Definition:Morphism Property

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 $$\phi$$.