Difference-Restriction Algebras

• Difference-restriction algebras are algebras of partial functions defined by operations like graph difference and domain restriction, which encode domain interplay and convert principal downsets into Boolean algebras.
• They can be finitely axiomatized using key subtraction and domain-restriction laws that ensure the representability and injectivity of partial functions.
• Atomicity in these algebras guarantees complete representation via maximal filters, linking abstract algebraic properties to topological duality and program algebra frameworks.

Difference-restriction algebras are algebras of partial functions whose primitive operations are relative complement and domain restriction. In the foundational representation-theoretic formulation, a difference-restriction algebra is an algebra over the signature $\{-,\rest\}$, where $-$ is graph difference and $\rest$ restricts the second argument to the domain of the first; later papers also write domain restriction as $\setminus$, $\perp$, or $o$, depending on the surrounding formalism (Borlido et al., 2020, Borlido et al., 5 Aug 2025, Borlido et al., 2020, Jackson et al., 2021). The subject studies which abstract algebras arise from partial functions, how such algebras are represented and completely represented, how atomicity controls completeness, and how the resulting structures interact with duality theory, compatible completion, restriction semigroups, and additional operators (Borlido et al., 2020, Borlido et al., 5 Aug 2025, Borlido et al., 2020, Kudryavtseva, 5 Nov 2025).

1. Signature, semantics, and internal order

A partial function $f$ from $X$ to $Y$ is a binary relation $f \subseteq X \times Y$ such that $-$0 and $-$1 imply $-$2. Its domain is

$-$3

In an algebra of partial functions with common base $-$4, the two basic operations are interpreted as

$-$5

and

$-$6

Thus $-$7 is set-theoretic difference of graphs, while $-$8 is $-$9 restricted to the domain of $\rest$0 (Borlido et al., 2020).

A derived meet operation is defined equationally by

$\rest$1

In algebras of partial functions this is graph intersection. The zero element is definable by $\rest$2, independently of the choice of $\rest$3, and the natural order is

$\rest$4

With this order, the $\rest$5-reduct has a meet-semilattice structure, and each principal downset

$\rest$6

is a Boolean algebra with bottom $\rest$7, top $\rest$8, meet $\rest$9, and complement in $\setminus$0 given by $\setminus$1 (Borlido et al., 2020).

Compatibility is a central derived relation. Abstractly, $\setminus$2 and $\setminus$3 are compatible iff $\setminus$4. In the partial-function semantics this means that the two partial functions agree on their shared domain. This relation is reflexive, symmetric, and downward closed (Borlido et al., 5 Aug 2025).

The theory repeatedly exploits the fact that domain information is implicit in the order-restriction interaction. The preorder

$\setminus$5

captures domain inclusion: in algebras of partial functions, $\setminus$6 iff $\setminus$7 (Borlido et al., 2020).

2. Finite axiomatisation and algebraic laws

The class of representable difference-restriction algebras is a variety axiomatized by five equations:

$\setminus$8

$\setminus$9

$\perp$0

$\perp$1

$\perp$2

Here (Ax.1)–(Ax.3) are Schein-style subtraction laws, while (Ax.4)–(Ax.5) control the interaction of domain restriction with the induced meet structure (Borlido et al., 2020).

These equations have several immediate consequences. In any subtraction algebra, hence in every difference-restriction algebra,

$\perp$3

In restriction semilattices, and therefore in difference-restriction algebras, restriction is associative and order-preserving:

$\perp$4

$\perp$5

In the full difference-restriction setting one also has

$\perp$6

These identities are used both in the finite equational theory and in the representation proofs (Borlido et al., 2020).

The same five equations axiomatize the class of algebras representable by injective partial functions. The equational theory therefore does not distinguish arbitrary partial functions from injective partial functions in this signature (Borlido et al., 2020).

Jackson and Stokes place the same structures in a broader program-algebraic landscape. Using $\perp$7 for domain restriction and $\perp$8 for graph difference, they record a finite equational axiomatization for the signature $\perp$9 and show that this signature remains finitely axiomatizable when composition is added, yielding a finitely axiomatized variety for $o$0 as well (Jackson et al., 2021).

