Intuitionistic Modal Algebras

. Recent research on algebraic models of quasi-Nelson logic has brought new attention to a number of classes of algebras which result from enriching (subreducts of) Heyting algebras with a special modal operator, known in the literature as a nucleus . Among these various algebraic structures, for which we employ the umbrella term intuitionistic modal algebras , some have been studied since at least the 1970s, usually within the framework of topology and sheaf theory. Others may seem more exotic, for their primitive operations arise from algebraic terms of the intuitionistic modal language which have not been previously considered. We shall for instance investigate the variety of weak implicative semilattices , whose members are (non-necessarily distributive) meet semilattices endowed with a nucleus and an implication operation which is not a relative pseudo-complement but satisﬁes the postulates of Celani and Jansana’s strict implication. For each of these new classes of algebras we establish a representation and a topological duality which generalize the known ones for Heyting algebras enriched with a nucleus.


Introduction
A nuclear Heyting algebra is obtained by enriching a Heyting algebra H; ∧, ∨, →, 0, 1 with a unary modal operator satisfying the following identity: x → y = x → y.
(One can equivalently require to satisfy either the properties stated in Definition 2.5 or Definition 2.6; see below.)Such an operator is also known in the literature as a nucleus or multiplicative closure operator 1 .Many natural constructions give rise to nuclei.For instance, having fixed an element a ∈ H of a Heyting algebra, we can obtain a nucleus by setting either x := a → x or x := a ∨ x, or x := (x → a) → a. So, in particular, the identity map, the constant map x → 1 and the double negation map also define nuclei (see [1,18] for further examples).
The class of nuclear Heyting algebras (and some of its subreducts) has been studied since the 1970s, usually within the framework of topology and sheaf theory [2,3,18,20].A more recent paper [14] proposed a logic based on nuclear Heyting algebras (called Lax Logic) as a tool in the formal verification of computer hardware.Even more recently, another connection between nuclear Heyting algebras and logic emerged within the study of the algebraic semantics of quasi-Nelson logic [29,30].The latter may be viewed as a common generalization of both intuitionistic logic and Nelson's constructive logic with strong negation [22] obtained by deleting the double negation law.
As shown in [25,26,29], there exists a formal relation between the algebraic counterpart of quasi-Nelson logic and the class of nuclear Heyting algebras which parallels the well-known connection between Nelson algebras and Heyting algebras (see e.g.[32]).This relation-which, as we shall see, concerns the algebras in the full language as well as some of their subreductsprovides, in our view, further motivation for the study of nuclear Heyting algebras from a logical as well as an algebraic point of view.It is interesting to note that, with the notable exception of [1], studies of this kind are scant in the literature-perhaps owing to the mainly topological interest in this class of algebras.The purpose of the present contribution is to fill in this gap, at least partly, and at the same time to draw attention to certain subreducts of nuclear Heyting algebras whose interest is motivated by the recent developments in the theory of quasi-Nelson logic.
Since a nuclear Heyting algebra is usually presented in the language {∧, ∨, →, , 0, 1}, fragments that appear to be of natural interest (from a logico-algebraic perspective) are, for instance, the implication-free one {∧, ∨, } -perhaps enriched with the lattice bounds 0 and 1-and the implicational one {→, }.The former, whose models are distributive lattices enriched with a modal operator, is in fact the main object of [1], while the latter-whose models are Hilbert algebras, the algebraic counterpart of the purely implicational fragment of intuitionistic logic, expanded with a modal operator-was studied, mainly from a topological perspective, as far back as in [18], and as recently as in [12].Other less obvious but, in our opinion, also interesting classes of algebras emerged in the course of our recent investigations on quasi-Nelson logic and its algebraic counterpart, the variety of quasi-Nelson algebras.An interest in these classes of algebras, however, can also be motivated within the limits of the traditional framework of nuclear Heyting algebras, as we shall try to explain below.

