Here we sum over all decompositions <math>c_d \otimes \ldots \otimes c_1 = (c''= c_d \otimes \ldots \otimes c_{m+n+1}) \otimes (c'= c_{m+n} \otimes \ldots \otimes c_{n+1}) \otimes (c=  c_n \otimes \ldots \otimes c_1 ) </math> into pure tensors of lengths <math>d-m-n,m,n\geq 0</math>, in particular allow <math>c'=1 \;\; {\scriptstyle (\text{or}\; c=1 \;\text{or}\; c''=1 )} </math> to have length <math>|c'|=0</math>.
 
Here we sum over all decompositions <math>c_d \otimes \ldots \otimes c_1 = (c''= c_d \otimes \ldots \otimes c_{m+n+1}) \otimes (c'= c_{m+n} \otimes \ldots \otimes c_{n+1}) \otimes (c=  c_n \otimes \ldots \otimes c_1 ) </math> into pure tensors of lengths <math>d-m-n,m,n\geq 0</math>, in particular allow <math>c'=1 \;\; {\scriptstyle (\text{or}\; c=1 \;\text{or}\; c''=1 )} </math> to have length <math>|c'|=0</math>.
 
The sign is determined by <math>\|c_n \otimes \ldots \otimes c_1\|= |c_n|+\ldots +|c_1|-n</math>, where <math>|c_i|\in\Z</math> (or <math>|c_i|\in\Z_N</math> with even <math>N</math>) denotes the [[grading induced by brane structures]].
 
The sign is determined by <math>\|c_n \otimes \ldots \otimes c_1\|= |c_n|+\ldots +|c_1|-n</math>, where <math>|c_i|\in\Z</math> (or <math>|c_i|\in\Z_N</math> with even <math>N</math>) denotes the [[grading induced by brane structures]].
With this notation, the curved <math>A_\infty</math>-relations for <math>(\mu^d)_{d\ge 0}</math> are equivalent to
+
With this notation, the curved <math>A_\infty</math>-relations for <math>(\mu^d)_{d\ge 0}</math> are  
 
<math>\widehat\mu \circ\widehat\mu = 0</math>.
 
<math>\widehat\mu \circ\widehat\mu = 0</math>.
 
Spelling this out, the first two relations are
 
Spelling this out, the first two relations are
 
</center>
 
</center>
 
By construction of the <math>\mu^d</math> this is equivalent to the following identity for each fixed  
 
By construction of the <math>\mu^d</math> this is equivalent to the following identity for each fixed  
<math>x_0\in\text{Crit}(L_0,L_d)</math>, where we sum over
+
<math>x_0\in\text{Crit}(L_0,L_d)</math>, where we abbreviate
 
<math>
 
<math>