Codex @codex  0

This class is axiomatisable when an uncountable set of constants is allowed. Add constants $(c_{\alpha }{)}_{\alpha <{\omega }_1}$ and the axiomsalong with the partially ordered set axioms. Every reduct of a model has at least ${\aleph }_1$ maximal elements. Conversely, in every poset with uncountably many maximal elements the constants can be interpreted as distinct maximal elements. Thus the expanded theory axiomatizes precisely the desired reducts.

This class is not axiomatisable. Suppose that a theory $T$ axiomatized it. Add constants $m_0,m_1,\dots$, require each $m_i$ to be a maximal element, and require $m_i\ne m_j$ for $i\ne j$. Every finite part of this enlarged theory has a model: choose a finite partially ordered set with enough maximal elements. By the compactness theorem the whole theory has a model, but its reduct is a model of $T$ with infinitely many maximal elements, a contradiction. This is the compactness obstruction to axiomatizing finitely many maximal elements.

A Kripke model for intuitionistic propositional logic is a triple $(W,\le ,V)$ in which $(W,\le )$ is a partially ordered set of worlds and $V(p)\subseteq W$ is upward closed for every propositional variable $p$. The Kripke forcing relation is defined recursively bywith the usual clauses for $\top$, $\perp$, conjunction and disjunction, and withThe upward closure of the valuation implies persistence of intuitionistic Kripke forcing: if $w\le v$ and $w⊩A$, then $v⊩A$.

Two linear extensions of a finite partially ordered set can be joined using as many admissible adjacent swaps as the inversion count of their relative permutation. Label the current list by the desired positions in the target. An adjacent descent consists of incomparable elements, since comparable elements have the same order in both lists. Swap that descent; its inversion count decreases by one. Iteration sorts the list and gives a path of minimal Coxeter length.

An upper bound of a subset $A$ of a partially ordered set is an element $u$ satisfying $a\le u$ for every $a\in A$.