User:CJKG/Eigenbox: Difference between revisions

Revision as of 16:36, 27 September 2014

(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

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

(row-preserving law).

A magma $(M;\circ )$ is termed

(5 a) right static

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

(right-staticity law).

(5 b) II static

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

(II-staticity law).

(5 c) III static

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

(III-staticity law).

(5 d) left static

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

(left-staticity law).

(5 e) V static

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

(V-staticity law).

(5 f) VI static

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

(VI-staticity law).

(5 g) VII static

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

(VII-staticity law).

(5 h) VIII static

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

(VIII-staticity law).

@@ Line 1: / Line 1: @@
-a) <math>\mathcal{M}</math> is termed '''right-equiponent magma''' or '''R-equiponent magma''' if and only if
+'''(1 a)''' <math>\mathcal{M}</math> is termed '''right-equiponent magma''' or '''R-equiponent magma''' if and only if
 <math>\exists M, \circ</math> with <math>((M; \circ) = \mathcal{M} \; \wedge \; \circ: M \times M \to M)</math> such that
-<math>\forall a \in M \, \exists b \in M \, \forall c \in M \; (c \circ a) \circ c = b</math> (''right-equiponency law'').
+: <math>\forall a, b \in M \; (b \circ a) \circ b = (a \circ a) \circ a</math> (''right-equiponency law'').
-b) <math>\mathcal{M}</math> is termed '''left-equiponent magma''' or '''L-equiponent magma''' if and only if
+Equivalent
+'''(1 b)''' <math>\exists M, \circ</math> with <math>((M; \circ) = \mathcal{M} \; \wedge \; \circ: M \times M \to M)</math> such that
+: <math>\forall a, b, c \in M \; (a \circ c) \circ a = (b \circ c) \circ b</math>
+Equivalent
+'''(1 c)''' <math>\exists M, \circ</math> with <math>((M; \circ) = \mathcal{M} \; \wedge \; \circ: M \times M \to M)</math> such that
+: <math>\forall a \in M \, \exists b \in M \, \forall c \in M \; (c \circ a) \circ c = b</math>
+'''(2 a)''' <math>\mathcal{M}</math> is termed '''left-equiponent magma''' or '''L-equiponent magma''' if and only if
 <math>\exists M, \circ</math> with <math>((M; \circ) = \mathcal{M} \; \wedge \; \circ: M \times M \to M)</math> such that
-<math>\forall a \in M \, \exists b \in M \, \forall c \in M \; c \circ (a \circ c) = b</math> (''left-equiponency law'').
+: <math>\forall a, b \in M \; b \circ (a \circ b) = a \circ (a \circ a)</math> (''left-equiponency law'').
+Equivalent
+'''(2 b)''' <math>\exists M, \circ</math> with <math>((M; \circ) = \mathcal{M} \; \wedge \; \circ: M \times M \to M)</math> such that
+: <math>\forall a, b c \in M \; a \circ (c \circ a) = b \circ (c \circ b)</math>
+Equivalent
+'''(2 c)''' <math>\exists M, \circ</math> with <math>((M; \circ) = \mathcal{M} \; \wedge \; \circ: M \times M \to M)</math> such that
+: <math>\forall a \in M \, \exists b \in M \, \forall c \in M \; c \circ (a \circ c) = b</math>
+Remark to 1 and 2: <math>M = \emptyset</math> is allowed.
-Remark: <math>M = \emptyset</math> is allowed.
-c) <math>\mathcal{M}</math> is termed '''column-preserving magma''' if and only if <math>\mathcal{M}</math> is a magma and
+'''(3)''' <math>\mathcal{M}</math> is termed '''column-preserving magma''' if and only if <math>\mathcal{M}</math> is a magma and
 <math>\exists M, \circ</math> with <math>(M; \circ) = \mathcal{M}</math> such that
-<math>\forall a, b \in M \; b \circ (b \circ a) = a</math> (''column-preserving law'').
+: <math>\forall a, b \in M \; b \circ (b \circ a) = a</math> (''column-preserving law'').
-d) <math>\mathcal{M}</math> is termed '''row-preserving magma''' if and only if <math>\mathcal{M}</math> is a magma and
+'''(4)''' <math>\mathcal{M}</math> is termed '''row-preserving magma''' if and only if <math>\mathcal{M}</math> is a magma and
 <math>\exists M, \circ</math> with <math>(M; \circ) = \mathcal{M}</math> such that
-<math>\forall a, b \in M \; (a \circ b) \circ b = a</math> (''row-preserving law'').
+: <math>\forall a, b \in M \; (a \circ b) \circ b = a</math> (''row-preserving law'').
 A magma <math>(M;\circ)</math> is termed
-: e)  '''right static''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; a \circ (b \circ c) = c</math> (''right-staticity law'').
+: '''(5 a)'''  '''right static''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; a \circ (b \circ c) = c</math> (''right-staticity law'').
-: f)  '''II static''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; (b \circ c) \circ a = c</math> (''II-staticity law'').
+: '''(5 b)'''  '''II static''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; (b \circ c) \circ a = c</math> (''II-staticity law'').
-: g)  '''III static''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; a \circ (c \circ b) = c</math> (''III-staticity law'').
+: '''(5 c)'''  '''III static''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; a \circ (c \circ b) = c</math> (''III-staticity law'').
-: h)  '''left static''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; (c \circ b) \circ a = c</math> (''left-staticity law'').
+: '''(5 d)'''  '''left static''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; (c \circ b) \circ a = c</math> (''left-staticity law'').
-: i)  '''V static''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; b \circ (a \circ c) = c</math> (''V-staticity law'').
+: '''(5 e)'''  '''V static''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; b \circ (a \circ c) = c</math> (''V-staticity law'').
-: j)  '''VI static''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; (a \circ c) \circ b = c</math> (''VI-staticity law'').
+: '''(5 f)'''  '''VI static''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; (a \circ c) \circ b = c</math> (''VI-staticity law'').
-: k)  '''VII static''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; b \circ (c \circ a) = c</math> (''VII-staticity law'').
+: '''(5 g)'''  '''VII static''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; b \circ (c \circ a) = c</math> (''VII-staticity law'').
-: l)  '''VIII static''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; (c \circ a) \circ b = c</math> (''VIII-staticity law'').
+: '''(5 h)'''  '''VIII static''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; (c \circ a) \circ b = c</math> (''VIII-staticity law'').