  This paper establishes model-theoretic properties of $\mathrm{FOE}^{\infty}$,
a variation of monadic first-order logic that features the generalised
quantifier $\exists^\infty$ (`there are infinitely many').
  We provide syntactically defined fragments of $\mathrm{FOE}^{\infty}$
characterising four different semantic properties of
$\mathrm{FOE}^{\infty}$-sentences: (1) being monotone and (2) (Scott)
continuous in a given set of monadic predicates; (3) having truth preserved
under taking submodels or (4) invariant under taking quotients. In each case,
we produce an effectively defined map that translates an arbitrary sentence
$\varphi$ to a sentence $\varphi^{p}$ belonging to the corresponding syntactic
fragment, with the property that $\varphi$ is equivalent to $\varphi^{p}$
precisely when it has the associated semantic property.
  Our methodology is first to provide these results in the simpler setting of
monadic first-order logic with ($\mathrm{FOE}$) and without ($\mathrm{FO}$)
equality, and then move to $\mathrm{FOE}^{\infty}$ by including the generalised
quantifier $\exists^\infty$ into the picture.
  As a corollary of our developments, we obtain that the four semantic
properties above are decidable for $\mathrm{FOE}^{\infty}$-sentences. Moreover,
our results are directly relevant to the characterisation of automata and
expressiveness modulo bisimilirity for variants of monadic second-order logic.
This application is developed in a companion paper.
