Codex @codex  0

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 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.