Double Negation Semantics for Generalisations of Heyting Algebras

. This paper presents an algebraic framework for investigating proposed translations of classical logic into intuitionistic logic, such as the four negative translations in-troduced by Kolmogorov, G¨odel, Gentzen and Glivenko. We view these as variant semantics and present a semantic formulation of Troelstra’s syntactic criteria for a satisfactory negative translation. We consider how each of the above-mentioned translation schemes behaves on two generalisations of Heyting algebras: bounded pocrims and bounded hoops. When a translation fails for a particular class of algebras, we demonstrate that failure via speciﬁc ﬁnite examples. Using these, we prove that the syntactic version of these translations will fail to satisfy Troelstra’s criteria in the corresponding substructural logical setting.


Introduction
Schemes for translating classical logic into intuitionistic logic have been studied since the 1920s and are important for understanding the computational content of classical logic. These so-called negative translations or double negation translations such as those proposed by Kolmogorov, Gödel, Gentzen and Glivenko are generally presented as syntactic translations and are studied by mainly syntactic methods (e.g., see [9,11]). In this paper we use an algebraic framework for investigating proposed double negation translations.
The arguments justifying the syntactic Kolgomorov and Gödel translations do not need the rule of contraction and hence we develop our framework in the context of two generalisations of Heyting algebras: bounded pocrims and bounded hoops. In logical terms these correspond to the conjunctionimplication fragment of intuitionistic affine logic and what we call intuitionistic Lukasiewicz logic, respectively. We view a translation as a variant semantics for the logical language and we give a semantic formulation of Troelstra's criteria for a satisfactory translation.
The algebras that correspond to classical logic are called involutive (i.e., they satisfy ¬¬x = x). We associate with each bounded pocrim A two involutive pocrims: • a bounded pocrim A C called the involutive core of A, whose universe is a subset of the universe of A, and • a bounded pocrim A R called the involutive replica of A, whose universe is a quotient of the universe of A.
A generalisation of the first construction (involutive core) was studied in [20], where it is called a c-retraction. The injection ι : A C → A and the projection π : A → A R are not necessarily homomorphisms when A is a general bounded pocrim, but they are homomorphisms when A is a bounded hoop. The involutive core and the involutive replica turn out to be naturally isomorphic via the composite π • ι. The two constructions give complementary ways of viewing the double negation operation δ(x) = ¬¬x.
Using the involutive core and the involutive replica, we show that the Kolmogorov and Gödel translations satisfy our algebraic formulation of Troelstra's criteria for a satisfactory negative translation in any reasonable class of bounded pocrims. We also show by explicit finite examples, that the Gentzen and Glivenko translations fail to satisfy our algebraic formulation of Troelstra's criteria in general. The proofs that the Gentzen and Glivenko translations fail are based on specific finite classes of finite bounded pocrims. Using these counter-examples we can prove that the syntactic versions of these translations fail to satisfy Troelstra's formulation of his criteria.
For bounded hoops, the situation is much simpler. The double negation operation is a homomorphism implying that all reasonable double negation translation schemes are equivalent and hence satisfy our formulation of Troelstra's formulation. The results for bounded hoops is dependent on certain algebraic identities, some of which are not easy to derive from the axioms for this class of algebra. We use an indirect semantic method to verify the harder identities (see Section 4.2).

Related work
Cignoli and Torrell [8] investigate Glivenko's negative translation scheme in the setting of bounded BCK algebras, the algebraic models of the implicative fragment of intuitionistic affine logic. They study an analogue for BCK algebras of what we call the involutive core of a bounded pocrim, and discuss extensions of their results on the Glivenko translation to bounded pocrims and bounded hoops. In the present paper, we are interested in negative translation schemes in general and give a framework for comparing different translations.
