Category:Epimorphism Preserves Properties
This category contains pages concerning Epimorphism Preserves Properties:
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.
Pages in category "Epimorphism Preserves Properties"
The following 2 pages are in this category, out of 2 total.