. , a n ).
Denoting this algebra by H , we observe that the universe H can equivalently be defined as the nucleus image { a : a ∈ H} of H.While H is indeed a nuclear Heyting algebra, it is a very special one on which the operator is the identity map.This very fact, in turn, is essential in ensuring that H has a Heyting algebra reduct; for instance we have, for all a, b ∈ H , guaranteeing that ∧ is a meet semilattice operation on H .A similar reasoning applies to the other operations, although the join ∨ (computed in H ) does not coincide with the join ∨ (computed in H), i.e.H is not a subalgebra of H.This construction is easily seen to be a generalization of Glivenko's result relating Heyting and Boolean algebras (the latter corresponding to the case where x = ¬¬x).
Thus, although nothing prevents one from considering each operation f as defined on the whole universe H, in general ∧ and ∨ will not be semilattice operations on H, and → will not be a Heyting (i.e.relative pseudo-complement) implication on H (on the other hand, we always have = and 1 = 1.By definition, these new operations will be generalizations of the intuitionistic ones, which can be retrieved by requiring to be the identity map on H.In this respect natural questions to ask are, in our opinion, (1) which properties each generalized operation f retains, and (2) whether some particular choice of f has any independent interest that may justify further study.
A first answer to the latter question may be sought within the theory of quasi-Nelson logic.Indeed, as shown in the papers [25][26][27]29], some of the above-defined operations of type f naturally arise within the study of fragments of the quasi-Nelson language.From this standpoint, it is also interesting to observe that the classes of algebras one obtains through the twist representation (see below) combine the original Heyting operations with the new ones.Thus, for instance, one of the classes of algebras arising in this way (see Definition 3.1) retains the original meet semilattice operation (and the lattice bounds) while replacing the Heyting implication with a generalized counterpart: that is, we are looking at the {∧, → , 0, 1}-subreducts of nuclear Heyting algebras.We stress that these new algebras are not the result of an arbitrary choice of operations, but arise as twist factors in the representation of subreducts of quasi-Nelson algebras, as we now proceed to explain.
A quasi-Nelson algebra may be defined as a commutative integral bounded residuated lattice (see e.g.[16] for formal definitions of these terms) A = A; , , * , ⇒, ⊥ that (upon letting ∼ x := x ⇒ ⊥) satisfies the Nelson identity: Quasi-Nelson algebras arise as the algebraic counterpart of quasi-Nelson logic, which can be viewed either as a generalization (i.e. a weakening) common to Nelson's constructive logic with strong negation and to intuitionistic logic, or as the extension (i.e.strengthening) of the well-known substructural logic F L ew (the Full Lambek Calculus with Exchange and Weakening) by the Nelson axiom: We refer to [30] for further details on quasi-Nelson logic, as well as for other equivalent characterizations of the variety of quasi-Nelson algebras (which can e.g. also be obtained as the class of (0, 1)-congruence orderable commutative integral bounded residuated lattices).
Formally, every Heyting algebra may be viewed as a quasi-Nelson algebra (on which ∧ = * , ∨ = , → = ⇒ and 0 = ⊥) and, as noted earlier, the double negation map defines a modal operator on every Heyting algebra H.If we replace H by a quasi-Nelson algebra A, then the double negation map still gives us a nucleus in the sense of Definition 2.5 but no longer in the sense of Definition 2.6, for item (ii) may fail to be satisfied (indeed, the equivalence of both definitions breaks down outside the setting of Heyting algebras).The double negation can, however, be used to obtain a nucleus on a special quotient H(A), which is the (Heyting) algebra canonically associated to each quasi-Nelson algebra A via the twist construction.
Given a quasi-Nelson algebra A, consider the map given, for all a ∈ A, by a → a * a.The kernel θ of this map is a congruence of the reduct A; , , * which is also compatible with the double negation operation and with the weak implication ⇒ 2 given by x ⇒ 2 y := x ⇒ (x ⇒ y).Letting (x/θ) := ∼ ∼ x/θ, we thus have a quotient algebra H(A) = A/θ; , , ⇒ 2 , , ⊥ , which is a nuclear Heyting algebra (where * = ).Moreover, A embeds into a twist-algebra over H(A), defined as follows.
Definition 1.1 suggests that certain term operations of the language of nuclear Heyting algebras may be of particular interest in the study of fragments of the quasi-Nelson language.Consider, for instance, the monoid operation ( * ).In order to define it, on a quasi-Nelson algebra A ≤ H , we need two operations on H: the semilattice operation ∧ (for the first component) and, for the second component, an implication-like operation (henceforth denoted by ) which can be given by x y := x → y.The latter claim may not be obvious, but using the properties of the twist construction and the modal operation, it is not hard to verify the following equalities: These observations led to the introduction of the class of algebras dubbed -semilattices in [27], where it is shown in particular that the { * , ∼}subreducts of quasi-Nelson algebras are precisely the algebras representable as twist-algebras over -semilattices.Similar considerations motivated the introduction of other term operations of the language of nuclear Heyting algebras, such as the following: x y := (x ∧ y) x ⊕ y := (x ∨ y).
The previous considerations suggest the above-mentioned classes of modal algebras as mathematical objects that may be of interest both in themselves and in relation to the study of non-classical logics, in particular Nelson's logics 2 .The aim of the present paper is to improve our understanding of these classes of algebras from an algebraic and a topological point of view.
The rest of the paper is organized as follows.Section 2 recalls the definitions of (subreducts of) Heyting algebras and of the modal operators known as nuclei.
In Sect. 3 we introduce the main classes of algebras of interest.The first is the variety of weak implicative semilattices (Subsect.3.1), which is a variety of semilattices enriched with an implication operation that, while not being necessarily a relative pseudo-complement, satisfies the postulates of Visser's strict implication [9].We then introduce the variety of nuclear Hilbert semigroups (Subsect.3.2), whose members consist of bounded Hilbert algebras (the purely implicational subreducts of Heyting algebras) enriched with a pseudo-meet operation giving rise to a semigroup that is not necessarily a semilattice.The variety of nuclear implicative semilattices (Subsect.3.3), which is not new (see e.g.[2]), mostly interests us as a basis for our definition of ⊕-implicative semilattices (Subsect.3.4): the latter are bounded implicative semilattices (i.e. the join-free subreducts of Heyting algebras) endowed with a nucleus and a join-like operation ⊕ which forms a semigroup but not necessarily a semilattice.
In Sect. 4 we recall or establish some simple facts about congruences and homomorphisms of the above-mentioned classes of algebras; these will be useful for characterizing the morphisms in the corresponding categories.
In Sect. 5 we introduce topological dualities for our new classes of intuitionistic modal algebras: in order to do so we shall build on the existing dualities for Lax Hilbert algebras (i.e.Hilbert algebras expanded with a nucleus, Subsect.5.1) and for (non-necessarily distributive) meet semilattices (Subsect.5.3).The new dualities are established in Subsects.5.2, 5.4 and 5. 5.
Finally, the concluding Sect.6 contains some indications for future research.

Heyting Algebras, Subreducts and Nuclei
In this section we recall the main definitions of subreducts of Heyting algebras that are relevant to our study, as well as the definition(s) of nucleus.
The purely implicational subreducts of Heyting algebras are known in the literature as Hilbert algebras, or (positive) implication algebras.Definition 2.1.A Hilbert algebra is an algebra H; →, 1 of type 2, 0 that satisfies the following (quasi-)identities: Every Hilbert algebra has a natural order ≤ given, for all a, b ∈ H, by a ≤ b iff a → b = 1, having 1 as top element (as a matter of fact, the constant 1 need not be included in the language, for it is term definable by 1 := x → x).If the order ≤ also has a minimum element (denoted 0), we speak of a bounded Hilbert algebra.In such a case we include 0 in the algebraic signature, and one can define a negation operation ¬ by ¬x := x → 0. Bounded Hilbert algebras correspond to the {→, ¬, 0, 1}-subreducts of Heyting algebras.
We shall refer to item (ii) above as to the "property of the pseudocomplement".Pseudo-complemented semilattices form a variety whose only proper subvariety is the class of Boolean algebras [31, p. 305]; the latter can thus be relatively axiomatized by adding any identity that is not valid on all pseudo-complemented semilattices (for instance the involutive law ¬¬x = x).
(ii) x ∧ y ≤ z if and only if x ≤ y → z.
The property in item (ii) is known as residuation, and we shall say that ∧, → is a residuated pair.Implicative meet semilattices are precisely the ∨-free subreducts of Heyting algebras; in turn, the ∧-free reduct of every implicative meet semilattice forms a Hilbert algebra.A bounded implicative semilattice is one whose semilattice reduct has a least element 0. In such a case, by letting ¬x := x → 0, one obtains a pseudo-complemented semilattice.With the above definitions in mind, Heyting algebras may be introduced as follows.
The pseudo-complement negation ¬ is defined, on every Heyting algebra, by ¬x := x → 0, as in the preceding cases.
In the following sections we shall consider algebras that result from adding a modal-like operator to the subreducts of Heyting algebras introduced earlier.Such operators are known as nuclei (or modal operators, or multiplicative closure operators), and have been extensively studied in the literature on residuated lattices and Heyting algebras; for our purposes, the results contained in the dissertation [18] will be particularly useful.We shall consider two different but essentially equivalent definitions for a nucleus, which depend on which other operations are available on the algebra.Definition 2.5.Let A be an algebra having a reduct A; ∧, 0 that is a (meet) semilattice with order ≤ and minimum 0. We shall say that an operation : A → A is a nucleus on A if the following identities are satisfied: If 0 = 0, then we say that is dense.
Observe that the above properties entail that, if the order ≤ has a maximum element 1, then 1 = 1 (so the operator is indeed modal-like in that it preserves all finite meets).When the underlying algebra does not have a meet operation, we can define a nucleus as follows.
Following [12], a Hilbert algebra endowed with a nucleus, thus viewed as an algebra in the language {→, , 1}, will be called a Lax Hilbert algebra (an LH-algebra, for short).