Galatos and Ono [14] look at the Glivenko and Kolmogorov translations for substructural logics over the full Lambek calculus, taking again an algebraic approach studying involutive sub-structures of residuated lattices. In particular, they show that every involutive sub-structural logic has a minimal substructural logic that contains the first via a double negation interpretation. Commutativity is not assumed, so the paper has to deal with two forms of negation. A proof-theoretic presentation of the results in [14] for the Glivenko translation are then presented by Ono [20], looking at the weakest extension of full Lambek calculus needed to derive the Glivenko theorem for classical logic.
The work that is perhaps closest to ours is that of Farahani and Ono [10], where they also study various negative translations, analysing the role of the double negation shift principle in the treatment of the quantifiers in predicate logic. In their final section on "algebras" they discuss a construction (c-retraction), which can be viewed as a generalisation of our involutive core construction. In the present paper our goal is to create a general framework for negative translations, enabling us to identify situations where particular translation schemes fail to have the required algebraic properties for a negative translation. In our study we also an alternative to the cretraction/involutive core construction, the involutive replica, which turns out to fit more naturally in some cases.

Syntactic Negative Translations
As mentioned above, we are studying here classes of algebras that capture the semantics of some well-known logics. A formula is provable in the conjunction-implication fragment of intuitionistic affine logic iff it is valid in all bounded pocrims. Similarly, provability in the conjunction-implication fragment of GBL (the fragment that we call intuitionistic Lukasiewicz logic) is captured by validity in the algebraic class of hoops. The classical counterparts of these logics, i.e. the extension of these logics with the double negation elimination (DNE) principle A ⊥⊥ → A, can be also captured by the sub-class of involutive pocrims/hoops, i.e. bounded pocrims/hoops which satisfy x ⊥⊥ = x.
Negative translations provide a way to eliminate DNE from classical proofs of a formula A, turning these into intuitionistic proofs of the translation of A. Although various negative translations have been proposed in the literature [15,16,17,19], it is well known that all negative translations which satisfy Troelstra's criteria [22,Section 1.10] are intuitionistically equivalent. Formally, Troelstra calls a formula translation A → A N a negative translation if (i) A and A N are classically equivalent; (ii) If A is provable classically then A N is provable intuitionistically; (iii) A N is equivalent to a formula in the negative fragment (negated atomic formulas, implication and conjunction).
The point behind (iii) is that, for this negative fragment, classical and intuitionistic provability coincide, and in particular (A N 1 ) N 2 is intuitionistically equivalent to A N 1 . Assume then that two translations A N 1 and A N 2 satisfy the above. By (DNS1), we have that A N 1 → A holds classically. Hence, by (DNS2), (A N 1 → A) N 2 is intuitionistically valid. With a further assumption that these translations are modular (see [11]), we also have (

Pocrims
The most general class of algebras we consider is the class of pocrims: partially ordered, commutative, residuated, integral monoids [3]. Pocrims provide the natural algebraic models for the fragment of intuitionistic logic known as minimal affine logic, whose connectives are implication (φ ⇒ ψ) and a form of conjunction (φ ⊗ ψ) that is not required to be idempotent (so that the law of contraction need not hold). The underlying ordered set of a pocrim is bounded above but not necessarily below; bounded pocrims, i.e., those in which the order is bounded below provide the context for our study of negation.
Definition 2.1 (Pocrim). A pocrim is a structure for the signature ( , ·, →) of type (0, 2, 2) satisfying the following laws, in which x ≤ y is an abbreviation for x → y = : x · y ≤ z iff x ≤ y → z. [r] We will refer to the operations · and → as conjunction and residuation respectively. We adopt the convention that residuation associates to the right and has lower precedence than conjunction. So the brackets in x · ((x → y) → y) are all necessary while those in (x · z) →(y → z) may all be omitted.
