Proposition. The represented counting adjoint [fdip-001H]
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  • October 20, 2026

Let \(R\) be a commutative unital ring, \(V\) a finite set and \(H=\mathcal F_R(V)\). Equip \(H\) with the perfect counting duality \(b,\Phi ,\Psi \), with \(H^\vee =\operatorname {Hom}_R(H,R)\). For \(a\in \operatorname {End}_R(H)\), define \[\begin {aligned} a^\vee :H^\vee &\longrightarrow H^\vee , &a^\vee (\ell )&=\ell \circ a,\\ a^\dagger &=\Psi \circ a^\vee \circ \Phi :H\longrightarrow H. \end {aligned}\] The counting adjoint \(a^\dagger \) is the unique ordinary \(R\)-linear endomorphism satisfying \[b(f,a^\dagger g)=b(af,g)\qquad (f,g\in H).\] It gives an ordinary \(R\)-linear involution on the composition algebra \(\operatorname {End}_R(H)\). Products act with the right factor first.

The real incidence construction in Definition [fdip-0007] starts from \(D:\mathbb R^V\to \mathbb R^E\) and returns an adjoint in the opposite direction. Here \(a:H\to H\) is an endomorphism over the commutative ring \(R\). Both use the characterizing identity that moves the operator between the two counting-pairing slots. The existence and laws below follow from Lemma [fdip-001F], independently of the real incidence proof.

Proof. 

  • October 20, 2026

The composite \(\ell \circ a\) is linear, and precomposition is linear in \(\ell \); thus all three arrows defining \(a^\dagger \) are linear. Since \(\Phi \Psi =I\), evaluation at \(f\) gives the displayed adjoint identity. Injectivity of \(\Phi \) gives uniqueness for each \(g\). In particular, \[(a^\dagger g)(v)=b(a\delta _v,g).\] For \(a,c\in \operatorname {End}_R(H)\) and \(r\in R\), bilinearity and uniqueness give \[(a+c)^\dagger =a^\dagger +c^\dagger ,\quad (ra)^\dagger =ra^\dagger ,\quad I^\dagger =I.\] Indeed the right-hand sides represent respectively \(b((a+c)f,g)\), \(b((ra)f,g)\) and \(b(f,g)\).

For products, applying the defining identity twice yields \[b(f,c^\dagger a^\dagger g) =b(cf,a^\dagger g)=b(acf,g),\] so \((ac)^\dagger =c^\dagger a^\dagger \). Symmetry of \(b\) gives \[\begin {aligned} b(f,(a^\dagger )^\dagger g) &=b(a^\dagger f,g)=b(g,a^\dagger f)\\ &=b(ag,f)=b(f,ag). \end {aligned}\] Uniqueness therefore gives \((a^\dagger )^\dagger =a\).

The construction also applies when \(V\) is empty or \(R\) is the zero ring. Its representability is supplied by the explicit inverse to \(\Phi \), not by a field hypothesis or a positivity argument. Dual precomposition reverses composition before any pairing is chosen; the Hom convention is [stacks2017stacks, Section 10.10], Tag 0581. The proof here adds the specified symmetric counting identification with the dual.