Subtyping can have a simple semantics

Consider a first order typed language, with semantics $S$ for expressions and types. Adding subtyping means that a partial order $<$; on types is defined and that the typing rules are extended to the effect that expression $e$ has type $t$ whenever $e$ has type $s$ and $s<t$ We show how to adapt the semantics $S$ in a simple set theoretic way, obtaining a semantics $S'$ that satisfies, in addition to some obvious requirements, also the property that: $S'~s$ is included in $S'~t$, whenever $s < t$