Throughout this paper, we adopt the convention that if P is a structure then P is its universe. If P is a pocrim, the laws [m i ], [o j ] and [t] say that (P ; , ·; ≤) is a partially ordered commutative monoid with the identity as top element. Law [r], the residuation property, says that for any y and z the set {x | x · y ≤ z} is non-empty and has supremum y → z. It is an easy exercise in the use of the axioms to show that x → y is monotonic in y and antimonotonic in x.
A pocrim is said to be bounded if it has a (necessarily unique) annihilator, i.e., an element ⊥ such that for every x we have: [ann] Note that any finite pocrim P is bounded, the annihilator being given by x∈P x. In a bounded pocrim P, we have that ⊥ = x · ⊥ ≤ x · = x for any x, so that (M ; ≤) is indeed a bounded ordered set. We write ¬x for x → ⊥ (and give ¬ higher precedence than the binary operators).
Proof. The proofs are easy exercises in the use of the bounded pocrim axioms.
An element x of a bounded pocrim is said to be regular if it satisfies the double-negation identity: [dne] For example, and ⊥ are regular in any bounded pocrim. A bounded pocrim is said to be involutive if all its elements are regular. This class of algebras corresponds to the ( , ⊥, ⇒, ∧)-fragment of classical affine logic. See [21] for further information about pocrims in general and involutive pocrims in particular.
We will often write δ(x) for ¬¬x.
Lemma 2.3. The following are valid in all bounded pocrims: Proof. Let us prove part 6: using [r] several times, we have that ( * ) x·¬y ≤ ¬(x → y), whence: The proofs of the other parts are similar exercises in the use of the bounded pocrim axioms together with Lemma 2.2 and the monotonicity properties of · and → as necessary.
Example 2.4. There is a unique pocrim B with two elements. It is involutive and provides the standard model for classical Boolean logic.
Thus the order type of C ⊕ D is the concatenation of the partial orders (C \{ }; ≤) and (D; ≤).
Remark 2.6. As alluded to in Section 1.2 the equational theory of pocrims can be viewed as a logical theory, where a term t is viewed as a formula that holds in a pocrim P iff t = under all assignments of variables in t to values in P . Conversely, as x = y in a pocrim iff (x → y) · (y → x) = , the equational theory can be recovered from the logical theory. In the sequel, we concentrate on the case of bounded pocrims. If C is a class of bounded pocrims, we write Th(C) for the logical theory of C, i.e., the set of all terms t over the signature ( , ⊥, ·, →) of a bounded pocrim with variables drawn from the set Var = {v 1 , v 2 , . . .}, such that t = under any assignment Var → P taking values in a member P of C. It can be shown that a deductive system called intuitionistic affine logic, which we will refer to as AL i is sound and complete for the logical theory of all bounded pocrims. AL i is essentially the usual intuitionistic propositional logic IL without the rule of contraction.

Involutive pocrims
In general, N is not closed under conjunction and hence is not a subpocrim and δ does not respect either · or →: There is a bounded pocrim U with elements > a > b > c > ⊥ and with ·, → and δ as follows: However, in the above example, if we define x· y = δ(x · y), we find that N = (N ; ,·, →, ⊥) is an involutive pocrim whose residuation agrees with that of U. Dually, we find that the equivalence relation whose equivalence classes form the partition is an involutive pocrim where· is induced from · by the monoid congruence. Using the following lemma, we will see that these constructions generalise to all bounded pocrims.
Let the relation θ be defined on P by x θ y iff δ(x) = δ(y). Then θ is a congruence on the monoid (P, , ·).
Lemma 2.8 justifies the following definition: Definition 2.9. Given a bounded pocrim P we define the following structures over the signature of a bounded pocrim: • P C , the involutive core of P, is (P C , , ⊥,·,→) where P C = im(δ) ⊆ P , where and ⊥ are as in P and where· and→ are defined as follows: We write ι : P C → P for the inclusion.
where P R is the quotient P/θ of P by the equivalence relation defined by x θ y iff δ(x) = δ(y) and where, writing [x] for the equivalence class in P R of x ∈ P , we define· and→ as follows: We write π : P → P R for the projection.
