A Categorical Approach to Rough Equality Algebras via Approximation Functors

A CATEGORICAL APPROACH TO ROUGH EQUALITY ALGEBRAS VIA APPROXIMATION FUNCTORS JOAQUIM REIZI HIGUCHI Abstract. In this paper we develop a categorical framework for congruence-based rough set theory in the setting of equality algebras. We introduce the category of approximation equality algebras, denoted by AppEqAlg , whose objects are pairs ( E, θ ) consisting of an equality algebra E and a congruence relation θ on E , and whose morphisms are congruence-preserving homomorphisms. For a fixed ( E, θ ) we analyse the associated rough approximations on P ( E ), showing that the upper and lower approximations give rise, respectively, to a closure operator and an interior operator on the powerset lattice, and that the upper approximation restricts to a closure operator on the subalgebra lattice Sub ( E ). Within this setting we identify the exact (i.e. θ -definable) subalgebras and prove that their complete lattice is isomorphic to the subalgebra lattice of the quotient algebra E/θ . On the categorical side, we define the quotient functor U : AppEqAlg →EqAlg and the diagonal embedding G : EqAlg →AppEqAlg , and establish an adjunction U⊣G . Finally, we show that the forgetful functor V : AppEqAlg → EqAlg is topological, so that limits and colimits in AppEqAlg are obtained from those in EqAlg by equipping the underlying algebra with canonical initial and final congruences. This provides a unified categorical perspective on roughness in equality algebras. 1. Introduction The study of logical algebras has long been a central theme in non-classical logic, providing the algebraic semantics for various logical systems. Among these, equality algebras were introduced by Jenei [ 1 ] as a specialized structure for fuzzy type theory. Unlike residuated lattices or MV-algebras which rely primarily on implication, equality algebras take the equality connective (fuzzy equivalence) as the primitive operation. This shift in perspective has led to a fruitful area of research, exploring their topological and algebraic properties (e.g., [2]). Parallel to these developments, rough set theory, proposed by Pawlak [ 3 ], has established itself as a powerful mathematical tool for handling uncertainty and indiscernibility. While originally formulated in a set-theoretic context, the theory has been extensively applied to algebraic structures. The notion of “rough algebraic structures” investigates how algebraic operations interact with approximation spaces. Significant contributions have been made in this Date: November 23, 2025. 2020 Mathematics Subject Classification. 06D20, 03G25, 18B30, 54B30. Key words and phrases. Equality algebra, Rough sets, Approximation functors, Adjunction, Topological category, Closure operator. 1 2 JOAQUIM REIZI HIGUCHI direction, applying roughness to groups, rings, and various logical algebras (see, for instance, [4]). Most of the existing work on rough algebraic structures is local in nature: one fixes an algebra A together with an equivalence or congruence relation and then studies, for example, whether the lower approximation of a subalgebra is again a subalgebra, or what the properties of rough ideals or rough filters are. While such results are fundamental, they do not by themselves provide a global picture of how roughness behaves across the whole variety. In particular, a systematic categorical framework in which rough approximations are treated not only as operators on a given algebra, but as part of functorial constructions between categories, has not been fully developed in the specific setting of equality algebras. In this paper we take a step in this direction by establishing a categorical foundation