3. Representation by maximal filters and quotient structure

Representability means embeddability into an algebra of partial functions with the intended semantics. The basic representation theorem proceeds through the semilattice of filters and, more specifically, maximal filters of the derived meet-semilattice (Borlido et al., 2020).

The preorder

$o$1

induces an equivalence relation $o$2, and the quotient $o$3 is a meet-semilattice with

$o$4

In a difference-restriction algebra the quotient also carries a subtraction-algebra structure:

$o$5

Each principal quotient downset $o$6 is a Boolean algebra isomorphic to $o$7 via $o$8 (Borlido et al., 2020).

Let $o$9 be the set of filters of the underlying semilattice and $f$0 the set of maximal filters. The canonical representation is defined by

$f$1

where $f$2 is the filter-level equivalence corresponding to $f$3. In any restriction semilattice, $f$4 preserves $f$5 and $f$6; in a difference-restriction algebra it also preserves $f$7, because

$f$8

For difference-restriction algebras this representation is injective, yielding an embedding into a partial-function algebra and hence the converse direction of the axiomatisation theorem (Borlido et al., 2020).

A related representation produces injective partial functions:

$f$9

Because

$X$0

the injective case inherits the same equational theory (Borlido et al., 2020).

The subtraction laws are essential here. The maximal-filter map $X$1 need not be injective for a mere restriction semilattice; the paper gives a four-element example of identity partial functions on $X$2 where $X$3 fails to separate two elements. Adding the subtraction axioms restores injectivity. This isolates the genuinely difference-theoretic content of the representation theorem (Borlido et al., 2020).

4. Complete representations, atomicity, and examples

A representation $X$4 of the underlying ordered structure is meet complete if it preserves existing nonempty meets as intersections, and join complete if it preserves existing joins as unions. For difference-restriction algebras these notions coincide: if $X$5 is meet complete, then it is join complete, and if $X$6 is join complete, then it is meet complete. Complete representation is therefore a single notion in this setting (Borlido et al., 2020).

Atomicity provides the exact structural criterion. An atom is a minimal nonzero element, and an algebra is atomic if every nonzero element lies above an atom. A representation is atomic if every point in the image of an element lies in the image of some atom below that element. For representations of difference-restriction algebras by partial functions, atomicity of the representation is equivalent to completeness of the representation. Consequently, a difference-restriction algebra is completely representable by partial functions if and only if it is representable and atomic (Borlido et al., 2020).

Because each principal downset is a Boolean algebra, atomicity implies atomisticity: every element is the join of the atoms below it. For an atomic algebra, a canonical complete representation is obtained from the atom set:

$X$7

This representation is injective, preserves $X$8 and $X$9, and is complete. The same characterization holds for complete representability by injective partial functions (Borlido et al., 2020).

Atomicity can be expressed in the $Y$0-language by defining

$Y$1

and then writing

$Y$2

Together with (Ax.1)–(Ax.5), this gives a universal–existential–universal axiomatisation of the completely representable classes. The same paper also proves that no existential–universal–existential axiomatisation exists, by reduction to atomic Boolean algebras (Borlido et al., 2020).

The standard examples separate representability from complete representability:

Model	Key property	Consequence
Full algebra $Y$3	atomic	completely representable
Identity functions on a full powerset algebra	atomic	completely representable
Identity functions on the clopen algebra of the Cantor space	representable but not atomic	not completely representable

For $Y$4, every nonzero partial function contains a singleton graph $Y$5 below it, so the algebra is atomic. By contrast, identity-function algebras arising from atomless Boolean algebras are representable but fail complete representability because they fail atomicity (Borlido et al., 2020).

5. Discrete and topological duality, completion, and operators

A major later development studies atomic representable difference-restriction algebras categorically. The discrete duality paper considers the category of atomic representable DR-algebras with complete homomorphisms and a dual category of set quotients $Y$6 with partial maps satisfying fibrewise injectivity and fibrewise surjectivity conditions. The contravariant functor

$Y$7

and the functor

$Y$8

form an adjunction $Y$9. The induced monad $f \subseteq X \times Y$0 yields the compatible completion of any atomic representable DR-algebra, and the adjunction restricts to a duality on compatibly complete atomic representable DR-algebras (Borlido et al., 2020).