We will write≤ and≤ for the order relation on P C and P R respectively.
Theorem 2.10. Let P be a bounded pocrim. Then: 1. P C is an involutive pocrim and the inclusion of (P C ,≤) in (P, ≤) is strictly monotonic (x≤ y iff x ≤ y).
2. P R is an involutive pocrim and the projection of (P, 3. P C and P R are isomorphic bounded pocrims via the composition of the inclusion ι : P C → P and the projection π : P → P R . Proof. 1. Noting that→ is the restriction to P C of →, the claim about strong monotonicity is clear and we can write ≤ for≤. The bounded pocrim axioms are then easily proved with the exception of [m 1 ] (associativity of·) and [r] (residuation). For associativity, we have: So x·(y· z) = δ(x · y · z) and similarly (x· y)· z = δ(x · y · z), giving us the associativity of·. For residuation, the right-to-left direction is clear: if x ≤ y→ z = y → z, then x · y ≤ z and then x· y = δ(x · y) ≤ δ(z) by Lemma 2.3, part 3. But z = δ(z) since z ∈ P C = im(δ). Hence, x· y ≤ z. For the converse, assume x· y ≤ z, i.e. δ(x · y) ≤ z. By Lemma 2.3, part 5, we have that x · δ(y) ≤ δ(x · y), and hence x · δ(y) ≤ z. But y = δ(y) since y ∈ P C = im(δ), so we have x · y ≤ z, which by residuation in P gives x ≤ y → z. To conclude the proof of part 1, we must show that P C is involutive, but this is clear since negation in P C is the restriction to P C = im(δ) of the negation in P and all the elements of im(δ) are regular by Lemma 2.
We have Finally, we must show that P R is involutive.
, so negation and hence, also, double negation commute with the projection of P onto P R . As, by construction [δ(x)] = [x], P R is indeed involutive. 3. We must show that π • ι is one-to-one, onto and respects the pocrim operations. To see that π • ι is one-to-one, let x, y ∈ P C , so that x = δ(x) and y = δ(y), and assume iff δ(x→ y) = δ(x → y), which holds by definition.
Remark 2.11. For any bounded pocrim P, ι : P C → P is a homomorphism of the ( , ⊥, →)-reduct of P, and π : P → P R is a homomorphism of the ( , ⊥, ·)-reduct of P. In general, however, neither map is a pocrim homomorphism (see the discussion of the bounded pocrim U in Example 2.7).
As P C and P R are isomorphic pocrims, one could focus attention on one of the two constructions, and several authors work solely with their analogue of P C . We prefer to have both constructions available, since, in some contexts it is convenient for the ( , ⊥, →)-structure to be respected, while in other contexts it is more convenient for the ( , ⊥, ·)-structure to be respected (cf. the proofs of Theorems 3.5 and 3.6).

Generalised and Double Negation Semantics
Beginning with Kolmogorov [19], logicians have studied double negation translations (or negative translations) that represent classical logic in intuitionistic logic. Kolmogorov's translation inductively replaces every subformula of a formula by its double negation. Other authors have devised more economical translations: Gödel's translation [17] applies double negation to the right-hand operands of implications and at the outermost level; Gentzen's translation [15] applies double negation to atomic formulas only; and Glivenko's translation [16] is the most economical off all and just applies double negation once at the outermost level. In this section we undertake an algebraic study of these translations.

Generalised semantics
We wish to undertake an algebraic analysis of translations such as the various double negation translations. We will view the translations as variant semantics and so we need a framework to compare different semantics.
