On the Exactness and Categorical Properties of the Category of Basic Hoops

On the Exactness and Categorical Properties of the Category of Basic Hoops Joaquim Reizi Higuchi November 23, 2025 Abstract We consider the category BH of basic hoops, that is, integral commutative residuated monoids satisfying divisibility and prelinearity. As a finitary variety, BH is known to be complete, cocomplete, regular and Barr-exact. The aim of this paper is to give a self-contained and concrete description of these categorical properties in terms of filters and congruences. We first review the algebraic structure of basic hoops and describe limits and colimits in BH at the level of operations. We then develop in detail the correspondence between filters and congruences, and we characterize filter quotients and kernel filters. Using this machinery, we present explicit filter-based proofs of regularity and exactness of BH and identify internal equivalence relations with congruences. We also discuss the relation with BL-algebras and other residuated structures, and we outline some universal-algebraic and logical questions suggested by this description. In this way the paper provides a structural note and partial survey on the categorical algebra of basic hoops. Keywords: basic hoops, BL-algebras, regular category, Barr-exact category, filters, congruences. MSC 2020: Primary 06F05; Secondary 03G10, 18B15, 18E10. 1 Introduction The algebraic semantics of substructural and fuzzy logics has led to a number of important varieties of residuated structures, among which hoops and BL-algebras play a central role (see, e.g., [5,2,6]). From the logical point of view, these algebras provide semantics for a wide range of multiple-valued logics based on continuous t-norms and their residua; from the algebraic point of view, they are integral, commutative, residuated monoids satisfying suitable divisibility and prelinearity conditions. Basic hoops form a distinguished subvariety of hoops, obtained by adding a prelinearity axiom. They arise as the implicational reducts of H´ajek’s BL-algebras, and thus algebraize the implicational fragment of basic logic and of several of its extensions, including many standard t-norm based logics. While the algebraic and logical aspects of hoops and BL-algebras have been extensively investigated, the categorical structure of the category BH of basic hoops and homomorphisms has not been analyzed in a systematic way. On the one hand, general results of universal algebra and categorical algebra imply that any finitary variety of algebras forms 1 a Barr-exact category in the sense of Barr [1,3]. On the other hand, in algebraic logic and many-valued semantics it is often crucial to work with explicit descriptions of limits, colimits, regular epimorphisms and effective equivalence relations in terms of the underlying algebraic structure, since these translate directly into the behaviour of deductive filters, Lindenbaum algebras, and quotient logics. The aim of this paper is to provide a detailed and self-contained categorical analysis of the category BH, with particular emphasis on its role as an algebraic semantics for basic logic and related many-valued logics. We focus on the interaction between filters, congruences, quotients, and the regular and exact structure of BH, and we formulate all our constructions in a filter-based language that has a direct logical interpretation in terms of deductive systems. Although the final conclusion that BH is Barr-exact is a special case of the general theory of varieties, the concrete filter-based description we obtain is tailored to the many-valued logic setting and designed to be used as a tool in the study of algebraizable and t-norm-based logics. Main contributions. The main results of the paper can be summarized as follows. (1) We give explicit constructions of limits and colimits in BH. In particular, we describe products and equalizers as cartesian products and subalgebras determined by equalizing conditions, and coproducts and coequalizers as free basic hoops modulo suitable congruences. As a consequence, we show that BH is complete and cocomplete, and that all small limits and colimits can be obtained from products, coproducts, equalizers and coequalizers (Theorem 3.7). This provides a concrete description of the categorical constructions that underlie standard model-theoretic operations on basic-hoop semantics. (2) We establish a precise filter–congruence correspondence for basic hoops: filters on a basic hoop Aare in bijection with congruences on A, via explicit maps F7→ θF and θ7→ Fθ(Theorem 4.13). Using this correspondence, we show that surjective homomorphisms in BH are exactly quotients by filters, and we obtain a filter-based formulation of the first isomorphism theorem (Theorem 4.19). Logically, this recovers and systematizes the correspondence between deductive filters, congruences and Lindenbaum algebras for the implicational fragment of basic logic and its extensions. (3) We characterize regular epimorphisms in BH as precisely the surjective homomorphisms, and prove that they are stable under pullback. Combined with the existence of finite limits, this shows that BH is a regular category in the sense of Barr [1,3] (Theorem 5.7). From the viewpoint of algebraic logic, this yields a clean factorization theory for semantic consequence maps between basic hoops, with kernel filters and image subalgebras playing the role of algebraic invariants of deductive systems. (4) We give an explicit description of internal equivalence relations in BH in terms of congruences and filters, and prove that every such relation is effective: it arises as the kernel pair of the corresponding quotient homomorphism A→A/θ or A→ A/F. In particular, we show that BH is a Barr-exact category (Theorem 6.6). This identifies internal equivalence relations with algebraic counterparts of provable equivalence in basic logic and related many-valued systems. 