User:CJKG/Eigenbox

a) ${\mathcal {M}}$ is called right-equiponent magma or R-equiponent magma if and only if $\exists M,\circ$ with $((M;\circ )={\mathcal {M}}\;\wedge \;\circ :M\times M\to M)$ such that

$\forall a\in M\,\exists b\in M\,\forall c\in M\;(c\circ a)\circ c=b$ (R-equiponency law).

b) ${\mathcal {M}}$ is called left-equiponent magma or L-equiponent magma if and only if $\exists M,\circ$ with $((M;\circ )={\mathcal {M}}\wedge \circ :M\times M\to M)$ such that

$\forall a\in M\,\exists b\in M\,\forall c\in M\;c\circ (a\circ c)=b$ (L-equiponency law).

Remark: $M=\emptyset$ is allowed.

c) ${\mathcal {M}}$ is called column-preserving magma if and only if ${\mathcal {M}}$ is a magma and $\exists M,\circ$ with $(M;\circ )={\mathcal {M}}$ such that

$\forall a,b\in M\;b\circ (b\circ a)=a$ (column-preserving law).

d) ${\mathcal {M}}$ is called row-preserving magma if and only if ${\mathcal {M}}$ is a magma and $\exists M,\circ$ with $(M;\circ )={\mathcal {M}}$ such that

$\forall a,b\in M\;(a\circ b)\circ b=a$ (row-preserving law).

e) A magma $(M;\circ )$ is called static $:\Leftrightarrow$

$\forall a\in M\,\exists b\in M\,\forall c\in M\;a\circ (b\circ c)=c$ (staticity law).