Finitary Compatible Completion

Updated 8 July 2026

Finitary compatible completion is a construction that embeds difference–restriction algebras into structures with finite compatible joins, ensuring a minimal and dense expansion.
It uses an adjunction between difference–restriction algebras and Hausdorff étale spaces, with the completion realized by the monad induced through this duality.
The construction features a universal property guaranteeing uniqueness up to isomorphism and extends naturally to frameworks like generalized Boolean algebras and operator expansions.

Finitary compatible completion is, in its explicit formal use, a completion construction for difference–restriction algebras. It enlarges an algebra of partial-function-like operations just enough to add finite compatible joins, rather than arbitrary finite joins, because for partial functions the union $f\cup g$ is a partial function iff $f$ and $g$ agree wherever both are defined. In that setting, the completion is an embedding $\iota:A\hookrightarrow C$ into a finitarily compatibly complete algebra whose image is finite-join dense, and it is realized by the monad induced by an adjunction with Hausdorff étale spaces (Borlido et al., 5 Aug 2025).

1. Formal definition in difference–restriction algebras

A difference–restriction algebra is an abstract $\{-,\rest\}$-algebra isomorphic to an algebra of partial functions closed under relative complement and domain restriction. For partial functions $f,g \subseteq X\times Y$ ,

$f-g=\{(x,y)\in X\times Y : (x,y)\in f \text{ and } (x,y)\notin g\},$

and

$f\rest g=\{(x,y)\in X\times Y : x\in \dom(f)\text{ and } (x,y)\in g\}.$

The intended order is inclusion, the derived meet is

$a\cdot b \coloneqq a-(a-b),$

and the induced order is

$a\le b \iff a=a\cdot b.$

Compatibility is the central notion. Two elements $f$ 0 are compatible exactly when

$f$ 1

For actual partial functions, this says that they agree on the overlap of their domains. A poset with a compatibility relation is finitarily compatibly complete when it has joins of every finite set of pairwise-compatible elements. For difference–restriction algebras, having finite compatible joins is equivalent to having joins of each pair of compatible elements.

A subset $f$ 2 of a poset is finite-join dense if every element is the join of some finite subset of $f$ 3. A finitary compatible completion of a difference–restriction algebra $f$ 4 is then an embedding

$f$ 5

such that $f$ 6 is a finitarily compatibly complete difference–restriction algebra and $f$ 7 is finite-join dense in $f$ 8 (Borlido et al., 5 Aug 2025).

2. Construction via Hausdorff étale spaces and the monad $f$ 9

The construction is organized by an adjunction

$g$ 0

For a difference–restriction algebra $g$ 1, $g$ 2 is the Hausdorff étale space

$g$ 3

where $g$ 4 is the set of maximal filters of $g$ 5. Its topology has basis

$g$ 6

Conversely, for a Hausdorff étale space $g$ 7, $g$ 8 is the difference–restriction algebra of compact open subsets $g$ 9 such that $\iota:A\hookrightarrow C$ 0 is injective. The operations are

$\iota:A\hookrightarrow C$ 1

These are compact-open local sections, and compatibility becomes geometric: compatible local sections have union still a local section.

The unit of the adjunction is

$\iota:A\hookrightarrow C$ 2

The completion theorem states that this map is the finitary compatible completion of $\iota:A\hookrightarrow C$ 3. Concretely, elements of $\iota:A\hookrightarrow C$ 4 are compact-open subsets of $\iota:A\hookrightarrow C$ 5 on which $\iota:A\hookrightarrow C$ 6 is injective, and every such compact-open local section is a finite union of basic opens

$\iota:A\hookrightarrow C$ 7

Thus the completed algebra is generated by finite compatible joins of the image of $\iota:A\hookrightarrow C$ 8, with compatible joins realized as unions (Borlido et al., 5 Aug 2025).

3. Universal property, uniqueness, and duality

Finitary compatible completions are unique up to unique isomorphism over the original algebra. If

$\iota:A\hookrightarrow C$ 9

are finitary compatible completions, then there is a unique isomorphism

$\{-,\rest\}$0

satisfying

$\{-,\rest\}$1

The completion also admits equivalent universal formulations. It is simultaneously the smallest finitarily compatibly complete extension of $\{-,\rest\}$2 and the largest finite-join-dense extension of $\{-,\rest\}$3: embeddings from $\{-,\rest\}$4 into any finitarily compatibly complete algebra factor through it, and any finite-join-dense embedding of $\{-,\rest\}$5 factors into it.

