User:CJKG/Eigenbox

(1 a) ${\mathcal {M}}$ is termed 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,b\in M\;(b\circ a)\circ b=(a\circ a)\circ a

(right-equiponency law).

Equivalent

(1 b) $\exists M,\circ$ with $((M;\circ )={\mathcal {M}}\;\wedge \;\circ :M\times M\to M)$ such that

\forall a,b,c\in M\;(a\circ c)\circ a=(b\circ c)\circ b

Equivalent

(1 c) $\exists M,\circ$ 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

(2 a) ${\mathcal {M}}$ is termed 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,b\in M\;b\circ (a\circ b)=a\circ (a\circ a)

(left-equiponency law).

Equivalent

(2 b) $\exists M,\circ$ with $((M;\circ )={\mathcal {M}}\;\wedge \;\circ :M\times M\to M)$ such that

\forall a,bc\in M\;a\circ (c\circ a)=b\circ (c\circ b)

Equivalent

(2 c) $\exists M,\circ$ 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

Remark to 1 and 2: $M=\emptyset$ is allowed.

(3) ${\mathcal {M}}$ is termed 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).

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