isomorphism theorems on algebraic systems
In this entry, all algebraic systems are of the same type; they are all $O$algebras^{}. We list the generalizations^{} of three famous isomorphism theorems, familiar to those who have studied abstract algebra in college.
Theorem 1.
If $f\mathrm{:}A\mathrm{\to}B$ is a homomorphism^{} from algebras $A$ and $B$. Then
$$A/\mathrm{ker}(f)\cong f(A).$$ 
Theorem 2.
If $B\mathrm{\subseteq}A$ are algebras and $\mathrm{C}$ is a congruence^{} (http://planetmath.org/CongruenceRelationOnAnAlgebraicSystem) on $A$, then
$$B/{\u212d}_{B}\cong {B}^{\u212d}/\u212d,$$ 
where ${\mathrm{C}}_{B}$ is the congruence restricted to $B$, and ${B}^{\mathrm{C}}$ is the extension^{} of $B$ by $\mathrm{C}$.
Theorem 3.
If $A$ is an algebra and $\mathrm{C}\mathrm{\subseteq}\mathrm{D}$ are congruences on $A$. Then

1.
there is a unique homomorphism $f:A/\u212d\to A/\U0001d507$ such that
$$\text{xymatrix}\mathrm{\&}A\text{ar}{[dl]}_{{[\cdot ]}_{\u212d}}\text{ar}{[dr]}^{{[\cdot ]}_{\U0001d507}}\mathrm{\&}A/\u212d\text{ar}{[rr]}^{f}\mathrm{\&}\mathrm{\&}A/\U0001d507$$ where the downward pointing arrows are the natural projections^{} of $A$ onto the quotient algebras (induced by the respective congruences).

2.
Furthermore, if $ker(f)=\U0001d507/\u212d$, then

–
$\U0001d507/\u212d$ is a congruence on $A/\u212d$, and

–
there is a unique isomorphism^{} ${f}^{\prime}:A/\u212d\to (A/\u212d)/(\U0001d507/\u212d)$ satisfying the equation $f={[\cdot ]}_{\U0001d507/\u212d}\circ {f}^{\prime}$. In other words,
$$(A/\u212d)/(\U0001d507/\u212d)\cong A/\U0001d507.$$

–
Title  isomorphism theorems on algebraic systems 

Canonical name  IsomorphismTheoremsOnAlgebraicSystems 
Date of creation  20130322 16:45:28 
Last modified on  20130322 16:45:28 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Theorem 
Classification  msc 08A05 