Categorically, the full subcategory $\{-,\rest\}$6 of finitarily compatibly complete difference–restriction algebras is reflective in $\{-,\rest\}$7, with reflector given by the completion $\{-,\rest\}$8. On the topological side, the adjunction restricts to a duality between the finitarily compatibly complete difference–restriction algebras and the locally compact zero-dimensional Hausdorff étale spaces. This is the precise sense in which the completion theorem is also a Stone-type duality statement (Borlido et al., 5 Aug 2025).

4. Specialization to generalised Boolean algebras and expansion by operators

The Boolean case appears as a specialization. In the subtraction-algebra situation, $\{-,\rest\}$9 becomes meet, compatibility becomes trivial, and finite compatible joins become ordinary finite joins. In that case, having all finite compatible joins is equivalent to being a generalised Boolean algebra, and the duality above restricts to the classical duality between $f,g \subseteq X\times Y$ 0 and $f,g \subseteq X\times Y$ 1. This explains why the difference–restriction construction is presented as a generalization of the adjunction between generalised Boolean algebras and Hausdorff spaces, and of the duality between generalised Boolean algebras and locally compact zero-dimensional Hausdorff spaces.

The construction also extends to algebras with additional compatibility preserving operators. An $f,g \subseteq X\times Y$ 2-ary operation $f,g \subseteq X\times Y$ 3 is compatibility preserving when pairwise-compatible inputs are sent to compatible outputs, and it is an operator when it is also normal in each coordinate and additive whenever a compatible join exists. In that expanded setting, the monad $f,g \subseteq X\times Y$ 4 again yields the finitary compatible completion, and uniqueness up to unique isomorphism persists (Borlido et al., 5 Aug 2025).

5. Categorical and algebraic analogues

Several other completion constructions are close analogues, though not the same definition. Garner shows that the passage from a finitary monad on $f,g \subseteq X\times Y$ 5 to its associated Lawvere theory is the free completion under the finitary class $f,g \subseteq X\times Y$ 6 of absolute colimits, and Garner–Power generalize this to arbitrary locally finitely presentable bases as free completion under absolute tensors in $f,g \subseteq X\times Y$ 7. In both cases, the completion is “finitary” because only finite-power structure is adjoined, and “compatible” because the relevant colimits are absolute (Garner, 2013, Garner et al., 2017).

A second analogue comes from algebra–coalgebra completion. For finitary, and more generally precontinuous, set functors, the initial algebra $f,g \subseteq X\times Y$ 8 and terminal coalgebra $f,g \subseteq X\times Y$ 9 carry a canonical partial order with the same ideal $f-g=\{(x,y)\in X\times Y : (x,y)\in f \text{ and } (x,y)\notin g\},$ 0-completion and a canonical ultrametric with the same Cauchy completion. For free algebras and free completely iterative algebras, the free cia is the conservative completion of the free algebra in the bicontinuous case, and in the general finitary case the two structures share the same conservative completion. This suggests a broader pattern in which finitary approximation data are completed so that algebraic and coalgebraic structure extend continuously (Adámek, 2019, Adamek, 2019).

The phrase should not be conflated with several neighboring constructions. Miller and Ng’s finitary computable reducibility preserves the full pairwise equivalence pattern on each finite tuple, and is therefore a rigorous framework for finite compatibility preservation, but it is not a completion operator (Miller et al., 2014). Wrigley’s geometric completion of a doctrine is topology-parameterized, idempotent, and semantically invariant; coherent and existential completions appear there as compatible subcompletions, but the main completion is geometric rather than purely finitary (Wrigley, 2023).

An order-theoretic analogue appears in meet-completions of isotone poset expansions. There a standard closure operator yields a meet-completion carrying lifted finite-arity isotone operations, and many equations are preserved; however, full algebraic compatibility can fail, since associativity, $f-g=\{(x,y)\in X\times Y : (x,y)\in f \text{ and } (x,y)\notin g\},$ 1, and $f-g=\{(x,y)\in X\times Y : (x,y)\in f \text{ and } (x,y)\notin g\},$ 2 can fail in the ordered-domain-algebra completion (Egrot et al., 2015). In logic programming, ordered completion captures stable models of arbitrary programs only for finite models, and an extension with level mappings handles infinite stable models; in rewriting, abstract completion and ordered completion use compatibility with a reduction order, but these are different completion procedures again (Heuer, 19 Apr 2025, Hirokawa et al., 2018).

A plausible unifying description is therefore narrower than a single universal definition: finitary compatible completion refers most precisely to the completion of difference–restriction algebras by finite compatible joins, while more broadly naming a family of constructions that adjoin only the finite or admissible structure needed for a given notion of compatibility (Borlido et al., 5 Aug 2025).