Intuitionistic Modal Algebras
In this section we introduce a number of classes of algebras that, as observed earlier, arise from the twist representation of (subreducts of) quasi-Nelson algebras.All these carry a nucleus operator together with one or more algebraic operations that are term definable in the language of Heyting algebras enriched with the nucleus.

Weak Implicative Semilattices
As mentioned in the introduction, the algebras introduced in the next definition were first considered in [27], under the name of -semilattices, as factors in the twist representation of { * , ∼}-subreducts of quasi-Nelson algebras.Here we introduce the more suggestive term "weak implicative semilattices", which we shall explain in a moment.Definition 3.1.A weak implicative semilattice (abbreviated WIS) is an algebra S = S; ∧, , 0, 1 satisfying the following properties (we abbreviate x := 1 x): (i) S; ∧, 0, 1 is a bounded semilattice (with order ≤).
(vii) x ≤ y z if and only if x ∧ y ≤ z.
(viii) x y = x y.
Items (ii)-(v) of the preceding definition entail that the operation given by x := 1 x indeed realizes a nucleus (in the sense of Definition 2.5) on every weak implicative semilattice S. Therefore, whenever convenient, we shall consider weak implicative semilattices as algebras in the language that includes the nucleus thus defined.
The operation can be thought of as a generalized (intuitionistic) implication in the following sense.For every algebra having a bounded implicative semilattice reduct S; ∧, →, 0, 1 as per Definition 2.3 and a nucleus , we can obtain a weak implicative semilattice by letting x y := x → y (cf.Definition 3.12 and Example 3.13).As a nucleus we can for example take the double negation (which gives us x y = x → ¬¬y) or the identity function on S (which gives us = →).The following remark should explain the name chosen for our algebras.Consider the class of weak Heyting algebras (or WH-algebras), which arose in [9] from the study of strict implication fragments of modal logics.Formally, a WH-algebra is a bounded distributive lattice L; ∧, ∨, 0, 1 further endowed with a binary operation which satisfies the following identities [9, Def.3.1]: It is not hard to verify that every weak implicative semilattice satisfies all the above identities, except of course (WH2).Conversely, the ∨-free reduct of every WH-algebra satisfies all the properties listed in Definition 3.1 except perhaps item (v).Those WH-algebras that satisfy item (v) are known as basic algebras [9, Def.3.3], and form the algebraic counterpart of Visser's logic [34].
On every weak implicative semilattice S (or, more generally, on every algebra having a nucleus ), we can consider the set of -fixpoints, which can be given in either of the following ways: It is easy to verify that, for every weak implicative semilattice S = S; ∧, , 0, 1 , the set S is the universe of a subalgebra S = S ; ∧, , 0, 1 , which is a bounded implicative semilattice.The case where S = S is characterized in the following proposition.(ii) S; , 0, 1 is a (bounded) Hilbert algebra.
Regarding the preceding proposition, we observe that the algebras on which the nucleus is the identity function (which are then just ordinary subreducts of Heyting algebras) are of special interest within the theory of quasi-Nelson algebras, for they correspond precisely to subreducts of Nelson algebras (i.e.involutive quasi-Nelson algebras).
As expected, every implicative semilattice with a nucleus S = S; ∧, →, , 0, 1 (Definition 3.12) may be endowed with a weak implicative semilattice structure by letting x y := x → y.Given this definition, we have the following result, which was used in [27] to show that the class of twist-algebras over weak implicative semilattices coincides with the { * , ∼}-subreducts of quasi-Nelson algebras.
It was shown in [2] that the class of bounded nuclear implicative semilattices is locally finite.This result, together with Proposition 3.6, entails that weak implicative semilattices are also locally finite.
Proposition 3. 7. The class of weak implicative semilattices is a variety.Proof.It suffices to verify that item (vii) of Definition 3.1, which is the one non-equational condition, can be equivalently replaced by the following two: It is easy to verify that (I) and (II) are satisfied by all weak implicative semilattices.Indeed, by item (vii) of Definition 3.1, we have 1 ≤ x x iff 1 ∧ x ≤ x, which is true by item (v).Similarly, we have x ≤ y x iff x ∧ y ≤ x, which is certainly true.
For the other direction of the equivalence, let S = S; ∧, , 0, 1 be an algebra that satisfies all items of Definition 3.
To justify the last equality, recall that is a nucleus, and in particular c = c, because (by item (ii) of Definition 3.1) Remark 3.8.As observed earlier, for each weak implicative semilattice S = S; ∧, , 0, 1 , the nucleus image S is the universe of a subalgebra S = S ; ∧, , 0, 1 which is a bounded implicative semilattice.Conversely, from a pair S, I , where S is a bounded meet semilattice and I is a bounded implicative semilattice related by suitable maps (if one wishes, I may simply be taken to be a subset of S), we can obtain a weak implicative semilattice as follows.Let S = S; ∧ S , 0 S , 1 S and I = I; ∧ I , → I , 0 I , 1 I be as above and let n : S → I and p : I → S be maps satisfying the following properties: (i) both n and p preserve finite meets and the bounds; Then, letting x y := p(n(x) → n(y)), we have that S; ∧ S , , 0 S , 1 S is a weak implicative semilattice.Indeed, one can prove that every weak implicative semilattice S = S; ∧, , 0, 1 , arises in this way from the pair S; ∧, 0, 1 , S ; ∧, , 0, 1 by letting n = and p = Id S .