Typically, these translations are defined by recursion over the syntactic structure of a term, sometimes composed with an additional top-level transformation. See, for example, [11] where top-level transformations are handled by redefining the provability relation. Here, rather than working with syntax, we prefer to think of a syntactic term t as its denotation viewed as a family of maps α → x, where x ranges over the universe of a bounded pocrim P and α is an assignment of values in P to the free variables of t. The modularity properties of a translation scheme which are needed for our proofs (see, for example, Theorem 3.11) are then captured by the following definition: Definition 3.1. Let Poc ⊥ be the category of bounded pocrims and homomorphisms and let Set be the category of sets. Given any set X, let H X : Poc ⊥ → Set be the functor that maps a pocrim P to Hom Set (X, P ), i.e., the set of all functions from X to P , and maps a homomorphism h : P → Q to f → h • f : Hom Set (X, P ) → Hom Set (X, Q). Now let Ass = H Var and Sem = H L where L is the set of all terms over the signature ( , ⊥, ·, →) of a bounded pocrim with variables drawn from the set Var = {v 1 , v 2 , . . .}. We define a semantics to be a natural transformation µ : Ass → Sem.
So given a bounded pocrim P, Ass(P) denotes the set of assignments α : Var → P , while Sem(P) denotes the set of all possible functions s : L → P . A semantics µ is a family of functions µ P indexed by bounded pocrims P such that µ P : Ass(P) → Sem(P) and such that for any homomorphism f : P → Q the following diagram commutes.

Ass(P)
The standard semantics µ S is the one that simply uses the given assignment α : Var → P to give values to the variables in a term in L and then calculates its value interpreting the operations in the obvious way: Kolmogorov translation corresponds to a semantics µ Kol defined like µ S , but applying double negation to everything in sight: The Gödel translation 1 corresponds to a semantics that applies double negation to the right operands of residuation and at the outermost level. We define it using an auxiliary semantics µ * .
The Gentzen and Glivenko translations correspond to semantics obtained by composing the standard semantics with double negation: where δ X denotes the natural transformation from H X = Hom Set (X, ·) to itself with δ X P = f → δ • f .

Double negation semantics
Definition 3.2 (Double negation semantics). Let C be a class of bounded pocrims, we say that a semantics µ is a double negation semantics for C if the following conditions hold: (DNS1) If P ∈ C is involutive, then µ P = µ S P . (DNS2) Given a term t, if, for every involutive Q ∈ C and every β : Var → Q, we have: µ S Q (β)(t) = , then, for every P ∈ C and every α : Var → P , we have: (DNS3) δ L • µ P = µ P , for every P ∈ C.
Note that these condition are trivially true if C is empty. If C is nonempty but does not contain any involutive pocrim, the conditions only hold if µ P (α)(t) = for every P ∈ C, assignment α : Var → P and term t. Remark 3.3. Subject to one proviso, the above definition can be seen to agree with the usual syntactic definition of a double negation translation due to Troelstra, as summarised in Section 1.2. The proviso is that we must have Th(I) = Th(C) + [dne], where I comprises the involutive pocrims in C and where Th(C) + [dne] denotes the smallest set of terms that contains Th(C) that is closed under rewriting with equations that either hold in every member of C or have one of the forms ¬¬x = x or x = ¬¬x.
Definition 3.4. We say a class C of bounded pocrims is inv-closed if whenever P ∈ C, then there is Q ∈ C such that Q is isomorphic to the involutive core, or equivalently the involutive replica, of P.
Theorem 3.5. The Kolmogorov semantics, µ Kol , is a double negation semantics for any inv-closed class C of bounded pocrims.
Proof. (DNS1) and (DNS3) are easy to verify. As for (DNS2), let P ∈ C and let t be a term such such that µ S Q (β)(t) = for every assignment β : Var → Q when Q is involutive. Then, if α : Var → P , it is easy to see by induction on the structure of any term s that the Kolmogorov semantics of s in P under an assignment α agrees with the standard semantics of s on P C , the involutive core of P, under the assignment δ • α: (For the inductive step for residuation use the identity δ(δ(x) → δ(y)) = δ(x) → δ(x), which follows from Lemma 2.3 parts 3 and 6.) Now δ • α is an assignment into the involutive pocrim P C , which by assumption is isomorphic to some Q ∈ C, via some isomorphism φ : P C → Q. Hence, using our hypothesis on involutive members of C, and the fact that µ S is a natural transformation, we have: completing the proof of (DNS2).
