Submagma (mathematica)

In mathematica, ${\displaystyle {\boldsymbol {N}}}$ es un submagma de ${\displaystyle {\boldsymbol {M}}}$ ${\displaystyle :\Leftrightarrow }$ ${\displaystyle {\boldsymbol {N}}\leq {\boldsymbol {M}}}$ ${\displaystyle :\Leftrightarrow }$ ${\displaystyle {\boldsymbol {M}}}$ es un magma ${\displaystyle \wedge }$ ${\displaystyle {\boldsymbol {N}}}$ es un magma ${\displaystyle \wedge }$ ${\displaystyle \exists M,N,\ast ,\diamond }$ con ${\displaystyle ((M;\ast )={\boldsymbol {M}}\;\wedge \;(N;\diamond )={\boldsymbol {N}})}$ tal que ${\displaystyle (N\subseteq M\;\wedge \;\forall a,b\in N\;a\diamond b=a\ast b)}$

In iste caso, ${\displaystyle {\boldsymbol {M}}}$ es un supermagma de ${\displaystyle {\boldsymbol {N}}}$: ${\displaystyle {\boldsymbol {M}}\geq {\boldsymbol {N}}}$.

${\displaystyle {\boldsymbol {N}}}$ es un submagma proprie de ${\displaystyle {\boldsymbol {M}}}$ ${\displaystyle :\Leftrightarrow }$ ${\displaystyle {\boldsymbol {N}}<{\boldsymbol {M}}}$ ${\displaystyle :\Leftrightarrow }$ ${\displaystyle {\boldsymbol {N}}\leq {\boldsymbol {M}}\;\wedge \;{\boldsymbol {N}}\neq {\boldsymbol {M}}}$.

Sia ${\displaystyle {\boldsymbol {M}}}$ un magma. ${\displaystyle {\boldsymbol {N}}}$ es un submagma trivial de ${\displaystyle {\boldsymbol {M}}}$ ${\displaystyle :\Leftrightarrow }$ (${\displaystyle {\boldsymbol {N}}}$ es le magma vacue) ${\displaystyle \vee }$ ${\displaystyle {\boldsymbol {N}}={\boldsymbol {M}}}$.