for rough equality algebras based on congruence-induced approximation. Our starting point is the observation that, once a congruence θ is fixed on an equality algebra E , the associated rough approximations have two complementary incarnations: • on the algebraic side, θ determines upper and lower rough approximations on the power set P ( E ), and in particular a closure operator on the lattice of subalgebras of E together with a dual interior operator on P(E); • on the categorical side, the same congruence gives rise to a quotient homomorphism E→E/θ , and quotients by congruences assemble into a functor U from a category of “rough equality algebras” to the base category of equality algebras. The aim of the paper is to organise these facts in a uniform categorical framework. More precisely, we introduce a category whose objects are equality algebras equipped with a chosen congruence (thought of as an approximation structure) and whose morphisms are congruence-preserving homomorphisms. Inside this category we identify: • aquotient functor U sending ( E, θ ) to the quotient equality algebra E/θ; • adiagonal embedding G which sends an equality algebra Y to ( Y, ∆ Y ), where ∆Yis the diagonal congruence; • aforgetful functor V discarding the congruence and remembering only the underlying equality algebra. We show that U and G form an adjoint pair U⊣G , thus expressing the universal property of the quotient E/θ purely categorically. The forgetful functor V turns out to be topological in the sense of Ad´amek, Herrlich and Strecker, so that limits and colimits in our rough category can be constructed by equipping the corresponding limits and colimits in EqAlg with canonical initial and final congruences. Main Contributions. The main contributions of this paper can be summarised as follows: (1) We introduce the category AppEqAlg , whose objects are pairs ( E, θ ) consisting of an equality algebra E and a congruence relation A CATEGORICAL APPROACH TO ROUGH EQUALITY ALGEBRAS 3 θ on E , and whose morphisms are congruence-preserving homomorphisms. This category formalises the universe of congruence-based approximation structures over equality algebras. (2) For a fixed equality algebra E and congruence θ , we study the induced rough approximations on P ( E ). We show that the upper and lower approximations define, respectively, a closure operator and an interior operator on the complete lattice P ( E ), and that the upper approximation restricts to a closure operator on the subalgebra lattice Sub ( E ). Within this setting we identify the exact (i.e. θ -definable) subalgebras and show that they form a complete sublattice of Sub(E). (3) On the categorical side, we define the quotient functor U:AppEqAlg →EqAlg and the diagonal embedding functor G:EqAlg →AppEqAlg . We prove that U is the left adjoint of G ( U⊣G ). This adjunction provides a precise categorical formulation of the passage from a rough structure ( E, θ ) to its crisp quotient E/θ and connects the unit of the adjunction with the lattice of exact subalgebras. (4) We prove that the forgetful functor V:AppEqAlg →EqAlg is topological. We explicitly construct initial and final lifts of structured sources and sinks, and deduce that (co)limits in AppEqAlg are obtained by equipping (co)limits in EqAlg with canonical initial or final congruences. This situates rough equality algebras within the general theory of topological concrete categories. Structure of the Paper. The remainder of this paper is organised as follows. Section 2 reviews the necessary preliminaries on equality algebras and rough set theory. Section 3 is devoted to the algebraic study of congruenceinduced rough approximations on an equality algebra: we describe upper and lower