Elementary equivalence is an important concept in mathematical logic. Two models $\mathfrak{M}$ and $\mathfrak{N}$ of the same signature are elementarily equivalent, written $\mathfrak{M} \equiv \mathfrak{N}$, provided they make true the same formulae: $$ \mathfrak{M} \equiv \mathfrak{N} \qquad\text{iff}\qquad \{\phi\ |\ \mathfrak{M} \models \phi\} = \{\phi\ |\ \mathfrak{N} \models \phi\} $$
Obviously, one can also define an asymmetric version of this: $$ \mathfrak{M} \preceq \mathfrak{N} \qquad\text{iff}\qquad \{\phi\ |\ \mathfrak{M} \models \phi\} \subseteq \{\phi\ |\ \mathfrak{N} \models \phi\} $$ Now my question: what is the name of this partial order $\preceq$ in the literature and has it been discussed anywhere?
Edit following FPE's suggestions below: $\preceq$ is not terribly interesting a relation, because in classical logic, $\mathfrak{M} \preceq \mathfrak{N}$ already implies that $\mathfrak{M} \equiv \mathfrak{N}$. However I'm investigating a logic without negation where $\{\phi\ |\ \mathfrak{M} \models \phi\}$ is not a complete theory.
Suppose that $\mathcal M\preceq \mathcal N$ (with your definition, I'm not meaning elementary substructure), and let $\phi$ be such that $\mathcal N\models \phi$. Then $\mathcal N\not\models \neg\phi$, i.e. by assumption $\mathcal M\not\models \neg\phi$, which means that $\mathcal M\models \phi$! Hence taking your definition, we have that $\mathcal M\preceq \mathcal N$ implies $\mathcal M\equiv \mathcal N$. This comes from the fact that the theory $T := \mathrm{Th}(\mathcal M)$ of a structure is a complete theory, therefore any theory that contains $T$ is in fact equal to it.