nH-Semigroups
The class of algebras considered in this section was introduced to provide a twist representation for the (weak) implication-negation subreducts of quasi-Nelson algebras [25,29]; the latter class, in turn, is the equivalent algebraic semantics of the algebraizable fragment of quasi-Nelson logic studied in [19].
We recall that, according to the twist construction of quasi-Nelson algebras (Definition 1.1), the weak implication is given by: This suggests that the factor algebras corresponding to the weak implicationnegation subreducts will need to carry, besides the nucleus operator, only an intuitionistic implication → and a "pseudo-meet" operation such that x y = (x ∧ y); this motivates the following definition.Definition 3.9.([25], Def.4.5).A bounded nuclear Hilbert semigroup (nHsemigroup for short) is an algebra S = S; , →, 0, 1 such that: is a commutative semigroup.
(iii) The operation given by x := x x is a dense nucleus on S; →, 0, 1 in the sense of Definition 2.6.
It is clear that the {→, , 1}-reduct of every nH-semigroup is a Lax Hilbert algebra in the sense of [12], an observation that we shall exploit later on.Observe that nH-semigroups generalize bounded implicative semilattices, for every bounded implicative semilattice A; ∧, →, 0, 1 is an nHsemigroup where ∧ = and is the identity map.As in the case of weak implicative semilattices, it is also easy to verify that the algebra of -fixpoints S ; , →, 0, 1 of every nH-semigroup S = S; , →, 0, 1 is a bounded implicative semilattice.The following examples should provide some further insight on nH-semigroups.

Nuclear Implicative Semilattices
The following definition is easily seen to be equivalent to the one adopted in [2] which is based, for the nucleus, on our Definition 2.5.
As in the preceding cases, the algebra of -fixpoints S = S ; ∧, →, 0, 1 of a nuclear implicative semilattice S = S; ∧, →, , 0, 1 is a bounded implicative semilattice.Thus each class of algebras K introduced so far is nuclear in the sense of [3], that is, for every member A ∈ K, we have A ∈ K.
The following example should help clarify the relationship among the abovementioned classes.

⊕-Implicative Semilattices
As mentioned earlier, nH-semigroups arise as factors in the twist representation of the weak implication-negation subreducts of quasi-Nelson algebras.If we enrich the latter with the quasi-Nelson meet operation, we obtain a variety of algebras (dubbed quasi-Nelson semihoops in [27]) which can be represented as twist-algebras over the class of ⊕-implicative semilattices introduced below.Indeed, Definition 1.1 suggests that, in order to represent the quasi-Nelson meet ( ), one only needs to introduce a further binary operation (here denoted ⊕) such that x ⊕ y = (x ∨ y).
As observed in [27], the quasi-Nelson monoid operation ( * ) is definable as follows: Thus every quasi-Nelson semihoop also carries the quasi-Nelson monoid operation; this is in keeping with the observation that, according to Definition 1.1, the operation * can be defined on twist-algebras using only the implicative semilattice operations (and the nucleus) of the factor algebra.Definition 3. 14.A ⊕-implicative semilattice is an algebra S = S; ∧, ⊕, →, 0, 1 such that: (i) S; ∧, →, , 0, 1 is a bounded nuclear implicative semilattice whose nucleus is given by x := x ⊕ x (Definition 3.12).
(iii) x ⊕ 1 = 1. (iv Let us illustrate the preceding definition with an example. Example 3. 15.Let H; ∧, ∨, →, , 0, 1 be a nuclear Heyting algebra.Then, upon defining x ⊕ y := (x ∨ y), the algebra H; ∧, ⊕, →, 0, 1 is a ⊕implicative semilattice.Thus, in particular, every Heyting algebra may be viewed as a ⊕-implicative semilattice where, taking the nucleus to be the identity map, we have that ⊕ coincides with the lattice join, whereas taking to be the double negation map we have x ⊕ y = ¬(¬x ∧ ¬y).
The following properties will be useful later on, in particular for giving a characterization of congruences of ⊕-implicative semilattices.

Congruences and Homomorphisms
In this section we take a look at congruences and homomorphisms of the above-introduced classes of intuitionistic modal algebras.As we shall see, similarly to the case of Heyting algebras endowed with a nucleus, in most cases the congruences of each intuitionistic modal algebra coincide with those of its purely implicational reduct.The next well-known fact (see e.g.[21]) will be a key ingredient in subsequent proofs.
Proposition 4.2.Let A be either (i) a Lax Hilbert algebra, (ii) an nHsemigroup, (iii) a nuclear implicative semilattice or (iv) a ⊕-implicative semilattice.In all these cases, the congruences of A coincide with those of the Hilbert algebra reduct of A.

