This paper establishes model-theoretic properties of ME∞, a variation of monadic first-order logic that features the generalised quantifier ∃ ∞ (‘there are infinitely many’). We will also prove analogous versions of these results in the simpler setting of monadic first-order logic with and without equality (ME and M, respectively). For each logic L∈ { M, ME, ME∞} we will show the following. We provide syntactically defined fragments of L characterising four different semantic properties of L-sentences: (1) being monotone and (2) (Scott) continuous in a given set of monadic predicates; (3) having truth preserved under taking submodels or (4) being truth invariant under taking quotients. In each case, we produce an effectively defined map that translates an arbitrary sentence φ to a sentence φp belonging to the corresponding syntactic fragment, with the property that φ is equivalent to φp precisely when it has the associated semantic property. As a corollary of our developments, we obtain that the four semantic properties above are decidable for L-sentences
