| Version 3 |
Version 2 |
| 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. |
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. |
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| \begin{thm} If $f:A\to B$ is a homomorphism from algebras $A$ and $B$. Then $$A/\ker(f)\cong f(A).$$ |
\begin{thm} If $f:A\to B$ is a homomorphism from algebras $A$ and $B$. Then $$A/\ker(f)\cong f(A).$$ |
| \end{thm} |
\end{thm} |
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| \begin{thm} If $B\subseteq A$ are algebras and $\mathfrak{C}$ is a congruence on $A$, then $$B/\mathfrak{C}_B\cong B^{\mathfrak{C}}/\mathfrak{C},$$ |
\begin{thm} If $B\subseteq A$ are algebras and $\mathfrak{C}$ is a congruence on $A$, then $$B/\mathfrak{C}_B\cong B^{\mathfrak{C}}/\mathfrak{C},$$ |
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where $\mathfrak{C}_B$ is the congruence restricted to $B$, and $B^{\mathfrak{C}}$ is the extension of $B$ by $\mathfrak{C}$.
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where $\mathfrak{C}_B$ is the congruence restricted to $B$ and $B^{\mathfrak{C}}$ is the extension of $B$ by $\mathfrak{C}$.
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| \end{thm} |
\end{thm} |
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| \begin{thm} If $A$ is an algebra and $\mathfrak{C}\subseteq \mathfrak{D}$ are congruences on $A$. Then there is a unique homomorphism $f:A/\mathfrak{C}\to A/\mathfrak{D}$ such that |
\begin{thm} If $A$ is an algebra and $\mathfrak{C}\subseteq \mathfrak{D}$ are congruences on $A$. Then there is a unique homomorphism $f:A/\mathfrak{C}\to A/\mathfrak{D}$ such that |
| $$\xymatrix{ |
$$\xymatrix{ |
| & A \ar[dl]_{[\cdot]_{\mathfrak{C}}} \ar[dr]^{[\cdot]_{\mathfrak{D}}} & \\ |
& A \ar[dl]_{[\cdot]_{\mathfrak{C}}} \ar[dr]^{[\cdot]_{\mathfrak{D}}} & \\ |
| A/\mathfrak{C} \ar[rr]^f && A/\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). Furthermore, if $ker(f)=\mathfrak{D}/\mathfrak{C}$, then $\mathfrak{D}/\mathfrak{C}$ is a congruence on $A/\mathfrak{C}$ and $$(A/\mathfrak{C})/(\mathfrak{D}/\mathfrak{C})\cong A/\mathfrak{D},$$ where the isomorphism $f'$ is induced by $f$, in the equation $f=[\cdot]_{\mathfrak{D}/\mathfrak{C}}\circ f'$. |
where the downward pointing arrows are the natural projections of $A$ onto the quotient algebras (induced by the respective congruences). Furthermore, if $ker(f)=\mathfrak{D}/\mathfrak{C}$, then $\mathfrak{D}/\mathfrak{C}$ is a congruence on $A/\mathfrak{C}$ and $$(A/\mathfrak{C})/(\mathfrak{D}/\mathfrak{C})\cong A/\mathfrak{D},$$ where the isomorphism $f'$ is induced by $f$, in the equation $f=[\cdot]_{\mathfrak{D}/\mathfrak{C}}\circ f'$. |
| \end{thm} |
\end{thm} |
