In this paper, we introduce several co-associative laws and the notion of a pre- $B$-algebra. We show that every $B$-algebra is both a pre- $B$-algebra and a $\perp$-algebra. We apply the notions of a post groupoid and a pre-semigroup of a groupoid to the set $(\mathbb{N},+)$ of all nonnegative integers, and we prove that the groupoid $(\mathbb{N},+)$ cannot be a post groupoid of a $B$-algebra or an edge $d$-algebra.