A denotational semantic for the lambda calculus can be given by solving the domain equation
$$ D \simeq [D →_c D]_\bot $$ in the category of $\omega$-complete CPOs, where $→_c$ denotes the space of $\omega$-continuous functions. The existence of an (initial) solution to this domain equation is a classic result by Scott (or Plotkin?).
One reason for the restriction to continuous functions here is that the equation $$ D \simeq [D → D]_\bot $$ (with $->$ denoting all functions) doesn’t have a solution, due to cardinality constraints.
But what about the middle ground where we allow monotonic, but possibly non-continuous functions: $$ D \simeq [D →_m D]_\bot $$ Does this have a solution in the category of partial ordered sets with monotone functions?
(Or maybe in the category of CPOs, but still allowing non-continuous functions in $\to_m$)