Proof. (i).
Let A = A; →, , 1 be a Lax Hilbert algebra, and let θ be a congruence of the Hilbert algebra reduct A; →, (iii).Let A = A; ∧, →, , 1 be a nuclear implicative semilattice.Then the reduct A; →, , 1 is a Lax Hilbert algebra, and by item (i) we know that Con( A; →, ) = Con( A; → ).To conclude the proof, it remains to show that Con( A; ∧, → ) = Con( A; → ).Let us then assume that θ ∈ Con( A; → ) and a, b ∈ θ.Thus, by Lemma 4.1, we have a → b, 1 , b → a, 1 ∈ θ.Since the identity x → (y ∧ z) = (x → y) ∧ (x → z) holds on every implicative semilattice [10], for all c ∈ A, we have (a∧c) A similar reasoning shows that (b ∧ c) → (a ∧ c), 1 ∈ θ as well, so we can apply Lemma 4.1 to conclude that a ∧ c, b ∧ c ∈ θ.Since we are in a semilattice, this is sufficient to establish that θ is compatible with ∧, as required.
(iv).Now let A = A; ∧, ⊕, →, 0, 1 be a ⊕-implicative semilattice and θ ∈ Con( A; → ).We know by item (iii) above that θ is compatible with ∧ (and with the nucleus , which is given by x := x ⊕ x).To conclude the proof, it suffices to show that θ is compatible with ⊕ as well.To this end, assume a, b ∈ θ and let c ∈ A. By Lemma 4.1, our assumptions imply ), so we can apply Lemma 3.17 (v) to compute: Given an algebra A with a partial order ≤ and maximum 1, we shall say that an element a ∈ A is the penultimate element of A if a = 1 and, for all b ∈ A such that b < 1, it holds that b ≤ a.
From Proposition 4.2 and Lemma 4.3 we immediately obtain the following result.
Corollary 4. 4. Let A be either a Lax Hilbert algebra, an nH-semigroup, an implicative semilattice (with a nucleus) or a ⊕-implicative semilattice.Then A is subdirectly irreducible if and only if its underlying order has a penultimate element.
Congruences correspond, of course, to special (i.e.surjective) homomorphisms; thus the result of Proposition 4.2 does not necessarily extend to arbitrary homomorphisms.This, however, does hold for nH-semigroups, as shown below.
Proposition 4. 5.A map h : S → S between nH-semigroups S and S is a homomorphism (i.e.preserves , →, 0 and 1) if and only if h is a bounded Lax Hilbert algebra homomorphism between the corresponding bounded Lax Hilbert algebra reducts (i.e.preserves , →, 0 and 1).From Proposition 4.5 and the observation that every implicative semilattice A; ∧, →, 0, 1 is an nH-semigroup where ∧ = and is the identity map, it is easy to adapt the preceding result to (not necessarily bounded) implicative semilattices (with a nucleus).

Corollary 4.6. A map h : S → S between implicative semilattices S and S is a homomorphism (i.e. preserves ∧ and →) if and only if h is a Hilbert algebra homomorphism between the corresponding Hilbert algebra reducts (i.e. preserves →).
Corollary 4. 7. A map h : S → S between (bounded) nuclear implicative semilattices S and S is a homomorphism (i.e.preserves ∧, →, , 1 and, if present, the bottom element 0) if and only if h is a Lax Hilbert algebra homomorphism between the corresponding (bounded) Lax Hilbert algebra reducts (i.e.preserves →, , 1 and, if present, the bottom element 0).
The preceding results suggest that, when we consider (most of) the preceding classes of algebra from a categorical point of view (see Sect. 5), the central notion we will have to look at with regards to morphisms will be that of Lax Hilbert algebras homomorphism.For weak implicative semilattices, we shall also consider the weaker notion of semi-homomorphism.
Thus a semi-homomorphism is a bounded meet semilattice homomorphism that further satisfies (SH4).We shall say that h is a homomorphism if h preserves all operations, that is, h is a semi-homomorphism that further satisfies:

(SH5) h(a) h(b) ≤ h(a b).
Lemma 4.9.A map h : S → S between weak implicative semilattices S and S is a semi-homomorphism if and only if h is a bounded meet semilattice homomorphism such that h( a) ≤ h(a) for all a ∈ S.
Let X, ≤ be a poset.For each ).We will write [y) and (y] instead of [{y}) and ({y}], respectively.We also write P(X) and Up(X) for the set of all subsets and upsets of X, respectively.We note that Up(X), ∩, ∪, ∅, X is a bounded distributive lattice.

Lax Hilbert Algebras
The papers [11] and [12] introduce topological dualities for categories associated to Hilbert algebras with a modal operator, which can be easily extended to nH-semigroups.
Let us begin by introducing Lax Hilbert spaces, which are special T 0 spaces X, τ K having a base of compact sets K enriched with a binary relation R ⊆ X × X. Recall that the saturation of a set Y ⊆ X is given by: and the closure of Y ⊆ X is given by: cl(Y ) := {U c : U ∈ K and Y ∩ U = ∅}.
We denote by ≤ K , or by ≤, the dual specialization order given, for all x, y ∈ X, by x ≤ K y iff y ∈ cl(x).Recall that X is a T 0 space if and only if ≤ is a partial order.Recall also that a subset Y ⊆ X is said to be irreducible when, for all closed sets A space is sober when, for every irreducible closed set Y ⊆ X, there exists a unique x ∈ X such that Y = cl(x).
Let X and Y be sets and R ⊆ X × Y a binary relation.For every (x, y) ∈ X ×Y , we consider the sets We define the map R : P(Y ) → P(X) given by Definition 5. 1. ([12], Def. 7).A Lax Hilbert space (or LH-space) is a structure X, τ K , Q such that: (i) X, τ K is a topological space having a base of compact sets K.
Let LHSp denote the category of Lax Hilbert spaces with LH-relations, and let LHSpF be the subcategory of LHSp having Lax Hilbert spaces as objects and functional LH-relations as morphisms.Correspondingly, let LHA denote the category having Lax Hilbert algebras as objects and Lax Hilbert algebra semi-homomorphisms as morphisms (Definition 5.3).Definition 5. 3 We denote by LHAH the subcategory of LHA having the same objects but requiring the morphisms to be algebraic homomorphisms, i.
Given a Lax Hilbert algebra H; →, , we define an (implicative) filter as a non-empty subset

The set of all irreducible filters of an algebra H is denoted by X(H).
Consider the map σ given by a → {x ∈ X(H) : a ∈ X} for all a ∈ H. Then the family: is the base for a topology τ K H .Further defining:

which the dual specialization order is the inclusion relation on X(H).
Given Lax Hilbert algebras H; →, , H ; → , and a semihomomorphism h : H → H , we have that

and X(h) is functional if and only if h is an LH-algebra homomorphism.
Conversely, given an LH-space X, τ K , Q , the Lax Hilbert algebra H(X); → K , Q has the set H(X) := {U c : U ∈ K} as universe and operations given, for all U, V ∈ H(X), as follows: given by H(R)(U ) := {x ∈ X : R(x) ⊆ U } for all U ∈ H(X ) is a semi-homomorphism of LH-algebras; furthermore, as expected, H(R) is an LH-algebra homomorphism if and only if R is functional.
For every Lax Hilbert algebra H; →, , the mapping σ : H → H(X(H)) is an isomorphism of Lax Hilbert algebras.Conversely, every LH-space X, τ K , Q is homeomorphic to the space X(H(X)), τ K H(X) , Q H(X) through the map H X given, for all x ∈ X, by: Joining together these observations we have the announced equivalence(s).

