Special identities of nonsymmetric bicommutative operations in associative algebra with an operator

Abstract

Let $A$ be an associative non-unital algebra with a linear operator $S$ satisfying \[aS(S(b)c)=S(aS(b))c,\] and define $a\prec b=aS(b)$ and $a\succ b=S(a)b$. We determine the special identities of these two operations. If $K_n$ is the arity-$n$ kernel of the corresponding operad morphism and $D_n$ is the space of consequences of smaller-arity kernel elements, then for every $n\ge 8$ the quotient $K_n/D_n$ has dimension $n-2$. A basis is given by the classes of $F_{n-2}$, $\rho F_{n-2}$, $G_{n-3}$, $\rho G_{n-3}$, $J_{n-4}$, $\rho J_{n-4}$ and $H_{p,n-p-1}$, $4\le p\le n-5$. The upper bound is proved by a direct reduction of singleton sectors, using pure normalization, rigid boundary contraction and the four-singleton corner. The lower bound is obtained from coefficient functionals and a corner projection. In particular, new independent special identities occur in every arity $n\ge 8$, and the operadic kernel is not finitely generated.

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