approximations as closure and interior operators on P ( E ), analyse the restriction of the upper approximation to the subalgebra lattice, and introduce the notion of exact subalgebras, showing that they are in bijective correspondence with subalgebras of the quotient algebra. Section 4 develops the categorical framework, defining the category AppEqAlg , the functors U and G , and proving the adjunction U⊣G . Section 5 investigates the topological aspects, proving that the forgetful functor V:AppEqAlg → EqAlg is topological and describing initial and final lifts. Section 6 provides a concrete example based on a three-element equality algebra to illustrate the constructions, and Section 7 concludes the paper with a brief discussion of possible directions for future work. 2. Preliminaries In this section, we review the basic definitions and properties of equality algebras and rough set theory that are essential for the subsequent discussions. For more details on equality algebras, we refer the reader to [1, 2]. 2.1. Equality Algebras. Equality algebras were introduced by Jenei as a structure dealing with the equality connective in fuzzy logic, inspired by the fact that fuzzy type theory relies heavily on equality rather than implication. 4 JOAQUIM REIZI HIGUCHI Definition 2.1. An algebra E = ( E, ∧,∼, 1) of type (2 , 2 , 0) is called an equality algebra if it satisfies the following conditions for all x, y, z ∈E: (E1) (E, ∧,1) is a commutative semilattice with top element 1. (E2) x∼y=y∼x(Commutativity of ∼). (E3) x∼x= 1. (E4) x∼1 = x. (E5) x≤y≤z = ⇒x∼z≤y∼z and x∼z≤x∼y , where x≤y if and only if x∧y=x. (E6) x∼y≤x∧z∼y∧z. (E7) x∼y≤(x∼z)∼(y∼z). We interpret the operation ∼ as a fuzzy equality (bi-implication). An equality algebra is bounded if there exists an element 0 ∈E such that 0 ≤x for all x∈E . In this paper, unless otherwise stated, we do not assume boundedness, but our examples typically involve bounded chains. Crucial to our study is the notion of congruence relations, as they form the basis of the approximation space. Definition 2.2. Let E = ( E, ∧,∼, 1) be an equality algebra. An equivalence relation θ on E is called a congruence relation if it is compatible with the operations: (x, y)∈θand (u, v)∈θ=⇒(x∧u, y ∧v)∈θand (x∼u, y ∼v)∈θ for all x, y, u, v ∈E. The set of all congruence relations on E is denoted by Con ( E ). It is a well-known result in universal algebra that Con ( E ) forms a complete lattice under set inclusion. 2.2. Rough Set Theory. Rough set theory, proposed by Pawlak [ 3 ], provides a formal tool for dealing with uncertainty arising from indiscernibility. Definition 2.3. An approximation space is a pair ( U, R ), where U is a non-empty set (the universe) and R is an equivalence relation on U . For any subset X⊆U , the lower approximation R ( X ) and the upper approximation R(X) are defined as: R(X) = {x∈U|[x]R⊆X}, R(X) = {x∈U|[x]R∩X=∅}, where [x]Rdenotes the equivalence class of xwith respect to R. A subset X is called exact (or crisp) with respect to R if R ( X ) = R ( X ). Otherwise, it is called rough. In recent years, this theory has been applied to various algebraic structures such as groups, rings, and logical algebras (e.g., [ 4 ]). In such algebraic contexts, the universe U is replaced by an algebraic structure A , and R is typically chosen to be a congruence relation θ . The main interest lies in whether the approximations of a subalgebra are again subalgebras. As we shall see in Section 3, equality algebras provide a rich setting for such investigations. A CATEGORICAL APPROACH TO ROUGH EQUALITY ALGEBRAS 5 3. Algebraic Properties of Rough Approximations In this section, we investigate the algebraic properties of rough approximations induced by a congruence relation on an equality algebra. While many of the constructions can be formulated in the general setting of universal algebras, we restrict ourselves to equality algebras, since this is the ambient variety for the categorical framework developed later. Throughout this section, let E = ( E, ∧,∼, 1) be an equality algebra and let θ be a congruence relation on E . We denote by P ( E ) the power set of E , and by Sub ( E ) the set of all subalgebras of E , which forms a complete lattice under set inclusion. 