2 Taken together, these results provide a concrete description of the regular and exact structure of BH in algebraic and filter-theoretic terms. From the point of view of algebraic logic and many-valued semantics, our analysis recasts the standard correspondence between filters, congruences and Lindenbaum algebras in a categorical language, emphasizing the role of coequalizers, regular epimorphisms, and effective equivalence relations in the algebraic semantics of basic logic and its extensions. In this way the paper offers a structural framework that can be used in the study of algebraizable multiple-valued logics based on basic hoops and their expansions. Structure of the paper. Section 2collects basic background on hoops, basic hoops, and categorical notions of regularity and exactness. In Section 3we construct limits and colimits in BH explicitly, and prove completeness and cocompleteness. The filter– congruence correspondence and the description of quotients modulo filters are developed in Section 4. In Section 5we prove that BH is a regular category, identifying regular epimorphisms with surjective homomorphisms and studying their stability under pullback. Section 6contains the proof that BH is Barr-exact: internal equivalence relations are identified with congruences and are shown to be effective via their associated filter quotients. Finally, Section 7discusses connections with BL-algebras and other residuated structures, comments on the logical interpretations of our results, and outlines several directions for further investigation in the setting of multiple-valued and fuzzy logics. 2 Preliminaries In this section we fix notation and recall the algebraic and categorical background needed in the rest of the paper. We assume basic familiarity with universal algebra and elementary category theory. Our presentation is mainly algebraic; categorical notions are recalled only to the extent needed to describe the structural properties of the category of basic hoops that will be applied later to algebraic semantics of many-valued logics. Throughout, we work in the algebraic language L={·,→,1} of type (2,2,0), and we write simply ·for the monoidal product and →for the residuation operation. 2.1 Basic hoops We begin by recalling the notion of a hoop, and then specialize to basic hoops. Definition 2.1 (Hoop).Ahoop is an algebra A= (A, ·,→,1) of type (2,2,0) such that the following identities hold for all x, y, z ∈A: (H1) (A, ·,1) is a commutative monoid: x·(y·z)=(x·y)·z, x ·y=y·x, x ·1 = x= 1 ·x; (H2) x→x= 1; (H3) (x·y)→z=x→(y→z); 3 (H4) x·(x→y) = y·(y→x). The equational axiomatization above is standard in the literature on hoops; in particular, it is equivalent to the more usual description of hoops as cancellative, integral, commutative, residuated monoids satisfying a suitable divisibility condition. Remark 2.2 (Natural order and residuation).Let Abe a hoop. Define a binary relation ≤on Aby x≤y:⇐⇒ x→y= 1. Then: (i) ≤is a partial order on A; (ii) (A, ·,1,≤) is an integral, commutative, residuated monoid: for all x, y, z ∈A, x·y≤z⇐⇒ x≤y→z; (iii) the identity x·(x→y) = y·(y→x) implies that the operation x∧y:= x·(x→y) = y·(y→x) defines the infimum of xand ywith respect to ≤, so (A, ∧,≤) is a meet-semilattice. We call ≤the natural order of A. In the standard many-valued interpretation, the natural order compares truth degrees, ·represents a form of conjunction based on a t-norm, and →represents the corresponding implication. The term-definable meet will play a role in our description of filters, congruences and quotients, which in turn correspond to deductive systems and quotient logics. Definition 2.3 (Basic hoop).Abasic hoop is a hoop A= (A, ·,→,1) satisfying the prelinearity axiom (x→y)→z≤((y→x)→z)→z(1) for all x, y, z ∈A, with respect to the natural order ≤of Remark 2.2. Equivalently, since u≤vis the same as u→v= 1, the prelinearity axiom can be written equationally as (x→y)→z→((y→x)→z)→z= 1,(2) for all x, y, z ∈A. We denote by Ho the class (indeed variety) of all hoops, and by BH the class of all basic hoops. Proposition 2.4. The class BH of basic hoops is a finitary variety of algebras in the language L={·,→,1}. Proof. The identities (H1)–(H4) and (2) form a finite set of equations in the language L. Thus BH is an equational class in the sense of Birkhoff, and is consequently closed under products, subalgebras, and homomorphic images. 4 Remark 2.5 (Connection to BL-algebras and basic logic).Basic hoops arise naturally as implicational subreducts of H´ajek’s BL-algebras and, more generally, as the implicational reducts of residuated structures associated with continuous t-norms on [0,1]. The variety BH is generated by such examples and algebraizes H´ajek’s basic logic at the level of its implicational fragment. In particular, filters on a basic hoop correspond to deductively closed sets of implications, and quotients by filters yield Lindenbaum basic hoops of corresponding many-valued logics. In the present paper we use this logical viewpoint mainly as motivation and as an interpretation of our categorical constructions; no specific proof system will be needed. 