The mod-two square algebra, admissible sequences, and excess

Assume AC, inherited from the cited bundle, cohomology, or operation suppliers. Let $T={\mathbb{F}}_2⟨s_1,s_2,\dots ⟩$ be the vector space with basis the finite words in the symbols $s_j$ for $j>0$, including the empty word, and with multiplication the bilinear extension of concatenation. Concatenation is associative and the empty word is its unit, so this explicitly constructs the free associative graded algebra, with $\mathrm{\mid }{s}_j\mathrm{\mid }=j$ and $s_0=1$. Define ${\mathcal{A}}_{\mathrm{A}\mathrm{d}\mathrm{e}\mathrm{m}}$ as its quotient by the two-sided ideal generated by

$s_a{s}_b+{\sum }_{j=0}^{\lfloor a\mathrm{/}2\rfloor }\left(\genfrac{}{}{0px}{}{b-j-1}{a-2j}\right)s_{a+b-j}{s}_j,0<a<2b\mathrm{.}$

A normalized sequence $I=(i_1,\dots ,i_k)$ has positive entries; the empty sequence denotes the unit. It is admissible if $i_j\ge 2i_{j+1}$. Write $\mathrm{\mid }I\mathrm{\mid }={\sum }_j{i}_j$, put $i_{k+1}=0$, and define

$d_j(I)=i_j-2i_{j+1},e(I)={\sum }_{j=1}^k{d}_j(I)=i_1-i_2-\cdots -i_k\mathrm{.}$

For the empty sequence set $\mathrm{\mid }I\mathrm{\mid }=e(I)=0$. The square algebra ${\mathcal{A}}_{\mathrm{S}\mathrm{q}}$ is the image of the free algebra under $s_j\mapsto S{q}^j$ in natural mod-two cohomology operations, allowing every nonnegative input degree. The published Adem theorem makes the induced map ${\mathcal{A}}_{\mathrm{A}\mathrm{d}\mathrm{e}\mathrm{m}}\to {\mathcal{A}}_{\mathrm{S}\mathrm{q}}$ well defined. The admissible-composites theorem proves it is an isomorphism.

Every relation is homogeneous. A quotient by a two-sided ideal is a unital associative graded algebra: products of cosets are independent of representatives since multiplying an ideal element on either side remains in the ideal. Its grading is the direct sum of homogeneous quotient spaces since the generating relations are homogeneous. The displayed finite sums defining degree and excess are unambiguous; admissibility makes each $d_j$ nonnegative. Composition and addition of the published natural linear operations preserve naturality, so the image algebra exists. Each square commutes with suspension by the published normalization/suspension proposition, so the resulting composites are stable wherever their source degrees are defined. No assertion that every stable operation is such a composite is needed for this definition.