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 refer to the morphism property as the homomorphism condition.

Also see

• Results about the morphism property can be found here.

Sources

• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.10$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): morphism
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): morphism