An example in the Singer category of algebras with coproducts at odd primes (Q331582)

Let \(p\) be any prime and let \(\mathfrak{B}(p)\) be the algebra of Steenrod operations on \(\mathrm{Ext}_{\Lambda}(\mathbb{Z}/p\mathbb{Z}, \mathbb{Z}/p\mathbb{Z})\), the cohomology of a graded cocommutative Hopf algebra \(\Lambda\) over \(\mathbb{F}_{p}\) (see [\textit{A. Liulevicius}, Mem. Am. Math. Soc. 42, 112 p. (1962; Zbl 0131.38101)]). Suppose \(p\) an odd prime, it is natural to ask if \(\mathfrak{B}(p)\) and its dual admit, as in the case \(p=2\) (see [\textit{W. M. Singer}, Bull. Lond. Math. Soc. 37, No. 4, 627--635 (2005; Zbl 1080.55014)]), a structure as \(\mathbb{F}_{p}\)-algebra with coproduct and \(\mathbb{F}_{p}\)-coalgebra with product, respectively. In this paper the authors give a positive answer to this question.