2.2 The category BH of basic hoops We now pass from the algebraic structure of basic hoops to their categorical organization, which will be our main tool for describing the algebraic semantics of basic logic and related many-valued logics. Definition 2.6 (The category BH).Let BH denote the category defined as follows. •Objects are basic hoops A= (A, ·,→,1). •Morphisms f:A→Bare homomorphisms of basic hoops, i.e. functions f:A→B preserving all operations and the constant: f(x·y) = f(x)·f(y), f(x→y) = f(x)→f(y), f(1A) = 1B, for all x, y ∈A. Composition and identities are taken in the usual way from the category of sets. We write Set for the category of sets and functions, and denote by U:BH −→ Set the forgetful functor sending a basic hoop to its underlying set and a homomorphism to its underlying function. The functor Uis faithful and reflects isomorphisms. Lemma 2.7 (Order-theoretic behaviour of morphisms).Let f:A→Bbe a morphism in BH. Then: (i) fis monotone with respect to the natural orders, i.e. x≤Ayimplies f(x)≤Bf(y); (ii) fpreserves the term-definable meet: f(x∧y) = f(x)∧f(y) for all x, y ∈A, where x∧y:= x·(x→y). Proof. (i) If x≤Ay, then x→y= 1A. Applying fand using preservation of →and 1 yields f(x)→f(y) = f(x→y) = f(1A) = 1B, so f(x)≤Bf(y). (ii) Using the definition of ∧and preservation of ·and →we obtain f(x∧y) = fx·(x→y)=f(x)·f(x→y) = f(x)·f(x)→f(y)=f(x)∧f(y), as required. 5 In the semantic reading, Lemma 2.7 says that homomorphisms between basic hoops are monotone w.r.t. truth degrees and preserve the interpretation of the (term-definable) meet, so they behave as truth-preserving maps between many-valued models. Example 2.8 (Terminal and zero object).The one-element algebra 1= ({1},·,→,1), with the only possible definitions 1 ·1 = 1 and 1 →1 = 1, is a basic hoop. For any basic hoop Athere is exactly one homomorphism A→1and exactly one homomorphism 1→A. Thus 1is both terminal and initial in BH, and BH is a pointed category. Since BH is a finitary variety, it inherits many structural properties from the general theory of universal algebra. Proposition 2.9 (Variety-theoretic properties).The category BH has the following properties: (i) U:BH →Set creates all small limits and all sifted colimits; (ii) BH is complete and cocomplete: it has all small limits and colimits; (iii) limits and sifted colimits in BH are computed on underlying sets and endowed with the pointwise basic hoop structure. Proof. This is standard for any finitary variety of algebras. The category BH is the Eilenberg–Moore category for the finitary monad on Set given by the free basic hoop construction. For such categories, the forgetful functor creates all limits and all sifted colimits, and the category is complete and cocomplete. We will not use Proposition 2.9 as a black box; instead, in Section 3we give explicit descriptions of products, equalizers, coproducts, and coequalizers in BH, and derive completeness and cocompleteness directly from those constructions. These concrete descriptions are later interpreted in terms of constructions on algebraic semantics of many-valued logics. 2.3 Regular and exact categories We briefly recall the notions of regular and exact categories in the sense of Barr. We refer to standard references in categorical algebra for further details. In our setting they will be used to formalize the behaviour of quotients, kernel pairs and equivalence relations on basic hoops, and hence of quotient logics and logical equivalence. Definition 2.10 (Kernel pair).Let Cbe a category with pullbacks and let f:X→Ybe a morphism in C. The kernel pair of fis the pair of morphisms p1, p2:R→Xobtained as the pullback of falong itself: R X X Y p2 p1f f The object Ris often regarded as an internal equivalence relation on X, consisting of pairs of elements identified by f. Definition 2.11 (Regular epimorphism).In a category Cwith coequalizers, a morphism e:X→Yis called a regular epimorphism if it is a coequalizer of some pair of morphisms, i.e. there exist r1, r2:W→Xsuch that e◦r1=e◦r2 and eis universal with this property. 6 Definition 2.12 (Regular category).A category Cis called regular if: (i) it has all finite limits; (ii) every morphism f:X→Yadmits a factorization Xe −→ Im −→ Y where eis a regular epimorphism and mis a monomorphism; (iii) regular epimorphisms are stable under pullback: if e:B→Cis a regular epimorphism and P B A C p2 p1e g is a pullback square, then p1is again a regular epimorphism. There are several equivalent formulations of regularity; the one above, based on image factorizations and pullback stability of regular epimorphisms, is convenient for our purposes. We next recall the categorical notion of an internal equivalence relation and of Barrexactness. Definition 2.13 (Internal equivalence relation).Let Cbe a category with finite limits and let Abe an object of C. An internal equivalence relation on Aconsists of a monomorphism m:R→A×Atogether with the induced projections r1, r2:R→A(obtained by composing with the first and second projections from A×A) such that: (i) (reflexivity) the diagonal morphism ∆A:A→A×Afactors through m; (ii) (symmetry)Ris stable under the flip τ:A×A→A×A,τ(x, y) = (y, x); (iii) (transitivity) the usual “composition” diagram, built as a pullback of Rover itself and then mapped back into R, is well-defined and satisfies appropriate associativity conditions. Intuitively, an internal equivalence relation is an ordinary equivalence relation on the underlying object together with the requirement that it be realized as a subobject of A×A. Definition 2.14 (Effective equivalence relation).Let Cbe a category with finite limits and coequalizers. An internal equivalence relation m:R→A×Ais called effective if there exists a morphism q:A→Qsuch that Ris (isomorphic to) the kernel pair of q. Definition 2.15 (Barr-exact category).A category Cis called Barr-exact (or simply exact) if: (i) Cis regular; (ii) every internal equivalence relation in Cis effective. It is a classical result that any finitary variety of algebras is Barr-exact. In this paper we revisit this general fact in the particular case of basic hoops, giving concrete constructions of: •limits and colimits in BH (Section 3); •quotients by congruences and by filters (Section 4); •image factorizations and regular epimorphisms (Section 5); •kernel pairs and effective equivalence relations (Section 6). 