3.1. Rough approximations on subsets and subalgebras. We first extend the definition of rough approximations to arbitrary subsets of E ; the case of subalgebras will then be obtained by restriction. Definition 3.1. For any subset X⊆E , the upper approximation θ ( X ) and the lower approximation θ ( X ) of X with respect to θ are defined by: θ(X) = {x∈E|[x]θ∩X=∅}, θ(X) = {x∈E|[x]θ⊆X}. If A is a subalgebra of E (denoted A≤ E ), we write θ ( A ) and θ ( A ) for the approximations of the underlying set A. These operators satisfy the usual Pawlak-type inequalities at the level of the power set. Lemma 3.2. For all X, Y ⊆E, the following hold: (i) θ(X)⊆X⊆θ(X). (ii) If X⊆Y, then θ(X)⊆θ(Y)and θ(X)⊆θ(Y). (iii) θ(θ(X)) = θ(X)and θ(θ(X)) = θ(X). Proof. (i) If x∈θ ( X ), then by definition [ x ] θ⊆X , and in particular x∈X . Conversely, if x∈X , then [ x ] θ∩X = ∅ (because x∈ [ x ] θ∩X ), hence x∈θ(X). (ii) Monotonicity is immediate: if X⊆Y and [ x ] θ⊆X , then [ x ] θ⊆Y , so x∈θ(Y); similarly, if [x]θ∩X=∅, then [x]θ∩Y=∅. (iii) For idempotence of θ , extensivity (i) gives θ ( X ) ⊆θ ( θ ( X )). For the converse inclusion, let x∈θ ( θ ( X )). Then there exists y∈θ ( X ) such that ( x, y ) ∈θ . By definition of θ ( X ), there exists z∈X with ( y, z ) ∈θ . By transitivity of θwe obtain (x, z)∈θ, so [x]θ∩X=∅and hence x∈θ(X). The dual argument gives idempotence of θ : if x∈θ ( X ), then for any y∈ [ x ] θ we have [ y ] θ = [ x ] θ⊆X , so y∈θ ( X ) and thus [ x ] θ⊆θ ( X ), i.e. x∈θ(θ(X)). Together with (i) this yields θ(θ(X)) = θ(X). □ Theorem 3.3 (Upper approximation as closure).The operator θ : P ( E ) → P ( E )is a closure operator on the complete lattice ( P ( E ) ,⊆ ). That is, for all X, Y ⊆E: (a) X⊆θ(X) (extensivity); (b) X⊆Y=⇒θ(X)⊆θ(Y) (monotonicity); (c) θ(θ(X)) = θ(X) (idempotence). 6 JOAQUIM REIZI HIGUCHI In particular, whenever A∈Sub ( E ), the set θ ( A )is a subalgebra of E , and the restriction of θ to Sub ( E )defines a closure operator on the subalgebra lattice Sub(E). Proof. The three axioms of a closure operator on P ( E ) are exactly Lemma 3.2 (i)–(iii). It remains to show that if A is a subalgebra of E , then θ ( A ) is again a subalgebra. Let A≤ E and let x, y ∈θ ( A ). By definition, there exist a, b ∈A such that ( x, a ) ∈θ and ( y, b ) ∈θ . Since θ is a congruence, it is compatible with the operations ∧and ∼, so (x∧y, a ∧b)∈θand (x∼y, a ∼b)∈θ. Because Ais a subalgebra, a∧b∈Aand a∼b∈A. Hence [x∧y]θ∩A=∅and [x∼y]θ∩A=∅, which shows x∧y∈θ ( A ) and x∼y∈θ ( A ). Moreover, 1 ∈A and (1 , 1) ∈θ , so 1 ∈θ ( A ). Therefore θ ( A ) is closed under ∧ , ∼ and contains 1, i.e. θ ( A ) is a subalgebra. Since θ is extensive, monotone and idempotent on P ( E ), and maps Sub ( E ) into itself, its restriction θ↾Sub(E):Sub(E)→Sub(E) is a closure operator on the complete lattice Sub(E). □ Theorem 3.4 (Lower approximation as interior).The operator θ : P ( E ) → P(E)is an interior operator on (P(E),⊆). That is, for all X, Y ⊆E: (a) θ(X)⊆X(contractivity); (b) X⊆Y=⇒θ(X)⊆θ(Y) (monotonicity); (c) θ(θ(X)) = θ(X) (idempotence). Proof. All three properties have already been verified in Lemma 3.2. Contractivity is Lemma 3.2(i), monotonicity is Lemma 3.2(ii), and idempotence is Lemma 3.2(iii). □ Remark 3.5. In contrast to the upper approximation, the lower approximation θ does not in general map Sub ( E ) into itself: for a subalgebra A≤ E , the set θ ( A ) is always a θ -saturated subset of A , but it need not be a subalgebra unless additional hypotheses are imposed. We therefore work with θ as a closure operator on Sub ( E ), while θ will be used at the level of P(E) and for characterizing exact (i.e. θ-saturated) subalgebras. 