Theorem 3.6. The Gödel semantics, µ Göd is a double negation semantics for any class C of inv-closed bounded pocrims.
Proof. We follow a similar line to the proof of Theorem 3.5 using the involutive replica in place of the involutive core. Again (DNS1) and (DNS3) are easy. For (DNS2), given a bounded pocrim P, we see by induction on the structure of a term s that for any assignment α : Var → P , we have: where π : P → P R is the natural projection onto the involutive replica. Hence if µ S Q (β)(t) = for every assignment β : Var → Q where Q is involutive, then, using our hypothesis on involutive members of C, and the fact that µ S is a natural transformation, we have: where φ : P R → Q is an isomorphism of P R with some involutive Q ∈ C. Now π(x) = π(y) iff δ(x) = δ(y), so δ(µ Göd P (α)(t)) = δ( ) = , but clearly δ • µ Göd = µ Göd and we have proved (DNS2).
We will now exhibit classes of bounded pocrims where the Gentzen and Glivenko semantics fail to give double negation semantics. These classes involve the pocrims defined in the following examples.
Example 3.7. The pocrim P 4 comprises the chain > p > q > ⊥. The operation tables for P 4 are as follows.
In P 4 , δ(q) = p, so P 4 is not involutive. However, the involutive core of P 4 is actually a subpocrim: namely the subpocrim with universe {⊥, p, } which (in anticipation of Example 4.4), we will refer to as L 3 .
Example 3.8. Consider the pocrim Q 6 with six elements > p > q > r > s > ⊥ and with ·, → and δ as shown in the following tables: · p q r s ⊥ p q r s ⊥ p p p r r s ⊥ q q r r r ⊥ ⊥ r r r r r ⊥ ⊥ s s s ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ → p q r s ⊥ p q r s ⊥ p q q s ⊥ q p s s r s s s q ⊥ δ p q q r q s s ⊥ ⊥ Q 6 is not involutive, as δ(x) = x fails for x ∈ {p, r}. In Q 6 , double negation is an implicative homomorphism: ¬¬x → ¬¬y = ¬¬(x → y) for all x, y. Double negation is not quite a conjunctive homomorphism in Q 6 : ¬¬x · ¬¬y = ¬¬(x · y) unless {x, y} ⊆ {q, r}, in which case ¬¬x · ¬¬y = r < q = ¬¬(x · y).
The involutive replica of Q 6 turns out to be a quotient pocrim: as indicated by the block decomposition of the above operation tables, there is a homomorphism h : Q 6 → Q 4 , where Q 4 is the involutive replica of Q 6 and comprises the chain > u > v > ⊥ with operation tables as follows: The kernel congruence of h has equivalence classes { , p}, {q, r}, {s} and {⊥} which are mapped by h to , u, v, respectively in Q 4 . Theorem 3.9. (i) The Gentzen semantics µ Gen is not a double negation semantics for any class of bounded pocrims that contains the pocrim Q 6 of Example 3.8. (ii) The Glivenko semantics µ Gli is not a double negation semantics for any class of bounded pocrims that contains the pocrim P 4 of Example 3.7.
Proof. By the remarks after Definition 3.2 we can assume that the class of bounded pocrims contains at least one involutive pocrim in both cases.