7 This not only provides an explicit algebraic proof that BH is Barr-exact, but also clarifies the role of filter quotients in the categorical structure of basic hoops and, via the algebraic correspondence between filters and theories, in the algebraic semantics of basic logic and related many-valued logics. 3 Limits and colimits in the category of basic hoops In this section we give explicit descriptions of limits and colimits in BH. Although completeness and cocompleteness of BH follow from the general theory of finitary varieties (see Proposition 2.9), we prefer to record concrete constructions of products, equalizers, coproducts and coequalizers, as these will be used later in our analysis of regular epimorphisms and exactness. 3.1 Products, equalizers, and pullbacks We first describe products and equalizers in BH, and then deduce a concrete description of pullbacks. Proposition 3.1 (Products).Let {Ai= (Ai,·,→,1i)}i∈Ibe a family of basic hoops. Define the cartesian product of the underlying sets A:= Y i∈I Ai and equip Awith operations and constant given componentwise by (x·y)(i) := x(i)·y(i),(x→y)(i) := x(i)→y(i),1A(i) := 1i, for all x, y ∈Aand i∈I. Then: (a) Ais a basic hoop; (b) for each i∈I, the projection πi:A→Ai, πi(x) := x(i), is a morphism in BH; (c) for any basic hoop Xand any family of morphisms fi:X→Ai(i∈I), there exists a unique morphism f:X→Asuch that πi◦f=fifor all i∈I. Consequently, Awith the projections (πi)i∈Iis the product Qi∈IAiin BH. Proof. (a) Since each Aisatisfies the defining equations of basic hoops and the operations on Aare defined componentwise, Aalso satisfies those equations. Thus Ais a basic hoop. (b) Each πiplainly preserves ·,→, and 1, hence is a homomorphism. (c) Given fi:X→Ai, define f:X→Aby f(x)(i) := fi(x). This is a homomorphism because the homomorphism identities hold coordinatewise. Uniqueness follows from the fact that any m:X→Awith πi◦m=fimust satisfy m(x)(i) = fi(x) for all i, hence m=f. Equalizers are realized as subalgebras determined by pointwise equalities. 8 Proposition 3.2 (Equalizers).Let f, g :A→Bbe morphisms in BH. Define E:= {x∈A|f(x) = g(x)}. Then: (a) Eis a subalgebra of A, hence a basic hoop with the induced operations; (b) the inclusion e:E ,→Ais the equalizer of fand gin BH. Proof. (a) If x, y ∈E, then f(x·y) = f(x)·f(y) = g(x)·g(y) = g(x·y), so x·y∈E. Similarly, f(x→y) = f(x)→f(y) = g(x)→g(y) = g(x→y), so x→y∈E. Finally, f(1A)=1B=g(1A), hence 1A∈E. Thus Eis closed under the basic operations and contains 1, so it is a subalgebra of A. (b) By construction, f◦e=g◦e. Let h:X→Abe any morphism with f◦h=g◦h. Then for each x∈X,f(h(x)) = g(h(x)), hence h(x)∈E. Thus hfactors uniquely through evia a homomorphism ¯ h:X→Egiven by ¯ h(x) := h(x) viewed as an element of E. This is the universal property of the equalizer. Pullbacks can be constructed from products and equalizers in the usual way. In our setting we can describe them concretely as follows. Corollary 3.3 (Pullbacks).Let f:X→Zand g:Y→Zbe morphisms in BH. Let X×Ybe the product in BH with projections π1:X×Y→Xand π2:X×Y→Y. Consider the pair of morphisms f◦π1, g ◦π2:X×Y⇒Z. Let e:P ,→X×Ybe their equalizer. Then Pcan be identified with the subalgebra P:= {(x, y)∈X×Y|f(x) = g(y)} of X×Y, equipped with the induced operations, and the morphisms p1:= π1◦e:P→X, p2:= π2◦e:P→Y form a pullback of fand gin BH. Proof. By Proposition 3.2, the equalizer e:P ,→X×Yof f◦π1and g◦π2is the subalgebra of X×Yconsisting of those (x, y) such that f(π1(x, y)) = g(π2(x, y)), i.e. f(x) = g(y). Thus Pis exactly the set displayed in the statement, with the induced operations. Now let h:W→Xand k:W→Ybe morphisms in BH such that f◦h=g◦k. Then the unique morphism ⟨h, k⟩:W→X×Yinto the product satisfies f◦π1◦ ⟨h, k⟩=f◦h=g◦k=g◦π2◦ ⟨h, k⟩, so ⟨h, k⟩equalizes f◦π1and g◦π2and therefore factors uniquely through eby a morphism u:W→Pwith e◦u=⟨h, k⟩. By definition of p1and p2, we then have p1◦u=h and p2◦u=k. Conversely, since the morphism urequired for p1and p2to satisfy the universal property of a pullback is uniquely determined by the condition e◦u=⟨h, k⟩, it follows that (P, p1, p2) is the pullback of fand g. 9 we have (x·(x→y), x ·1) = (x·(x→y), x)∈θ and (y·(y→x), y ·1) = (y·(y→x), y)∈θ. Since x·(x→y) = y·(y→x), these two pairs show that some common element c:= x·(x→y) is θ-equivalent to both xand y. By transitivity, (x, y)∈θ. Thus θFθ⊆θ, and the two inclusions together yield θFθ=θ. Theorem 4.13 (Filter–congruence correspondence).For a basic hoop A, the assignments F7−→ θF, θ 7−→ Fθ define mutually inverse, inclusion-preserving bijections between the set of filters on Aand the set of congruences on A. Proof. By Propositions 4.8 and 4.10, the assignments are well-defined. Propositions 4.11 and 4.12 show that FθF=Fand θFθ=θ, so the two maps are mutually inverse bijections. Monotonicity in each direction is immediate from the definitions. 