# Definition:Morphism Property

## Definition

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

Then $\circ$ has the morphism property under $\phi$ if and only if:

$\forall x, y \in S: \map \phi {x \circ y} = \map \phi x * \map \phi y$

## Also known as

Some sources call this property the homomorphism condition.