05/14/2014, 03:51 PM
first attempt to study the relation to associativity...
if \( * \) is associative then \( L_a^{\circ n}=L_{a^n} \), where \( a^n \) is defined as usual \( a^1:=a \) and \( a^{n+1}=a*a^n \)
but if it is not associative then the existence of an injection maybe depend on the existence of a binary function \( \cdot:G^2\rightarrow G \) witht his property
\( a*(b*x)=(a\cdot b)*x \)
or in other words
\( L_a \circ L_b=L_{a\cdot b} \)
someone knows when such operation can exist and how is called this property? When \( * \) is right invertible then \( a\cdot b=a*(b*x)\setminus_* x \) but I don't know when this operation becomes indipendent from \( x \)
if \( * \) is associative then \( L_a^{\circ n}=L_{a^n} \), where \( a^n \) is defined as usual \( a^1:=a \) and \( a^{n+1}=a*a^n \)
but if it is not associative then the existence of an injection maybe depend on the existence of a binary function \( \cdot:G^2\rightarrow G \) witht his property
\( a*(b*x)=(a\cdot b)*x \)
or in other words
\( L_a \circ L_b=L_{a\cdot b} \)
someone knows when such operation can exist and how is called this property? When \( * \) is right invertible then \( a\cdot b=a*(b*x)\setminus_* x \) but I don't know when this operation becomes indipendent from \( x \)
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)