4.3 Quotients modulo filters We now use the filter–congruence correspondence to express congruence quotients in terms of filters, i.e. in terms of deductive systems. Definition 4.14 (Quotient by a filter).Let Fbe a filter on A. The quotient of Amodulo Fis the quotient basic hoop A/F := A/θF, together with the canonical surjective homomorphism qF:A→A/F, qF(x) := [x]θF. Lemma 4.15. For any filter Fon A, q−1 F({1A/F }) = F. Proof. We have 1A/F = [1]θFby construction. For any x∈A, qF(x) = 1A/F ⇐⇒ [x]θF= [1]θF⇐⇒ (x, 1) ∈θF. As in the proof of Proposition 4.11, this is equivalent to x∈F. Thus filters on Aare exactly the inverse images of 1 under homomorphisms into quotient algebras, matching the logical intuition that theories are preimages of designated truth-values under semantics. Given a morphism f:A→B, we can describe its kernel congruence in terms of a filter, namely the preimage of 1. Definition 4.16 (Kernel filter).Let f:A→Bbe a morphism in BH. The kernel filter of fis ker1(f) := {x∈A|f(x) = 1B}. 16 Lemma 4.17. For any morphism f:A→Bin BH, the subset ker1(f)is a filter on A. Proof. We have 1A∈ker1(f) because f(1A)=1B. If x∈ker1(f) and x→y∈ker1(f), then f(x)=1Band f(x→y) = f(x)→f(y) = 1B→f(y) = f(y), so f(y) = 1Band y∈ker1(f). Thus ker1(f) is a filter. Proposition 4.18 (Kernel congruence via kernel filter).Let f:A→Bbe a morphism in BH. Then θker1(f)={(x, y)∈A2|f(x) = f(y)}, i.e. θker1(f)is the kernel congruence of f. Proof. If f(x) = f(y), then f(x→y) = f(x)→f(y) = 1B, f(y→x) = f(y)→f(x)=1B, so x→y, y →x∈ker1(f) and hence xθker1(f)y. Conversely, if xθker1(f)y, then x→y, y →x∈ker1(f). Thus f(x→y) = 1B, f(y→x) = 1B, which, using preservation of →, gives f(x)→f(y) = 1B, f(y)→f(x)=1B. Hence f(x)≤f(y) and f(y)≤f(x) in the natural order of B, so f(x) = f(y). Therefore θker1(f)={(x, y)|f(x) = f(y)}. The usual first isomorphism theorem for basic hoops can now be phrased in terms of filter quotients, making the logical interpretation as quotient logics by theories completely transparent. Theorem 4.19 (First isomorphism theorem).Let f:A→Bbe a morphism in BH, and let F:= ker1(f)be its kernel filter. Then: (a) the map ¯ f:A/F →B, ¯ f([x]θF) := f(x), is a well-defined homomorphism; (b) the image Im(f)⊆Bis a subalgebra of B, and ¯ finduces an isomorphism A/F ∼ =Im(f); (c) in particular, if fis surjective, then ¯ f:A/F →Bis an isomorphism. Proof. (a) Well-definedness: if [x]θF= [y]θF, then (x, y)∈θF=θker1(f)by Proposition 4.18, so f(x) = f(y). Thus ¯ fis well-defined. Preservation of ·,→, and 1 follows from preservation by fand the definition of A/F. (b) The image Im(f) is closed under the operations of Band contains 1B, so it is a subalgebra. Define e:A/F →Im(f), e([x]) := f(x). 17 By (a), eis a well-defined homomorphism and is surjective by definition of Im(f). If e([x]) = e([y]), then f(x) = f(y), so (x, y) lies in the kernel congruence of f, which equals θFby Proposition 4.18. Hence [x] = [y], and eis injective. Thus A/F ∼ =Im(f). (c) If fis surjective, then Im(f) = B, and the isomorphism in (b) identifies A/F with B. As an immediate corollary, surjective morphisms in BH are precisely quotient maps by filters (up to isomorphism). In logical terms, every surjective homomorphism between basic-hoop semantics arises as the quotient by a theory (filter), and conversely every such quotient semantics is induced by a surjective homomorphism. Corollary 4.20 (Surjective homomorphisms as filter quotients).Let f:A→Bbe a morphism in BH. Then fis surjective if and only if there exists a filter Fon Aand an isomorphism φ:A/F →Bsuch that f=φ◦qF. Proof. If fis surjective, take F= ker1(f) and let φ=¯ fbe the isomorphism of Theorem 4.19(c); then f=φ◦qF. Conversely, if f=φ◦qFwith φan isomorphism, then qF is surjective and so is f. In particular, combined with Proposition 3.6, this shows that coequalizers in BH are precisely quotients by filters (corresponding to the congruences generated by the parallel pair), so that the categorical construction of coequalizers coincides with the logical construction of quotient logics by deductive theories. 5 Regularity of the category BH In this section we prove that the category BH of basic hoops is regular in the sense of Definition 2.12. The key points are: •regular epimorphisms in BH coincide with surjective homomorphisms; •surjective homomorphisms are stable under pullback; •every morphism in BH admits a factorization as a regular epimorphism followed by a monomorphism. 5.1 Regular epimorphisms in BH Recall that in a category with coequalizers, a morphism is a regular epimorphism if it is the coequalizer of some pair of morphisms (Definition 2.11). In BH, coequalizers are realized as quotients by congruences (Proposition 3.6) and, equivalently, as quotients by filters (Corollary 4.20). Lemma 5.1. Every coequalizer in BH is a surjective homomorphism. Proof. Let f, g :A→Bbe morphisms in BH and let q:B→Qbe their coequalizer. By Proposition 3.6,Qis the quotient B/θ by the congruence θgenerated by R={(f(a), g(a)) |a∈A}, and qis the canonical quotient map b7→ [b]θ. The underlying function of qis surjective by construction of the quotient set B/θ, hence qis a surjective homomorphism. 18 Thus every regular epimorphism (being a coequalizer) is surjective. We now prove the converse. Lemma 5.2. Let Abe a basic hoop and θa congruence on A. Let qθ:A→A/θ be the canonical quotient. Then the kernel pair of qθis given by the subalgebra K:= {(x, y)∈A×A|qθ(x) = qθ(y)} of A×A, with the projections k1, k2:K→A, k1(x, y) = x, k2(x, y) = y. Moreover, Kcoincides with θas a subset of A×A. Proof. By definition of the kernel pair, Kis the pullback of qθalong itself: K A A A/θ k2 k1qθ qθ On underlying sets, Kis exactly {(x, y)∈A×A|qθ(x) = qθ(y)}, and k1, k2are the restrictions of the product projections, hence homomorphisms. The set Kis clearly closed under the operations of A×A, so it is a subalgebra. The equality qθ(x) = qθ(y) is equivalent to [x]θ= [y]θ, which in turn is equivalent to (x, y)∈θ. Thus Kcoincides with θas a subset of A×A. Proposition 5.3 (Surjective homomorphisms are regular epimorphisms).Let f:A→B be a surjective morphism in BH. Then fis a regular epimorphism. Proof. Let K:= {(x, y)∈A×A|f(x) = f(y)} be the kernel