Epimorphism Preserves Properties

Theorem

Epimorphism Preserves Associativity

Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an epimorphism.

Let $\circ$ be an associative operation.

Then $*$ is also an associative operation.

Epimorphism Preserves Commutativity

Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an epimorphism.

Let $\circ$ be a commutative operation.

Then $*$ is also a commutative operation.

Epimorphism Preserves Distributivity

Let $\struct {R_1, +_1, \circ_1}$ and $\struct {R_2, +_2, \circ_2}$ be algebraic structures.

Let $\phi: R_1 \to R_2$ be an epimorphism.

If $\circ_1$ is left distributive over $+_1$, then $\circ_2$ is left distributive over $+_2$.

If $\circ_1$ is right distributive over $+_1$, then $\circ_2$ is right distributive over $+_2$.

Consequently, if $\circ_1$ is distributive over $+_1$, then $\circ_2$ is distributive over $+_2$.

That is, epimorphism preserves distributivity.

Epimorphism Preserves Identity

Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an epimorphism.

Let $\struct {S, \circ}$ have an identity element $e_S$.

Then $\struct {T, *}$ has the identity element $\map \phi {e_S}$.

Epimorphism Preserves Inverses

Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an epimorphism.

Let $\struct {S, \circ}$ have an identity $e_S$.

Let $x^{-1}$ be an inverse element of $x$ for $\circ$.

Then $\map \phi {x^{-1} }$ is an inverse element of $\map \phi x$ for $*$.

That is:

$\map \phi {x^{-1} } = \paren {\map \phi x}^{-1}$

Epimorphism Preserves Semigroups

Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an epimorphism.

Let $\struct {S, \circ}$ be a semigroup.

Then $\struct {T, *}$ is also a semigroup.

Epimorphism Preserves Groups

Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an epimorphism.

Let $\struct {S, \circ}$ be a group.

Then $\struct {T, *}$ is also a group.

Warning

Note that this result is applied to epimorphisms.

For a general homomorphism which is not surjective, nothing definite can be said about the behaviour of the elements of its codomain which are not part of its image.