3.2. Exact subalgebras and the quotient algebra. We now isolate those subalgebras that are “crisp” with respect to the given congruence, i.e. unions of θ-classes. Definition 3.6. A subalgebra S∈Sub ( E ) is called exact (or θ -definable) if θ(S) = S. We denote the family of all exact subalgebras by Tθ(E) = {S∈Sub(E)|θ(S) = S}. If S is exact, then it is θ -saturated by definition: whenever x∈S and ( x, y ) ∈θ , we have [ y ] θ = [ x ] θ⊆θ ( S ) = S , so y∈S . Conversely, any θ -saturated subalgebra S satisfies θ ( S ) = S , since for every x∈θ ( S ) we have [x]θ∩S=∅, hence [x]θ⊆Sby saturation and thus x∈S. A CATEGORICAL APPROACH TO ROUGH EQUALITY ALGEBRAS 7 Lemma 3.7. If S∈ Tθ(E), then θ(S) = S=θ(S). Proof. We already observed that θ ( S ) = S by definition. For the lower approximation, recall that S is a union of θ -classes. If x∈S , then [ x ] θ⊆S , hence x∈θ ( S ). Thus S⊆θ ( S ). On the other hand, θ ( S ) ⊆S holds for every subset Sby contractivity, so θ(S) = S.□ The next theorem shows that exact subalgebras can be identified with subalgebras of the quotient algebra E/θ in a lattice-theoretic way. Theorem 3.8. Let E/θ be the quotient equality algebra. The complete lattice of exact subalgebras Tθ ( E )is isomorphic to the complete lattice of subalgebras of the quotient algebra, Sub(E/θ). Proof. Let q : E → E/θ be the canonical quotient homomorphism defined by q(x)=[x]θfor all x∈E. Step 1: Definition and basic properties of the maps. Define Ψ : Sub(E/θ)→ Tθ(E),Ψ(T) = q−1(T) = {x∈E|[x]θ∈T} for each subalgebra T≤ E/θ . Since q is a homomorphism of algebras, the inverse image of a subalgebra is a subalgebra. Moreover, if x∈q−1 ( T ) and ( x, y ) ∈θ , then q ( x ) = q ( y ), so q ( y ) ∈T and hence y∈q−1 ( T ). Thus q−1 ( T ) is θ-saturated, so Ψ(T)∈ Tθ(E). Conversely, define Φ : Tθ(E)→Sub(E/θ),Φ(S) = q(S) = {[x]θ|x∈S} for each exact subalgebra S∈ Tθ ( E ). As q is a surjective homomorphism and Sis a subalgebra, q(S) is a subalgebra of E/θ. Step 2: Φand Ψare inverse to each other. Let T∈Sub(E/θ). Then Φ(Ψ(T)) = q(q−1(T)) = T, since qis surjective. Thus Φ ◦Ψ = idSub(E/θ). Now let S∈ Tθ(E). Then Ψ(Φ(S)) = q−1(q(S)). Clearly S⊆q−1 ( q ( S )) for every subset S⊆E . To show the reverse inclusion, let x∈q−1 ( q ( S )). Then q ( x ) ∈q ( S ), so there exists s∈S with q ( x ) = q ( s ), i.e. ( x, s ) ∈θ . Since S is θ -saturated, we have x∈S . Hence q−1 ( q ( S )) ⊆S , and therefore Ψ(Φ(S)) = S. Thus Ψ ◦Φ = idTθ(E). Step 3: Preservation of arbitrary meets and joins. It remains to check that Ψ is a complete lattice isomorphism. Let ( Ti ) i∈I be a family of subalgebras of E/θ. Then Ψ\ i∈I Ti=q−1\ i∈I Ti=\ i∈I q−1(Ti) = \ i∈I Ψ(Ti), so Ψ preserves arbitrary meets (intersections). For joins, recall that the join in the lattice of subalgebras is the subalgebra generated by the union: _ i∈I Ti=⟨[ i∈I Ti⟩. 8 JOAQUIM REIZI HIGUCHI Using standard facts from universal algebra about inverse images of generated subalgebras under surjective homomorphisms, we obtain Ψ_ i∈I Ti=q−1[ i∈I Ti=q−1[ i∈I Ti =[ i∈I q−1(Ti)=_ i∈I q−1(Ti) = _ i∈I Ψ(Ti). Thus Ψ preserves arbitrary joins as well. Since Φ is the inverse map of Ψ, it follows that both Ψ and Φ are complete lattice isomorphisms between Sub(E/θ) and Tθ(E). □ This theorem shows that the “exact” (i.e. θ -definable) part of the rough structure on E is completely captured by the ordinary algebra of the quotient E/θ . In categorical terms, Tθ ( E ) can be viewed as the full sublattice of Sub ( E ) consisting of θ -closed subobjects, and Theorem 3.8 identifies this sublattice with the subobject lattice of the corresponding quotient in the category of equality algebras. 