Dyer-Lashof-Araki-Kudo operations

These are operations on the mod-$p$ homology of infinite loop spaces. They measure the failure of the Pontrjagin product to be commutative at the chain-level and are analogous to the Steenrod squares. The operations are linear maps of the form

$${\beta }^{ϵ}{Q}^r:H_n(X;{\mathbb{F}}_p)\to H_{n+2r(p-1)-ϵ}(X;{\mathbb{F}}_p)$$

for $ϵ\in \{0,1\}$ and $r\in {\mathbb{Z}}_{\ge ϵ}$.