The admissible square algebra is a connected bialgebra

Let T be the free associative graded algebra on symbols s_n, n>0, with s₀=1. Define an algebra homomorphism Δ̃:T→T⊗T, Δ̃(s_n)=Σ_{i+j=n}s_i⊗s_j, using the graded tensor-product multiplication; over F₂ all Koszul signs equal 1. Define ε̃(s₀)=1 and ε̃(s_n)=0 for n>0, extending multiplicatively. On generators, Δ̃ is coassociative because both iterates sum once over triples i+j+k=n. The two counit identities also hold on generators. Since all maps in these identities are algebra homomorphisms, the identities hold on T.

Let R₀ be the set of displayed Adem relation generators, and let I=(R₀) be their two-sided ideal. The admissible-basis identification $T\mathrm{/}I\to A$ identifies each R∈R₀ with the zero natural operation. For spaces X,Y and classes x∈H*(X), y∈H*(Y), repeated application of the published Cartan formula gives R(x×y)= (q⊗q)(Δ̃R)·(x⊗y), where q:T→A is the quotient map and the tensor action means external product after applying the two factors. The left side is zero because R∈R₀ is an Adem relation. Tensor faithfulness therefore gives (q⊗q)(Δ̃R)=0. To compute the kernel, use AC and basis extension to choose a complement C with T=I⊕C. Distribution of tensor products over this finite direct sum gives $T\otimes T=(I\otimes I)\oplus (I\otimes C)\oplus (C\otimes I)\oplus (C\otimes C)$.

The map q⊗q kills the first three summands and restricts to the isomorphism C⊗C→(T/I)⊗(T/I) on the last. Hence ker(q⊗q)=I⊗T+T⊗I; write J for this kernel. Since I is a two-sided ideal, J is a two-sided ideal in T⊗T. We have Δ̃(R)∈J for each R∈R₀. Every element of I is a finite sum of terms xRy with x,y∈T and R∈R₀, so multiplicativity gives Δ̃(xRy)=Δ̃(x)Δ̃(R)Δ̃(y)∈J. Therefore Δ̃(I)⊂J, and Δ̃ descends to Δ_A:A→A⊗A. Also ε̃(R)=0 for every R∈R₀ because each generator is homogeneous of positive degree. Since ε̃ is multiplicative, it vanishes on all of I and descends to A. The descended maps remain algebra homomorphisms, coassociative and counital. The basis theorem gives A₀=F₂·1 and A_d=0 for d<0. Thus A is a connected nonnegatively graded bialgebra, with Δ_A(Sqⁿ)=Σᵢ₌₀ⁿ Sqⁱ⊗Sqⁿ⁻ⁱ. No antipode is required by the Hopf-freeness supplier.