Example. Observables on a finite configuration set [fdip-0022]
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  • October 23, 2026

Let \(V\) be a finite vertex set and \(S\) a finite set of possible values. A configuration is a function \(\sigma :V\to S\); the configuration set is \(\Omega =S^V\). The set \(\Omega \) is distinct from the vertex set \(V\): an observable takes a whole configuration as its input. For a commutative unital ring \(R\), set \[A_\Omega =\operatorname {Map}(\Omega ,R).\] This is the pointwise function algebra: \((f+g)(\sigma )=f(\sigma )+g(\sigma )\), \((fg)(\sigma )=f(\sigma )g(\sigma )\) and \((rf)(\sigma )=rf(\sigma )\), with unit the constant function \(1_R\).

For each \(\sigma \in \Omega \), evaluation \(\operatorname {ev}_\sigma (f)=f(\sigma )\) is a unital \(R\)-algebra map and a normalized linear functional. The scalar map sends \(r\) to the constant function \(r\). The indicator \(\delta _\sigma \) is \(1_R\) at \(\sigma \) and \(0_R\) elsewhere. These are the same indicators used in counting duality for finite functions, now indexed by configurations.

Proof. 

  • October 23, 2026

Each algebra law follows by evaluating at every configuration and using the corresponding law in \(R\). Evaluation preserves addition, scalar multiplication, products and the unit by the displayed definitions. The indicators satisfy \[\delta _\sigma ^2=\delta _\sigma ,\qquad \delta _\sigma \delta _\tau =0\ (\sigma \ne \tau ),\qquad \sum _{\sigma \in \Omega }\delta _\sigma =1.\] These equalities hold pointwise, including when \(R\) is the zero ring.

If \(V=\varnothing \), then \(S^V\) contains the unique empty function, even when \(S=\varnothing \). If \(S=\varnothing \) and \(V\ne \varnothing \), then \(\Omega =\varnothing \); its function algebra is the zero algebra. There is then no evaluation at a configuration.

The underlying unital algebra convention is [stacks2017stacks, Section 22.2], Tag 09JF. Mathlib's Pi.algebra, Pi.evalAlgHom and Pi.constAlgHom provide the pointwise algebra, evaluation and constant algebra maps. The configuration interpretation and the empty-set cases above are direct consequences of these definitions.