(i): We show that (DNS2) does not hold for µ Gen in Q 6 . Let x, y ∈ Var and let t be the formula δ(x · y) → x · y. Clearly, µ S P (α)(t) = , for any involutive pocrim P and any α : Var → P . Thus (DNS2) requires µ Gen Q 6 (α)(t) = for any α : Var → Q 6 . However, if α(x) = α(y) = r, we have: (ii): we argue as in the proof of (i), but taking t to be δ(x) → x. Then, if α(x) = q, we have: Theorem 3.10. Let C 1 comprise the two bounded pocrims P 4 and L 3 of Example 3.7 and let C 2 comprise the two bounded pocrims Q 6 and Q 4 of Example 3.8. Then: (i) The Gentzen semantics, µ Gen , is a double negation semantics for C 1 , but the Glivenko semantics, µ Gli , is not.
(ii) The Glivenko semantics, µ Gli , is a double negation semantics for C 2 , but the Gentzen semantics, µ Gen , is not.
Proof. (i): By Theorem 3.9, µ Gli is not a double negation semantics for C 1 .
As for µ Gen , (DNS1) is easily verified. For (DNS3) and (DNS2), note that for any α : Var → P 4 , we have: where in the last expression we have identified L 3 with the bounded subpocrim of P 4 whose universe is im(δ). Thus evaluation under µ Gen with an assignment in any bounded pocrim in C 1 is equivalent to evaluation under the standard semantics, µ S , with an assignment in the involutive pocrim L 3 .
(ii): By Theorem 3.9, µ Gen is not a double negation semantics for C 2 . As for µ Gli , (DNS1) and (DNS3) are immediate from the definition of µ Gli . For (DNS2), let t be a formula, such that µ S Q 4 (α)(t) = , for any assignment α : Var → Q 4 . As Q 4 is the only involutive pocrim in C 2 , we must show that µ Gli P (α)(t) = for P ∈ C 2 under any assignment α : Var → P . This is easy to see for P = Q 4 , since the Glivenko semantics is the double negation of the standard semantics and Q 4 is involutive. As for P = Q 6 , let α : Var → Q 6 be given. As discussed in Example 3.8, there is a quotient projection h : Q 6 → Q 4 , so, as µ S is a natural transformation, the following diagram commutes: Hence, by the assumption on t, we have: So µ S Q 6 (α)(t) ∈ h −1 ( ) = { , p}. As δ( ) = δ(p) = , we can conclude: µ Gli Q 6 (α)(t) = δ(µ S Q 6 (α)(t)) = .
Theorem 3.11. There are extensions of intuitionistic affine logic AL i in which the syntactic Gentzen translation meets Troelstra's criteria for a double negation translation but the syntactic Glivenko translation does not and vice versa.

Hoops
If x and y are elements of a pocrim, x · (x → y) is a lower bound for x and y as is y · (y → x). Pocrims in which the two lower bounds coincide (and hence x · (x → y) is the meet of x and y) turn out to have many pleasant properties, motivating the following definition.
The following lemma provides some useful characterizations of hoops.
2. P is naturally ordered. I.e., for every x, y ∈ P such that x ≤ y, there is z ∈ P such that x = y · z.
3. For every x, y ∈ P such that x ≤ y, x = y · (y → x).
Proof. 1 ⇒ 2: Assume that P satisfies x · (x → y) = y · (y → x) and that x, y ∈ P satisfy x ≤ y, i.e., x → y = 1. Taking z = y → x, we have: 2 ⇒ 3: Assume that P is naturally ordered and that x, y ∈ P satisfy x ≤ y. Then x = y · z for some z. By the residuation property, we have z ≤ y → x, hence x = y · z ≤ y · (y → x) ≤ x and so x = y · (y → x). 3 ⇒ 4: assume that P satisfies x = y · (y → x) whenever x, y ∈ P and x ≤ y.
4 ⇒ 1: exchange x and y and use the fact that ≤ is antisymmetric.
The axiom [cwc] is often referred to as the axiom of divisibility in the literature, for reasons which become clear if one uses the alternative notation x/y for y → x, so that the formula of part 3 of Lemma 4.2 reads x = y ·(x/y).

