= Solution
This class is not axiomatisable. If $T$ axiomatized it, add constants $m_i$ and $n_i$ for $i\in\mathbb N$, requiring the $m_i$ to be distinct maximal elements and the $n_i$ to be distinct <minimal element of a partially ordered set>[minimal elements]. Every finite fragment has a model of $T$: make one of the two sets infinite and make the other finite but large enough to interpret the finitely many constants mentioned. Compactness would then give a model of $T$ having both infinitely many maximal and infinitely many minimal elements, contrary to the exclusive condition. This is the <compactness obstruction to an exclusive disjunction of infinitude>.
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