nH-Semigroups
The notion of filter of an nH-semigroup S = S; , →, 0, 1 can be taken to be the same as that of an (implicative) filter of the Hilbert algebra reduct S; →, 1 .This is suggested by the observation that any (implicative) filter F of S; →, 1 is closed under the semigroup operation of S. Indeed, assuming a, b ∈ F we have a, b ∈ F because F is increasing.Then we can apply Lemma 5.5 below to conclude a b ∈ F .Let X be a set.We recall that a binary relation R on Proposition 5. 6.Let S = S; , →, 0, 1 be an nH-semigroup, and let X(S), τ K S , Q S be the LH-space dual to the Lax Hilbert algebra reduct S; →, 1 .Then, (i) X(S) ∈ K S (hence, X(S), τ K S is compact).
(ii) Q S is serial (i.e. for all x ∈ X(S), there is y ∈ X(S) such that x, y ∈ Q S ). (iii For the first two items, see [11, p. 52 and Thm.4.5].To prove (iii), let U, V ∈ K S .Then there are a, b ∈ S such that U = (σ(a)) c and Observe that, on the one hand, we have ) iff there is y ∈ X(S) such that −1 (x) ⊆ y and either a / ∈ y or b / ∈ y.On the other hand, by Lemma 5.5, we have The required result then follows from [11,Lemma 3.5], which states that a / ∈ x iff there is y ∈ X(S) such that −1 (x) ⊆ y and a / ∈ y.
Definition 5. 7.An nH-space is an LH-space that further satisfies the three items in Proposition 5. 6.
By Definition 5.7, the space dual to (the Lax Hilbert algebra reduct of) every nH-semigroup S is an nH-space.Conversely, given an nH-space X, τ K , Q , the nH-semigroup H(X); Q , → K , Q is defined as for LHspaces, with the extra operation Q being given, for all U, V ∈ H(X), by: Proof.We know that H(X); → K , Q is a Lax Hilbert algebra that moreover (by item (i) of Proposition 5.6) is bounded.To complete the proof, it suffices to show that the operation Q is well defined on H(X) and satisfies the remaining identities of nH-semigroups.Regarding the former claim, let U, V ∈ H(X), so that U c , V c ∈ K.Then, by the property stated in item (iii) of Proposition 5.6, we have Let us now look at the properties in Definition 3.9.We know from the duality for Lax Hilbert algebras that items (i) and (iii) are satisfied.As for item (ii), commutativity of Q is clear; associativity easily follows from the observation that In fact, for all U, V, W ∈ H(X), we have: For item (iv), we need to show that, for all U, V ∈ H(X),

Let us verify that
as required.Item (vi) easily follows from the observation that Q is serial.Finally, item (vii) is immediate.Proposition 5.9.For every nH-semigroup S; , →, 0, 1 , the mapping σ : S → H(X(S)) is an isomorphism of nH-semigroups.Proof.Since we know that σ is a (Lax) Hilbert algebra isomorphism, it suffices to show that σ preserves 0 and the operation .Regarding the former, we have σ( 0 Let nH be the category of nH-semigroups with algebraic homomorphisms, and let nHSp be the category having nH-spaces as objects and nH-relations for morphisms, defined as follows. Definition 5. 10.We say that an LH-relation (Definition 5.2) R ⊆ X × X between nH-spaces X, τ K , Q and X , τ K , Q is an nH-relation iff R is functional and serial.
Given nH-spaces X, τ K , Q , X , τ K , Q and an nH-relation R ⊆ X ×X , we have that the map H(R) : H(X ) → H(X) defined as in the case of LH-algebras is an nH-semigroup homomorphism.Indeed, H(R) preserves the bounds (because R is serial) and the operation as well because of Proposition 4. 5.
The previous observations entail that every nH-space X, τ K , Q is homeomorphic to the space X(H(X)), τ K H(X) , Q H(X) through the map H X given as before.Joining together these observations we obtain the announced equivalence.
Theorem 5.11.The categories nHSp and nH are dually equivalent via the functors H and X.

