5, the order is compatible but not strongly compatible. It can be shown that if ? is compatible, then A(B) is either empty or an interval

A. Moreover, ) if and only if ? is ?-compatible. It follows that for strongly compatible orders, all achievable families are nonempty Boolean lattices. Achievable families play the rôle of maximal chains. As it can be easily checked, total orders on singletons are permutations on N , and all these orders are strongly compatible

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