Coassociative structures on self-injective algebras

For general finite-dimensional self-injective algebra $A$ we construct a family of injective coassociative coproducts $A\to A\otimes A$, all $A$-bimodule morphisms. In particular such structures always exist, confirming a conjecture of Hernandez, Walton and Yadav. The coproducts are indexed by subsets of $\{1,\cdots,m(i)\}\times \{1,\cdots,m(\nu^{-1}i)\}$, where $A\cong \mathrm{End}_{\Lambda}(M)$ is the general form of a self-injective algebra in terms of a basic Frobenius $\Lambda$, the $m(i)$, $1\le i\le n$ are the multiplicities of the indecomposable projective $\Lambda$-modules in $M$, and $\nu$ is the Nakayama permutation of $\Lambda$. We also characterize those among the coproducts introduced in this fashion, in terms this combinatorial data, which are counital.