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Homeisomorphism theorems on algebraic systems
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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 $f:A\to B$ is a homomorphism from algebras $A$ and $B$. Then
$A/\ker(f)\cong f(A).$ 
Theorem 2.
If $B\subseteq A$ are algebras and $\mathfrak{C}$ is a congruence on $A$, then
$B/\mathfrak{C}_{B}\cong B^{{\mathfrak{C}}}/\mathfrak{C},$ 
where $\mathfrak{C}_{B}$ is the congruence restricted to $B$, and $B^{{\mathfrak{C}}}$ is the extension of $B$ by $\mathfrak{C}$.
Theorem 3.
If $A$ is an algebra and $\mathfrak{C}\subseteq\mathfrak{D}$ are congruences on $A$. Then
1. there is a unique homomorphism $f:A/\mathfrak{C}\to A/\mathfrak{D}$ such that
$\xymatrix{&A\ar[dl]_{{[\cdot]_{{\mathfrak{C}}}}}\ar[dr]^{{[\cdot]_{{\mathfrak{% D}}}}}&\\ A/\mathfrak{C}\ar[rr]^{f}&&A/\mathfrak{D}}$ where the downward pointing arrows are the natural projections of $A$ onto the quotient algebras (induced by the respective congruences).
2. Furthermore, if $ker(f)=\mathfrak{D}/\mathfrak{C}$, then

$\mathfrak{D}/\mathfrak{C}$ is a congruence on $A/\mathfrak{C}$, and

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

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