Prime model

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In mathematics, and in particular model theory,^[1] a prime model is a model that is as simple as possible. Specifically, a model ${\displaystyle P}$ is prime if it admits an elementary embedding into any model ${\displaystyle M}$ to which it is elementarily equivalent (that is, into any model ${\displaystyle M}$ satisfying the same complete theory as ${\displaystyle P}$).