congruence of f, and let q:A→A/K be the canonical quotient. By Lemma 5.2, the kernel pair of qis Kseen as a subalgebra of A×Awith projections r1, r2:K→A,r1(x, y) = x,r2(x, y) = y, and qcoequalizes r1, r2. Define u:A/K →Bby u([x]) := f(x). This is well defined: if [x]=[y], then (x, y)∈Kand hence f(x) = f(y). It is a homomorphism since fis, and it is surjective because fis surjective. If u([x]) = u([y]), then f(x) = f(y), so (x, y)∈Kand [x]=[y]. Thus uis injective and hence an isomorphism A/K ∼ =B. We have f=u◦q. We now show that qis the coequalizer of its kernel pair (r1, r2). Certainly q◦r1=q◦r2 because r1(x, y) and r2(x, y) are K-equivalent. Let h:A→Cbe any morphism with h◦r1=h◦r2. Then h(x) = h(y) whenever (x, y)∈K, so his constant on K-equivalence classes. Define ¯ h:A/K →Cby ¯ h([x]) := h(x). This is well defined by the above and is a homomorphism because his. By construction, h=¯ h◦q. If h=¯ h′◦qfor another ¯ h′:A/K →C, then ¯ h′([x]) = ¯ h([x]) for all xsince qis surjective. Hence ¯ h′=¯ h. Thus qis the coequalizer of (r1, r2) and hence a regular epimorphism. Since fis the composite of qwith an isomorphism, fis also a regular epimorphism. Combining Lemma 5.1 and Proposition 5.3, we obtain: Corollary 5.4 (Regular epimorphisms in BH).In BH, a morphism is a regular epimorphism if and only if it is a surjective homomorphism. 19 5.2 Image factorizations and pullback stability We now show that every morphism in BH factors as a regular epimorphism followed by a monomorphism, and that regular epimorphisms are stable under pullback. Together with Theorem 3.7 and Definition 2.12, this will give regularity of BH. Proposition 5.5 (Image factorization).Let f:A→Bbe a morphism in BH. Then f factors as Ae −→ Im(f)m −→ B, where: (i) Im(f)is the subalgebra of Bwith underlying set {f(a)|a∈A}; (ii) e:A→Im(f)is a surjective homomorphism, hence a regular epimorphism; (iii) m: Im(f),→Bis the inclusion, which is a monomorphism. Proof. Define Im(f) to be the subset {f(a)|a∈A}of B. If b1=f(a1) and b2=f(a2) are in Im(f), then b1·b2=f(a1)·f(a2) = f(a1·a2)∈Im(f), and b1→b2=f(a1)→f(a2) = f(a1→a2)∈Im(f), and 1B=f(1A)∈Im(f). Thus Im(f) is closed under the operations and contains 1B, so it is a subalgebra of B. Define e:A→Im(f) by e(a) := f(a). This is a homomorphism and is surjective by definition of Im(f). By Corollary 5.4,eis a regular epimorphism. The inclusion m: Im(f),→Bis a homomorphism that is injective on underlying sets, hence a monomorphism in BH. Finally, f=m◦eholds by construction. We now prove stability of regular epimorphisms under pullback. Proposition 5.6 (Pullback stability of regular epimorphisms).Let e:B→C be a regular epimorphism in BH, and let g:A→Cbe any morphism. Form the pullback square P B A C p2 p1e g in BH. Then p1:P→Ais a regular epimorphism. Proof. By Corollary 5.4,eis a surjective homomorphism. The pullback Pcan be realized as the subalgebra P:= {(a, b)∈A×B|g(a) = e(b)} of A×B, with the induced operations and the projections p1(a, b) := a,p2(a, b) := b(see Corollary 3.3). We claim that p1:P→Ais surjective. Let a∈Abe arbitrary. Since eis surjective, there exists b∈Bwith e(b) = g(a). Then (a, b)∈Pand p1(a, b) = a. Thus every element of Alies in the image of p1, so p1is surjective. By Corollary 5.4,p1is a regular epimorphism. 20 We can now state the main result of this section. Theorem 5.7 (Regularity of BH).The category BH of basic hoops is a regular category. Proof. By Theorem 3.7,BH has all small limits, and in particular all finite limits. By Proposition 5.5, every morphism in BH admits a factorization as a regular epimorphism followed by a monomorphism. By Proposition 5.6, regular epimorphisms are stable under pullback. Thus the conditions of Definition 2.12 are satisfied, and BH is regular. 6 Exactness and effective equivalence relations In this section we prove that the regular category BH is Barr-exact (Definition 2.15). By Theorem 5.7,BH is regular; it therefore remains to show that every internal equivalence relation in BH is effective, i.e. arises as the kernel pair of some morphism. From the point of view of algebraic semantics for basic logic and related multiple-valued logics, this amounts to showing that every algebraic notion of equivalence compatible with the basic-hoop operations is induced by a quotient homomorphism, and hence by identifying elements that are semantically indistinguishable in an appropriate quotient logic. Because BH is a finitary variety, internal equivalence relations on a basic hoop Aare the same as congruences on A. We first make this identification explicit, and then show that every congruence is the kernel pair of the corresponding quotient homomorphism. In logical terms, this says that algebraic equivalence relations arising in the semantics of basic logic are exactly those determined by congruences (or filters) and the associated Lindenbaum algebras. 