User:CJKG/Eigenbox: Difference between revisions

Latest revision as of 12:27, 10 April 2016

(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).

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

(5 a) right static

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

(right-staticity law).

(5 b) left static

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

(left-staticity law).

(5 c) RC static

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

.

(5 d) LC static

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

.

(5 e) RT static

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

.

(5 f) LT static

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

.

(5 g) RCT static

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

.

(5 h) LCT static

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

.

(6 a) An REV-quasigroup is a right-equiponent and column-preserving magma.

(6 b) An LEH-quasigroup is a left-equiponent ans row-preserving magma.

(6 c) An AS-quasigroup is an abelian and (right or II or III or left or V or VI or VII or VIII) static magma.

Reference: Creation of technical terms in algebra and especially in koniology

@@ Line 1: / Line 1: @@
-a) <math>\mathcal{M}</math> is called '''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>\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> (''R-equiponency law'').
-b) <math>\mathcal{M}</math> is called '''left-equiponent magma''' or '''L-equiponent magma''' if and only if
+: <math>\forall a, b \in M \; (b \circ a) \circ b = (a \circ a) \circ a</math> (''right-equiponency law'').
-<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> (''L-equiponency law'').
-Remark: <math>M = \emptyset</math> is allowed.
+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, 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.
+'''(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'').
+'''(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'').
+'''(5)''' A magma <math>(M;\circ)</math> is termed
+: '''(5 a)'''  '''''right static''''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; (c \circ b) \circ a = c</math> (''right-staticity law'').
+: '''(5 b)'''  '''''left static''''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; a \circ (b \circ c) = c</math> (''left-staticity law'').
+: '''(5 c)'''  '''''RC static''''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; (b \circ c) \circ a = c</math>.
+: '''(5 d)'''  '''''LC static''''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; a \circ (c \circ b) = c</math>.
+: '''(5 e)'''  '''''RT static''''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; (c \circ a) \circ b = c</math>.
+: '''(5 f)'''  '''''LT static''''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; b \circ (a \circ c) = c</math>.
+: '''(5 g)'''  '''''RCT static''''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; (a \circ c) \circ b = c</math>.
+: '''(5 h)'''  '''''LCT static''''' <math>: \Leftrightarrow \forall a \in M \, \exists b \in M \, \forall c \in M \; b \circ (c \circ a) = c</math>.
+'''(6 a)''' An '''''REV-quasigroup''''' is a right-equiponent and column-preserving magma.
+'''(6 b)''' An '''''LEH-quasigroup''''' is a left-equiponent ans row-preserving magma.
+'''(6 c)''' An '''''AS-quasigroup''''' is an abelian and (right or II or III or left or V or VI or VII or VIII) static magma.
+'''Reference:''' [http://www.interlinguistik-gil.de/wb/media/beihefte/21/beiheft21-killing_guenkel.pdf ''Creation of technical terms in algebra and especially in koniology'']