The 2025 topological refinement replaces set quotients by Hausdorff étale spaces. For a difference-restriction algebra $f \subseteq X \times Y$1, one forms the space $f \subseteq X \times Y$2 of maximal filters of the meet-semilattice and the quotient map

$f \subseteq X \times Y$3

This produces a Hausdorff étale space. Conversely, for an étale space $f \subseteq X \times Y$4, the algebra $f \subseteq X \times Y$5 consists of compact open subsets $f \subseteq X \times Y$6 on which $f \subseteq X \times Y$7 is injective, with operations

$f \subseteq X \times Y$8

The resulting functors define an adjunction between the category of DRAs and the opposite of the category of Hausdorff étale spaces. The induced monad again gives a completion, now the finitary compatible completion. Restricting to finitarily compatibly complete DRAs yields a duality with locally compact zero-dimensional Hausdorff étale spaces (Borlido et al., 5 Aug 2025).

Both duality theories extend to enriched signatures. In the discrete setting, the added operations are required to be completely additive and compatibility preserving. In the étale-space setting, the added operations are arbitrary additional compatibility preserving operators, represented on the dual side by tight spectral relations. The examples explicitly named in the later paper include composition, domain, range, fixset, identity, and range-restriction; antidomain, override, and converse fall outside the compatibility-preserving class as stated there (Borlido et al., 2020, Borlido et al., 5 Aug 2025).

These constructions generalise the duality between generalized Boolean algebras and Hausdorff spaces, and, for complete objects, the duality between generalized Boolean algebras and locally compact zero-dimensional Hausdorff spaces. A plausible implication is that difference-restriction algebras function as a noncommutative or partial-function analogue of generalized Boolean algebras in the same way that Boolean algebras with operators serve as a base class for modal extensions (Borlido et al., 5 Aug 2025).

6. Relations to restriction semigroups, Booleanization, and program algebra

Difference-restriction algebras omit composition and isolate what one paper calls the “compatibility algebra” of partial functions. In this sense they sit adjacent to restriction semigroups and inverse semigroups, but are more order-theoretic: they capture agreement on shared domain through the equation $f \subseteq X \times Y$9, while the subtraction structure turns principal downsets into Boolean algebras (Borlido et al., 2020).

A topological realization of this perspective appears in the theory of restriction semigroups with local units. For such a semigroup $-$00, one forms the universal category $-$01 of germs of the spectral action on the character space of the projection semilattice. The compact slices of $-$02 form a Boolean restriction semigroup with local units, written $-$03. Set-theoretic difference on slices is built into this Booleanization; for slices of the form $-$04 and $-$05,

$-$06

This furnishes a canonical difference-restriction algebra associated to $-$07 and supports an embedding

$-$08

The same framework yields a topological ESN-type theorem, an extension of the Petrich–Reilly structure theorem for proper restriction semigroups, and an isomorphism between the semigroup algebra $-$09 and the convolution algebra $-$10 (Kudryavtseva, 5 Nov 2025).

Program algebra supplies another adjacent viewpoint. Jackson and Stokes treat domain restriction as a right normal band operation $-$11 and graph difference as $-$12, alongside minus, intersection, composition, override, and update. In their formulation,

$-$13

They show that algebras of partial functions with signature $-$14 are finitely axiomatized, and that finite axiomatisability persists in the presence of composition for the signature $-$15 (Jackson et al., 2021).

Across these settings, several structural points recur. The same equational theory governs arbitrary and injective partial-function representations; representability is strictly weaker than complete representability; and compatibility, rather than unrestricted union, is the mechanism that controls existence of joins. This suggests that the essential content of the theory is not merely set difference on graphs, but the interaction of difference with an intrinsic domain calculus (Borlido et al., 2020, Borlido et al., 2020, Borlido et al., 5 Aug 2025).