6.1 Internal equivalence relations and congruences Let Abe a basic hoop. Recall from Definition 2.13 that an internal equivalence relation on Ais a monomorphism m=⟨r1, r2⟩:R ,→A×A such that the induced structure on Rsatisfies reflexivity, symmetry, and transitivity conditions in the categorical sense. Since BH is a variety, monomorphisms are precisely injective homomorphisms, and subobjects of A×Aare identified with subalgebras of the product basic hoop A×A. Lemma 6.1. Let Abe a basic hoop. (i) If θ⊆A×Ais a congruence on A, then the inclusion mθ:θ ,→A×A (viewing θas a subalgebra of A×A) together with the coordinate projections r1, r2: θ→Adefines an internal equivalence relation on A. (ii) Conversely, if m:R ,→A×Ais an internal equivalence relation on A, then its image θ:= m(R)⊆A×A is a congruence on A. 21 Proof. (i) If θis a congruence on A, then by Definition 4.4 it is an equivalence relation on the underlying set Athat is closed under the basic operations. Thus θis a subalgebra of A×A, and the inclusion mθis a monomorphism in BH. The induced projections r1, r2:θ→Aare the restrictions of the product projections. The usual set-theoretic properties of an equivalence relation (reflexive, symmetric, transitive) translate into the categorical axioms for an internal equivalence relation in Definition 2.13. Hence mθis an internal equivalence relation. (ii) Conversely, let m:R ,→A×Abe an internal equivalence relation. Because mis a monomorphism in a variety, it identifies Rwith a subalgebra θ⊆A×Avia an isomorphism R∼ =θover A×A. The underlying relation θ⊆A×Ais an equivalence relation by the internal reflexivity, symmetry, and transitivity axioms. Closure of θunder the basic operations follows from the fact that θis a subalgebra. Thus θis a congruence on A. Proposition 6.2 (Internal equivalence relations vs. congruences).For each basic hoop A, the assignment θ7−→ mθ:θ ,→A×A from congruences on Ato internal equivalence relations on Ais a bijection, with inverse sending an internal equivalence relation m:R ,→A×Ato its image θ=m(R)⊆A×A. Proof. Lemma 6.1 shows that both directions are well-defined. If we start from a congruence θand form mθ, then the image of mθis exactly θitself. Conversely, if we start from m:R ,→A×Aand let θ=m(R), then mfactors as an isomorphism R∼ =θfollowed by mθ. Hence the two assignments are mutually inverse up to isomorphism of internal equivalence relations. In particular, to prove that every internal equivalence relation in BH is effective, it suffices to show that every congruence θon a basic hoop Ais the kernel pair of a suitable morphism q:A→Q. In logical terms, these congruences correspond to algebraic equivalences compatible with the connectives of the implicational fragment of basic logic. 6.2 Effectivity of congruences Let Abe a basic hoop and let θbe a congruence on A. We consider the canonical quotient homomorphism qθ:A→A/θ, qθ(x) := [x]θ, where A/θ is the quotient basic hoop defined in Section 4. We show that θis precisely the kernel pair of qθ. Lemma 6.3 (Kernel pair of the quotient).Let θbe a congruence on Aand qθ:A→A/θ the canonical quotient. Then the kernel pair of qθis the subalgebra K:= {(x, y)∈A×A|qθ(x) = qθ(y)} of A×A, with projections k1, k2:K→Agiven by k1(x, y) = x,k2(x, y) = y. Moreover, Kcoincides with θas a subset of A×A. 22 Proof. By Definition 2.10, the kernel pair of qθis given by the pullback of qθalong itself: K A A A/θ k2 k1qθ qθ In BH, this pullback is realized as the subalgebra Kof A×Aconsisting of those pairs (x, y) with qθ(x) = qθ(y), by Corollary 3.3. The projections k1, k2are obtained by restricting the product projections and are homomorphisms. By definition of the quotient, qθ(x) = qθ(y) if and only if [x]θ= [y]θ, which is equivalent to (x, y)∈θ. Thus K=θas subsets of A×A, and since both are subalgebras with the same underlying set and operations inherited from A×A, we can identify Kand θas objects of BH. Proposition 6.4 (Congruences are effective).Every congruence θon a basic hoop Ais an effective equivalence relation: it is the kernel pair of the canonical quotient homomorphism qθ:A→A/θ. Proof. By Lemma 6.3, the kernel pair of qθis given by the subobject θ ,→A×A, viewed as a subalgebra of A×Awith the coordinate projections. This is precisely the internal equivalence relation associated with θvia Proposition 6.2. Hence θis (isomorphic to) the kernel pair of qθand is effective in the sense of Definition 2.14. Combining this with the identification of internal equivalence relations and congruences, we obtain: Proposition 6.5 (Internal equivalence relations are effective).Every internal equivalence relation in BH is effective. Proof. Let Abe a basic hoop and let m:R ,→A×Abe an internal equivalence relation on A. By Proposition 6.2,mcorresponds to a congruence θon A, and midentifies R with the subalgebra θ⊆A×A. By Proposition 6.4,θis the kernel pair of qθ:A→A/θ. Kernel pairs are unique up to isomorphism, so Ris (isomorphic to) the kernel pair of qθ. Thus Ris effective. We can now state the main exactness result for BH. Theorem 6.6 (Exactness of BH).The category BH of basic hoops is Barr-exact. That is: (i) BH is a regular category (Theorem 5.7); (ii) every internal equivalence relation in BH is effective (Proposition 6.5). Proof. The first clause is Theorem 5.7, and the second is Proposition 6.5. These are exactly the requirements of Definition 2.15. Remark 6.7 (Logical interpretation).The filter–congruence correspondence (Theorem 4.13) shows that every internal equivalence relation on a basic hoop Ais determined by a filter F⊆A, and that its quotient is the filter quotient A/F. When Ais taken as a Lindenbaum basic hoop of a (possibly implicational) fragment of basic logic, filters correspond to deductive systems and the quotient A/F corresponds to a quotient logic in which 23 formulas are identified modulo provable equivalence relative to F. In this sense, exactness of BH may be viewed as a categorical reformulation of the classical description of congruence quotients in terms of filters, and it provides a structural explanation of why every algebraic notion of logical equivalence compatible with the operations of a basic hoop arises from a quotient by a deductive filter in the many-valued logic setting. 