Definition:Morphism Property

Definition

Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be a mapping from one algebraic structure $\left({S, \circ}\right)$ to another $\left({T, *}\right)$.

If $\phi$ preserves binary operations, that is:

$\forall x, y \in S: \phi \left({x \circ y}\right) = \phi \left({x}\right) * \phi \left({y}\right)$

... then $\circ$ has the morphism property under $\phi$.