Variant 11 as a Minimal Nontrivial Instance of the Common Support Interval.
This companion note clarifies the structural position of Variant 11 within the presentation-support framework established in Formal Frame, Syntactic Support, and Real Determination in Finite Boolean Functions and the subsequent Common Support Interval theorem. Variant 11 is the finite typed Boolean function f₁₁(p, q) = ¬q on the formal frame N = {p, q}, with determinational support R_{f₁₁} = {q}. The presentation-support theorem establishes that the exact syntactic supports of term presentations of a finite typed Boolean function form the interval between its determinational support and its formal frame. The Common Support Interval theorem further establishes that this family coincides with the corresponding factorization-support family. Applied to Variant 11, these results yield 𝔉_N(f₁₁) = 𝔖_N(f₁₁) = [{q}, {p, q}]_{⊆} = {{q}, {p, q}}. This equality concerns the families of admissible supports only. Coordinate factorization and term presentation remain structurally distinct constructions, even though their associated families of support sets coincide. The note also identifies the precise sense in which Variant 11 is minimal. Among nonconstant finite typed Boolean functions with proper determinational support, it attains the smallest possible ambient-frame size. Variant 11 is therefore a minimal nontrivial instance of the common support interval. No uniqueness or canonical-status claim is made.
Authors
- Piotr Dariusz Wójcik (ORCID: https://orcid.org/0009-0006-1439-4671)
Institutions
- Membrane Reactor Technologies (Canada) (CA)
Publication Details
- Journal
- Zenodo (CERN European Organization for Nuclear Research)
- Published
- 2026-10-20
- DOI
- https://doi.org/10.5281/zenodo.23485162
- Primary Topic
- Advanced Algebra and Logic
- Type
- preprint