Meet Semilattices
The next objective is to present a topological duality for the class of weak implicative semilattices.Since the underlying semilattice of these algebras is not distributive we need to appeal to another duality for semilattices that are not necessarily distributive.In this section we will use the duality developed in [13] for bounded semilattices.
Let S = S; ∧, 0, 1 be a bounded semilattice.A filter is a non-empty set F ⊆ S that is increasing with respect to the semilattice order and closed under finite meets.The set of all filters on S is denoted by Fi(S).
As before, a filter P of S is irreducible when, for all F 1 , F 2 ∈ Fi(S) such that P = F 1 ∩F 2 , one has P = F 1 or P = F 2 .The set of all irreducible filters on S is denoted by X(S).A filter P is prime when, for all F 1 , F 2 ∈ Fi(S) such that F 1 ∩ F 2 ⊆ P , one has F 1 ⊆ P or F 2 ⊆ P .Every prime filter is irreducible, while the converse need not hold (cf.Proposition 5.13 below).The following characterizations are quite useful in practice (see [7,8]).A filter F is irreducible iff for every a, b / ∈ F there exists f ∈ F and c / ∈ F such that a ∧ f ≤ c and b ∧ f ≤ c.A filter P is prime iff, for every a, b / ∈ F , there exists c / ∈ F such that a ≤ c and b ≤ c.An order ideal of S is a set I ⊆ S that is decreasing and such that for all a, b ∈ I, there exists c ∈ I with a, b ≤ c.It is easy to see that a filter F is irreducible iff F c = S − F is an order ideal.
The following result shows that every proper filter of a semilattice is the intersection of irreducible filters.
Theorem 5. 12. ([7], Thm. 8).Let S be a semilattice.Let F be a filter and let I be an order ideal of S such that F ∩ I = ∅.Then there exists P ∈ X(S) such that F ⊆ P and P ∩ I = ∅.
There are different generalizations of the notion of distributivity from lattices to semilattices.Here we are going to say that a semilattice S is distributive iff for all a, b, c ∈ S such that a ∧ b ≤ c, there exist a , b ∈ S such that a ≤ a , b ≤ b , and a ∧ b = c (see [8,13]).As expected, a bounded lattice L; ∨, ∧, 0, 1 is distributive (as a lattice) if and only if L, ∧, 1 is a distributive semilattice according to our definition.Proposition 5. 13. ([7]).Let S = S; ∧, 1 be a semilattice.The following conditions are equivalent: (iii) Every irreducible filter of S is prime.
Given a non-empty set X, consider a family K ⊆ P(X) such that X = K.In this section we denote by τ K the topology on X taking as subbase the family K.It is well known that τ K consists of ∅, X, all finite intersections of members of K and all arbitrary unions of these finite intersections.
Let X, τ K be a topological space.We consider the following collection of subsets of X: Let C K (X) be the closure system on X generated by S(X), i.e., C K (X) = { D : D ⊆ S(X)}.The closure operator associated to C K (X) is denoted by cl K .The elements of C K (X) are closed, but not every closed set is of this form, since K is only a subbase.Thus, in general cl(Y ) ⊆ cl K (Y ) for any Y ⊆ X, but cl(x) = cl K (x), for any x ∈ X.The elements of C K (X) will be called subbasic closed subsets of X.It is clear that S(X) is closed under finite intersections iff K is closed under finite unions.Thus, if K is closed under finite unions and ∅, X ∈ K, then S(X) = S(X), ∩, ∅, X is a bounded semilattice, called the dual semilattice of X, τ K .
Consider a topological space X, τ K .From now on we will always assume that the subbase K is closed under finite unions and that ∅, X ∈ K.
We are now going to define the dual spaces of (bounded) semilattices; the definition we propose does not coincide with the one given in [13], but it is easily seen to be equivalent (see Lemma 3.7 and Prop.3.8 of [13]).S(X); ∩, →, X .If the IS-space X, τ is compact, then S(X); ∩, →, ∅, X is a bounded implicative semilattice.Recall from [10] that a functional meetrelation is a meet-relation R between two IS-spaces X, Y such that, for every pair (x, y) ∈ X × Y , if (x, y) ∈ R, then there exists z ∈ X such that x ≤ z and R(z) = [y).
Let IS be the category of compact IS-spaces where the morphisms are the functional meet-relations.By the results of [8,10], we have that the category BIM bounded implicative semilattices whose morphisms are bounded implicative homomorphisms is dually equivalent to the category IS.
Given a semilattice S = S; ∧, 1 , we shall say that a map : S → S is a modal operator if it satisfies the identities (x ∧ y) = x ∧ y and 1 = 1 (so every nucleus is, in particular, a modal operator: cf.Definition 2.5).The algebra S; ∧, , 1 will be called a modal semilattice.A -homomorphism between modal semilattices S 1 , S 2 is a meet-homomorphism h : S 1 → S 2 such that h( 1 a) = 2 h(a), for all a ∈ S 1 .Since a modal operator on a semilattice S is a particular case of meet-homomorphism, we have that the relation R ⊆ X(S) × X(S) given by (x, y) ∈ R iff −1 (x) ⊆ y is a meetrelation.We are now going to characterize dually the -homomorphisms.

Proof. Let a ∈ S 1 and suppose that h( 1 a)
2 h(a).Then there exists x ∈ X(S 2 ) such that h( 1 a) ∈ x and 2 h(a) / ∈ x.As −1 2 (x) is a filter, there is y ∈ X(S 2 ) such that −1 2 (x) ⊆ y and h(a) / ∈ y.Again, since h −1 (y) is a filter of S 1 , there exists z ∈ X(S 1 ) such that h −1 (y) ⊆ z and a / ∈ z.Then (x, z) ∈ R 2 • r h .So, there exists k ∈ X(S 1 ) such that (x, k) ∈ r h and (k, z) ∈ R 1 .But as h( 1 a) ∈ x, we have 1 a ∈ k, and so a ∈ z, which is a contradiction.Thus h( 1 a) ≤ 2 h(a), for all a ∈ S 1 .The proof for the other inequality is similar.
Let S be a weak implicative semilattice with associated implicative semilattice S .We consider the following families of filters: Proposition 5. 23.For every weak implicative semilattice S, there exists an order isomorphism between the posets (Fi * (S), ⊆) and (Fi(S ), ⊆), which restricts to an order isomorphism between (X * (S), ⊆) and (X(S ), ⊆).Proof.For each F ∈ Fi * (S), we have that F ∩ S ∈ Fi(S ).And for each H ∈ Fi(S ) it is easy to see that −1 (H) ∈ Fi * (S).Moreover it easy to see that . Thus, the map α : Fi * (S) → Fi(S ) given by α(F ) = F ∩ S is an order isomorphism.
Let x ∈ X * (S).We prove that x ∩ S is an irreducible filter of S .Let S be a semilattice.For each D ⊆ S the filter generated by a subset D of S is denoted by Fg(D).
Our next aim is to identify the S-spaces (Definition 5.14) that correspond to weak implicative semilattices.
Theorem 5.28, together with the duality for bounded semilattices, motivates the following definition.Definition 5. 29.An implicative S-space is a triple X, R, τ K such that X, τ K is an S-space, and R ⊆ X × X is a relation such that: (IS1) R is dense, serial and R ⊆≤, where ≤ is the dual specialization order of X, τ K .
Then X * , τ X * is a compact S-space where K * is a base for τ X * .Thus, X * , τ X * is a compact DS-space, i.e., is the space of a bounded distributive semilattice [7,8].
Our next aim is to extend the representation of weak implicative semilattices through implicative S-spaces to a full categorical duality.To this end, we need to specify which are the morphisms between two objects in the respective categories.

2
. Then there exist We consider in S 2 the filter Fg(Z) ∩ (S 2 ) 2 where Assume otherwise.Then there exists