7 Further remarks and applications The results established in the previous sections show that the category BH of basic hoops is a well-behaved environment from the point of view of categorical algebra: it is complete and cocomplete (Theorem 3.7), regular (Theorem 5.7), and Barr-exact (Theorem 6.6). Moreover, all of these properties admit concrete descriptions in terms of filters, congruences and quotients. In this final section we collect several remarks and indicate how our results connect with the algebraic semantics of multiple-valued and fuzzy logics, as well as some directions for further research. 7.1 Connections with BL-algebras and related varieties Basic hoops appear naturally as implicational subreducts of BL-algebras and, more generally, of various classes of commutative integral residuated structures. Let BL denote the variety of BL-algebras, in the sense of H´ajek, and consider the forgetful functor Uimp :BL −→ BH that sends a BL-algebra to its basic hoop reduct (A, ·,→,1) and a BL-homomorphism to its underlying homomorphism of basic hoops. In logical terms, Uimp forgets all connectives except implication and the constant 1, and thus passes from a full BL-semantics to the implicational fragment of basic logic and its extensions. Since BL is a finitary variety, the arguments of Sections 3–6apply mutatis mutandis to show that BL is also complete, cocomplete, regular, and Barr-exact. In particular: •filters and congruences on BL-algebras correspond bijectively; •surjective BL-homomorphisms are precisely the regular epimorphisms; •internal equivalence relations on a BL-algebra are exactly its congruences, and they are effective. Thus the categorical picture developed for basic hoops extends to the full algebraic semantics of basic logic, and the filter-based description of quotients and internal equivalence relations carries over to BL-algebras without additional technical complications. From the viewpoint of many-valued logics based on continuous t-norms, this means that the standard constructions on theories and quotient logics (e.g. Lindenbaum algebras, conservative extensions, factor logics) can be organized categorically in terms of the regular and exact structure of BL and BH. The functor Uimp preserves and reflects finite limits, and preserves surjective homomorphisms. Hence it is a regular functor: it preserves finite limits and regular epimorphisms. As a consequence, much of the regular and exact structure of BL can be analyzed at the level of the simpler implicational reducts and then transported back along Uimp, providing a convenient way to separate those phenomena that are already visible at the implicational level from those that crucially involve additional connectives. 24 Similar remarks apply to other well-known subvarieties of hoops and BL-algebras (bounded hoops, involutive hoops, MV-algebras, etc.), which provide semantics for important families of multiple-valued logics. In each case, the variety-theoretic nature of the category guarantees regularity and exactness; the present paper suggests that it is useful to revisit these facts with explicit filter-based descriptions of quotients and effective equivalence relations in each setting, with an eye towards logical applications. 7.2 Towards homological and protomodular aspects Regular and exact categories provide a convenient framework for parts of homological algebra beyond the abelian context, especially in the presence of additional structure such as protomodularity or semi-abelianness. In turn, homological tools in algebraic categories often yield refined invariants for logical systems, for instance through central extensions, commutators, and derived functors associated with forgetful functors to algebraic semantics. It is therefore natural to ask to what extent the category BH fits into the “homological” side of categorical algebra. Recall that a semi-abelian category is, roughly speaking, a pointed, Barr-exact, protomodular category with binary coproducts. The category BH is pointed (Example 2.8), Barr-exact (Theorem 6.6), and has all small coproducts (Theorem 3.7). Thus a substantial portion of the semi-abelian axioms is already satisfied. What remains unclear is whether BH is protomodular (in the sense of Bourn) and whether short exact sequences in BH support a useful homological calculus with a meaningful logical interpretation. At present we do not attempt to settle these questions. Instead we record the following problems, which are motivated both by categorical considerations and by the prospect of developing “homological” invariants for basic logic and related multiple-valued logics. (P1) Is the category BH protomodular? If not, can one describe a simple categorical obstruction, possibly related to specific logical phenomena (e.g. failure of certain interpolation or amalgamation properties) in the associated logics? (P2) Identify natural subclasses of basic hoops (for example, chains, bounded or involutive basic hoops) whose full subcategories of BH exhibit stronger homological properties, such as protomodularity or semi-abelianness, and clarify the corresponding logical meaning (for instance, for linearly ordered or involutive semantics). A positive answer to (P2) would open the door to homological techniques (e.g. long exact sequences, derived functors) in the study of basic hoops and their logical counterparts, while a negative answer to (P1) would clarify the precise limitations of the categorical structure of BH and of the homological approach to these logics. 7.3 Logical interpretations From the point of view of algebraic logic, basic hoops provide an algebraic semantics for the implicational fragment of H´ajek’s basic logic and related t-norm based logics. In this context, filters on a basic hoop correspond to deductive systems, and quotients by filters are algebraic counterparts of Lindenbaum algebras. The categorical results of this paper can therefore be read as structural statements about theories and quotient logics. 25