Involutive hoops
Example 4.3. We write I for the involutive hoop whose universe is the unit interval [0, 1] and whose operations are defined by = 1 x · y = max(x + y − 1, 0) x → y = min(1 − x + y, 1) I provides an infinite model of classical Lukasiewicz logic, (which we refer to as LL c ).
Example 4.4. For n ≥ 2, let L n be the subhoop of I generated by 1 n−1 . It is easy to see that the universe of L n is L n = {0, 1 n−1 , 2 n−1 , . . . , n−2 n−1 , 1}. The hoops L n are involutive and provide natural finite models of classical Lukasiewicz logic LL c .
A hoop H is said to be Wajsberg, see [1,13], if it satisfies Proof. In a bounded Wajsberg hoop H we have therefore H is involutive. For the other direction, assume H is an involutive hoop and let x, y ∈ H. Since H is involutive, it is enough to show that ¬((x → y) → y) is symmetric in x and y which one may prove as follows: = (x → y) · (y → x) · ¬(y · (y → x)) Lemma 2.2, part 3 where the application of Lemma 4.2 uses that ¬y ≤ y → x. By [cwc], the last expression is symmetric in x and y.
There are, however, unbounded Wajsberg hoops, for instance: Example 4.6. Let O be the unbounded hoop whose universe is the half-open interval (0, 1] and whose operations are: = 1 x · y = xy x → y = min( y x , 1) O is easily seen to be a Wajsberg hoop because (x → y) → y = max(x, y).
Example 4.7. Apart from L 3 there is one other pocrim with 3 elements, namely G 3 = B ⊕ B. G 3 is the first non-Boolean example in the sequence of idempotent pocrims defined by the equations G 2 = B and G n+1 = G n ⊕ B. G n can be taken to be a set of n real numbers {1, x 1 , x 2 , . . . , x n−2 , 0} with = 1 > x 1 > x 2 . . . > x n−2 > 0 = ⊥ and with operations defined by The G n are finite Heyting algebras. They were used by Gödel to prove that intuitionistic propositional logic requires infinitely many truth values [18]. In G n , ¬x = ⊥ unless x = ⊥, so for n > 2, G n is not involutive.
It is easy to check from the definitions that C ⊕ D is a hoop iff both C and D are hoops.
Example 4.8. It can be shown that there are 7 pocrims with 4 elements: B × B, L 4 , G 4 , B ⊕ L 3 , L 3 ⊕ B, P 4 and Q 4 , where P 4 and Q 4 are as described in Examples 3.7 and 3.8 respectively. P 4 and Q 4 are the smallest pocrims that are not hoops: P 4 is not a hoop since it is not naturally ordered: there is no z with p · z = q. Likewise Q 4 is not a hoop, because there is no z with u · z = v.

De Morgan identities in hoops
In this section we prove two De Morgan identities for conjunction and residuation in bounded hoops. The proof of the identity for conjunction is elementary. The identity for residuation is proved using an indirect method captured in the following lemma. Here in each case S is subdirectly irreducible, Wajsberg and generated by the x i ∈ S. S is not necessarily bounded in cases (i) and (ii).
Proof. The proof uses Birkhoff's theorem (e.g., see [6,Theorem II.8.6]) to show that H is isomorphic to a subdirect product of subdirectly irreducible hoops and then uses the characterization of subdirectly irreducible hoops due to Blok and Ferreirim [1,Thorem 2.9]. Details are left to the reader.
Note that in case (i) of the lemma H is isomorphic to S and so is a bounded Wajsberg hoop and hence involutive.
Example 4.10. If 0 < k ∈ N, the identity ¬x k → δ(x) → x = clearly holds in any involutive hoop. It also holds in any hoop of the form B ⊕ S (since in such a hoop, either x = ⊥ or ¬x k = ⊥). This covers cases (i) and (iii) in Lemma 4.9. As the identity has only one variable, there is nothing to prove in case (ii). Hence, ¬x k → δ(x) → x = ⊥ holds in any bounded hoop.