Commit f718108b authored by Xin's avatar Xin

add 10, validated

parent be9e1563
\begin{mhmodnl}[creators=Xin]{homomorphism}{zhs}
\begin{definition}
$\defeq{\cA}{\mvstructure{A,o^A_1,\ldots,o^A_n}}$$\mvstructure{B,o^B_1,\ldots,o^B_n}$为两个相同类型的\mtrefi[algebraic-structure?algebraic-structure]{代数结构}并且$\fun{f}AB$\mtrefi[algebraic-structure?carrier]{承载子}间的\mtrefi[functions?function]{函数}。则称$f$\defi[name=preserves]{保存}一个\mtrefi[algebraic-structure?operation]{运算}$o$,当对于所有$\minsetli{a}1nA$$\nappa{f}{\nappli{o^A}a1n}=\nappa{o^B}{\nappa{f}{a_1},\ldots,\nappa{f}{a_n}}$。称$f$为一个\defi[name=homomorphism]{同态},当且仅当其保留了$\cA$中所有的运算。
\end{definition}
\end{mhmodnl}
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\begin{mhmodnl}[creators=Xin]{ideal}{zhs}
\begin{definition}[id=ideal.def]
\impdec{$\defeq\cR{\mvstructure{\rbaseset,\raddOp,\rzero,\rnegOp,\rmulOp,\rone}}$为一个带有\mtrefi[ring?additive-structure]{加法结构}\mtrefi[ring?ring]{}}$\cA$
一个$I$\mtrefi[subgroup?subgroup]{子群}被称为$\cR$\defi[name=right-ideal]{右理想},当其包含$\rbaseset$在右侧的\mtrefi[ring?ring-multiplication]{乘法},即对于所有$\inset{x}I$$\inset{r}R$$\inset{\rmul{x,r}}{I}$
类似地称$I$为一个$\cR$\defi[name=left-ideal]{左理想},当其包含$\rbaseset$在左侧的\mtrefi[ring?ring-multiplication]{乘法},即对于所有$\inset{x}I$$\inset{r}R$$\inset{\rmul{r,x}}{I}$
称一个\mtrefi[subsupset?subset]{子集}$I$$\cR$\defi[name=ideal]{双边理想}(通常简称为\defi[name=ideal]{理想}),当其既是左理想又是右理想。
\end{definition}
\end{mhmodnl}
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\begin{mhmodnl}[creators=Xin]{idempotent}{zhs}
\begin{definition}
$\defeq\cM{\mvstructure{\magmaset,\magmaopOp}}$为一个\mtrefi[magma?magma]{原群},称$\magmaopOp$(与$\cM$自身)\defi[name=idempotent]{幂等},当对于所有$\inset{a}\magmaset$$\magmaop{a}a=a$
\end{definition}
\end{mhmodnl}
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\begin{mhmodnl}[creators=Xin]{independence-system}{zhs}
\begin{definition}
一个\defi[name=independence-system]{幂等系统}$\defeq\cM{\mvstructure{X,\cI}}$是一个\mtrefi[finite-cardinality?finite]{有穷}集合$X$外加一个$X$上的\mtrefi[set-system?set-system]{集合族} $\sseteq\cI{\powerset{X}}$,使得
\begin{enumerate}
\item $\inset\eset\cI$
\item 对于所有$\inset{A}\cI$$\sseteq{A'}A$,则$\inset{A'}\cI$\defi[name=hereditary?property]{遗传性质})。
\end{enumerate}
$\cI$中的集合\defi[name=independent]{无关}$\cM$并且$X$不包含于$\cI$中的子集\defi[name=dependent]{相关}
\end{definition}
\end{mhmodnl}
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\begin{mhmodnl}[creators=Xin]{integral-domain}{zhs}
\begin{definition}
一个\defi[name=integral-domain]{整环}是一个没有\mtrefi[zero-divisor?zero-divisor]{零因子}\mtrefi[commutative]{交换环}
\end{definition}
\end{mhmodnl}
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\begin{mhmodnl}[creators=cdemirkiran,contributors=miko]{irreduciblepolynomial}{de}
\begin{definition}[id=irreduciblepolynomial.def]
Ein \mtrefi[polynomial?polynomial]{Polynom} $p$ hei"st \defi[name=irreducible]{irreduzibel},
wenn alle siner \mtrefi[polynomialring?factor]{Faktoren}
wenn alle seiner \mtrefi[polynomialring?factor]{Faktoren}
\mtrefi[polynomial?degree]{Grad} $0$ oder $\degree{p}$ haben.
\end{definition}
\end{mhmodnl}
......
\begin{mhmodnl}[creators=Xin]{irreduciblepolynomial}{zhs}
\begin{definition}[id=irreduciblepolynomial.def]
称一个\mtrefi[polynomial?polynomial]{多项式}$p$\defi[name=irreducible]{不可约},当其所有的\mtrefi[polynomialring?factor]{因数}都为$0$$\degree{p}$\mtrefi[polynomial?degree]{}
\end{definition}
\end{mhmodnl}
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\begin{mhmodnl}[creators=Xin]{lattice}{zhs}
\begin{definition}
\defi[name=lattice]{}是一个三元$\mvstructure{S,\joinOp,\meetOp}$,其中$\mvstructure{S,\joinOp}$$\mvstructure{S,\meetOp}$\mtrefi[semi-lattice?semi-lattice]{半格}并且满足两个\defi[name=absorption-law]{吸收律}
$\meet{a,\join{a,b}}=a$$\join{a,\meet{a,b}}=a$
\end{definition}
\end{mhmodnl}
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\begin{mhmodnl}[creators=Xin]{leadingcoefficient}{zhs}
\begin{definition}[id=leadingcoefficient.def]
$\defeq{p}{\genpoly{c}1nx}$为一个\mtrefi[polynomial?polynomial]{多项式},则称$\seqsel{c}n$$p$\defi[name=leading-coefficient]{首项系数}若其不为零。
\end{definition}
\end{mhmodnl}
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\begin{mhmodnl}[creators=Xin]{loop}{zhs}
\begin{definition}[id=loop.def]
\defi[name=loop]{幺拟群}(或\defi[name=loop]{})是一个\mtrefi[unital?unital]{有幺元的}\mtrefi[quasigroup?quasigroup]{拟群}.
\end{definition}
\end{mhmodnl}
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\begin{mhmodnl}[creators=Xin]{magma}{zhs}
\begin{definition}
称由一个\mtrefi[set?set]{集合}$\magmaset$\defi[name=base-set]{基集})和一个二元\mtrefi[functions?function]{函数}$\fun\magmaopOp{\magmaset,\magmaset}\magmaset$\defi[name=operation]{运算})组成的代数结构$\mvstructure{\magmaset,\magmaopOp}$\defi[name=magma]{原群}
\end{definition}
\end{mhmodnl}
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\begin{mhmodnl}[creators=Xin]{matroid}{zhs}
\begin{definition}
\defi[name=matroid]{拟阵}是一个\mtrefi[independence-system?independence-system]{独立系统}
$\defeq\cM{\mvstructure{X,\cI}}$,满足:
\begin{enumerate}
\item$\minset{A,B}\cI$ and $\natmorethan{\card{A}}{\card{B}}$,则有
$\inset{x}{\setdiff{A}B}$使得$\inset{\union{B,\set{x}}}\cI$
\defi[name=augmentation-property]{扩充特性}
\defi[name=augmentation-property]{独立集交换特性})。
\end{enumerate}
\end{definition}
\ednote{MK: REALM alarm - this is the definition Wikipedia uses, there are at least two
more, in Mathworld.}
\end{mhmodnl}
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