A Finite-Complement Congruence Topology for Partial Algebras

Let D be a partial algebra with carrier A, and let F(D) be its free compatible total extension. We consider compatible total extensions that fix A pointwise and contain only finitely many elements outside A. Their kernels define a descending family of congruences on F(D) and hence a Hausdorff congruence topology. For every finite subset of F(D) there is one such extension on which the canonical homomorphism is injective. This is an elementary Rees-congruence construction: retain its finite subterm closure and collapse the complement to one class. When A is finite, the resulting topology is exactly the ordinary profinite topology. The case of interest is therefore an infinite retained base, where the allowed target algebras may themselves be infinite. We prove a sufficient condition for the Hausdorff completion to be strictly larger than F(D). If D has an undefined base application and admits a nontrivial unary context whose orbits on A have uniformly bounded finite cardinality, then iteration from the undefined term yields a Cauchy sequence with no limit in F(D). We also exhibit an infinite partial algebra for which the induced topology is discrete, showing that partiality alone does not imply incompleteness. Preprint. This manuscript has not undergone peer review. Supplementary Lean 4 formalization and source materials are available at: https://github.com/Puhyuhy/ResolutionSemantics