4. Categorical Framework: The Adjunction In this section, we lift the local algebraic properties of Section 3 to a global categorical setting. We introduce the category of approximation equality algebras, denoted by AppEqAlg , in which objects are equality algebras equipped with a chosen congruence (the “approximation structure”). The passage from a rough object ( E, θ ) to its “crisp” quotient E/θ then becomes a functor U:AppEqAlg −→ EqAlg, and our main result is that this functor is left adjoint to a canonical embedding G:EqAlg −→ AppEqAlg. In particular, the universal property of the quotient E/θ is expressed purely categorically as an adjunction U⊣G . Combined with Theorem 3.8, this shows that the exact (i.e. θ -definable) subalgebras of ( E, θ ) are controlled by the unit of this adjunction. 4.1. The Category AppEqAlg .We first spell out the category of equality algebras endowed with a distinguished congruence relation. Definition 4.1. The category AppEqAlg of approximation equality algebras is defined as follows: (1) Objects: Pairs ( E, θ ), where E is an equality algebra and θ is a congruence relation on E. (2) Morphisms: A morphism f: (E1, θ1)−→ (E2, θ2) is a homomorphism of equality algebras f:E1→E2such that (x, y)∈θ1=⇒(f(x), f(y)) ∈θ2for all x, y ∈E1. We call such morphisms relation-preserving homomorphisms. (3) Composition is given by the usual composition of functions. (4) Identities are the identity homomorphisms idE:E→E. A CATEGORICAL APPROACH TO ROUGH EQUALITY ALGEBRAS 9 It is straightforward to verify that AppEqAlg is indeed a category: the identity on ( E, θ ) is relation-preserving, and the composite of relationpreserving homomorphisms is again relation-preserving. 4.2. Approximation (quotient) functors. We now define the basic functors relating AppEqAlg to the ambient category EqAlg of equality algebras and homomorphisms. Definition 4.2 (Quotient functor).We define a functor U:AppEqAlg −→ EqAlg by: •On objects: For any (E, θ)∈AppEqAlg, U(E, θ) = E/θ. That is, U sends an equality algebra together with a congruence to the corresponding quotient equality algebra. •On morphisms: For a morphism f: (E1, θ1)−→ (E2, θ2), we define U(f): E1/θ1→E2/θ2by U(f)([x]θ1)=[f(x)]θ2, x ∈E1. The map U ( f ) is well-defined: if [ x ] θ1 = [ y ] θ1 , then ( x, y ) ∈θ1 , and since f is relation-preserving, ( f ( x ) , f ( y )) ∈θ2 , so [ f ( x )] θ2 = [ f ( y )] θ2 . One easily checks that U ( id(E,θ) ) = idE/θ and U ( g◦f ) = U ( g ) ◦U ( f ), so U is a functor. Next we embed EqAlg into AppEqAlg by attaching the finest possible congruence, namely the diagonal. Definition 4.3 (Diagonal functor).We define a functor G:EqAlg −→ AppEqAlg by: •On objects: For an equality algebra Y∈EqAlg, G(Y) = (Y, ∆Y), where ∆Y={(y, y)|y∈Y}is the diagonal (equality) relation. • On morphisms: For a homomorphism g:Y1→Y2 in EqAlg , we set G(g) = g: (Y1,∆Y1)−→ (Y2,∆Y2). The map g is automatically relation-preserving: if ( y, y ) ∈ ∆ Y1 , then (g(y), g(y)) ∈∆Y2. Thus Gis indeed a functor G:EqAlg →AppEqAlg. 4.3. The adjunction theorem. We now show that the quotient functor U is left adjoint to the diagonal functor G. Theorem 4.4. The functor U:AppEqAlg →EqAlg is left adjoint to the functor G:EqAlg →AppEqAlg. That is, there is a natural bijection Φ(E,θ),Y : HomEqAlg(U(E, θ), Y )∼ =HomAppEqAlg((E, θ), G(Y)) for all ( E, θ ) ∈AppEqAlg and Y∈EqAlg , natural in both variables. Equivalently, U⊣G.