= Universal theory and substructures of elementarily equivalent models
{title2=$T\cup\operatorname{Th}_\forall(A)\text{ satisfiable}\iff B\models T,\ B\subseteq A^*\equiv A$}
If $B$ satisfies the universal <first-order theory>[theory] of $A$, every <existential sentence> true in $B$ is true in $A$, since its falsehood in $A$ would supply a universal <negation>. Apply the <existential-theory embedding criterion> to $B$ and the <complete theory> of $A$ to obtain $A^*$. Conversely universal truth descends to <substructure of a first-order structure>[substructures]. The <elementarily equivalent> <first-order model>[model] need not be the original structure itself.
Back to article page