⊕-Implicative Semilattices
To conclude the section, we shall extend the duality for bounded implicative semilattices to the case of the ⊕-implicative semilattices.
Proof.Since U ⊕∅ = R (U ), and R is serial and dense, R ⊆≤, we get that S(X), ∩, R , →, ∅, X is a bounded implicative semilattice with a nucleus R .It is immediate to see that the other conditions of Definition 3.14 are satisfied.By the duality for compact IS-spaces we have that the map H X : X → X(S(X)) is a homeomorphism.In the same way as what was done in Proposition 5.32 we can prove that H X satisfies the condition (x, y) ∈ R iff (H X (x), H X (y)) ∈ R R , for all x, y ∈ X.
Let S 1 and S 2 be two ⊕-implicative semilattices.A ⊕-homomorphism is a bounded implicative homomorphism h : We recall that if a map h : S 1 → S 2 between two implicative semilattices S 1 and S 2 is a bounded implicative homomorphism, then h −1 (F ) is a filter of S 1 for every filter F of S 2 .This fact will be used in the following result.
Proposition 5.45.Let S 1 and S 2 be two ⊕-implicative semilattices.Let h : S 1 → S 2 be a bounded implicative homomorphism.Then the following conditions are equivalent:

r h
Proof.
( Let S 1 and S 2 be two ⊕-implicative semilattices.Let h : S 1 → S 2 be a bounded implicative homomorphism.We shall say that h is a ⊕homomorphism if h(a ⊕ b) = h(a) ⊕ h(b) for all a, b ∈ A. Let IS⊕ be the category whose objects are ⊕-implicative semilattices and whose morphisms are ⊕-homomorphisms.
Let h : S 1 → S 2 be a ⊕-homomorphism.Since h is a bounded implicative semilattice, the relation r h ⊆ X(S 2 ) × X(S 1 ) given by (x, y) ∈ r h iff h −1 (x) ⊆ y, for all (x, y) ∈ X(S 2 )×X(S 1 ) is a functional meet-relation [10].Definition 5.47.Let X i , R i , τ i , for i = 1, 2, be ⊕-spaces.Let X * i = {x ∈ X i : (x, x) ∈ R i }.Let r ⊆ X 1 × X 2 be a relation.We shall say that r is a ⊕-relation if r is a functional meet-relation satisfying the following conditions: (OR2) For each x ∈ X * 1 there exists y ∈ X * 2 such that r(x) = [y).
Propositions 5.45 and 5.46 give us the following corollary.
Corollary 5.48.Let S 1 and S 2 be two ⊕-semilattices.Let h : S 1 → S 2 be a bounded implicative homomorphism.Consider the functional meet-relation r h ⊆ X(S 2 ) × X(S 1 ) defined by (x, y) ∈ r h iff h −1 (x) ⊆ y.Then h is a ⊕homomorphism iff r h is a ⊕-relation.
It is easy to see that the composition * between ⊕-relations given in Definition 5.3 is a ⊕-relation.Thus we have a category ISp⊕ whose objects are ⊕-spaces and whose morphism are ⊕-relations.
Theorem 5.49.The categories IS⊕ and ISp⊕ are dually equivalent.

Concluding Remarks
The present study has been proposed as a contribution towards a topological understanding of a few classes of intuitionistic modal algebras.These structures, as we have seen, arise as subreducts of nuclear Heyting algebras expanded with certain term definable operations whose definitions can be motivated within the study of fragments of quasi-Nelson logic.
We leave the above as suggestions for future research, and we take this opportunity to indicate two further fragments of the quasi-Nelson language (which have not been singled out yet) as potentially interesting ones: (vii) {∨, ⇒ 2 , ∼}, whose corresponding algebras (we anticipate) ought to arise as twist-algebras over nH-semigroups expanded with a join operation; (viii) { * , ∨, ∼}, whose corresponding algebras (we anticipate) ought to arise as twist-algebras over weak implicative semilattices expanded with a join, i.e. (basic) weak Heyting algebras.
We also believe that a deeper universal algebraic study of the varieties considered in the present paper-in particular a classification of their subvarieties-would also deserve further study.
Lastly, we should like to mention that an investigation of the logics associated to intuitionistic modal algebras might also turn out to be worth pursuing.For most of the classes of algebras at hand, the corresponding assertional logic will be algebraizable in the sense of Blok and Pigozzi, and may therefore be easily studied by algebraic means; in other cases, however-consider e.g. the logic of weak implicative semilattices, but also the order-preserving consequence relation associated to any of the above-mentioned varieties-it is not hard to see that the corresponding logic will not be algebraizable.In any case, the topological considerations developed in the present paper may provide a suitable setting for a study of these logics from the point of view of relational semantics.
1 except perhaps (vii).Given elements a, b, c ∈ S, assume a ≤ b c.Then a ∧ b ≤ b ∧ (b c), and item (vi) of Definition 3.1 gives us b ∧ (b c) = b ∧ c ≤ c, from which the result easily follows.Conversely, assume a ∧ b ≤ c.The latter means that a ∧ b = a ∧ b ∧ c.Then b (a ∧ b) = b (a ∧ b ∧ c), and by item (iii) of Definition 3.1 and (I), we have b (v) a b ≤ a.Proof.Items (i), (ii) and (iv) match those of[25, Lemma 4.9].Item (iii) is slightly more general than it appears on [25, Lemma 4.9], namely:a → (b c) = ( a → b) ( a → c).But the two formulations are easily seen to be equivalent by the identities x → y = x → y [25,Lemma 4.4] and x → (y z) = x → (y z), which is a consequence of item (i).Finally, regarding (v), using Definition 3.9 (v) we have (a b) → a = a → ( b → a) = 1.

Proof .
The only item not proved in [27, Lemma 5.5] is the last one.Let a, b, c ∈ S. Since (a → c) ∧ (b → c) ≤ (a → c) ∧ (b → c), it suffices to show that (a → c) ∧ (b → c) ≤ (a ⊕ b) → c.To see this, let us compute:
A similar reasoning allows us to conclude (b ⊕ c) → (a ⊕ c), 1 ∈ θ, so (again by Lemma 4.1) we have a ⊕ c, b ⊕ c ∈ θ.Since the operation ⊕ is commutative, this is sufficient to establish that θ is compatible with ⊕, as required.

Definition 4 . 8 .
A map h : S → S between weak implicative semilattices S and S is a semi-homomorphism if, for all a, b ∈ S,
) = ∅ because no proper implicative filter contains 0. Regarding the latter, we need to check that σ(a b) = σ(a) Q S σ(b) for all a, b ∈ S. Assume x ∈ σ(a b), so a b ∈ x.By Lemma 5.5, this gives us a, b ∈ x.Then, for every y ∈ Q S (x), we have a, b ∈ y, as required.On the other hand, assuming x / ∈ σ(a b), we can again apply Lemma 5.5 to conclude that a / ∈ x or b / ∈ x.Assume the former is the case (a similar reasoning applies if b / ∈ x).Then x / ∈ σ( a) = Q S σ(a).Thus, Q S (x) ⊆ σ(a), which means that there is y ∈ Q S (x) such that a / ∈ y.Hence, x / ∈ σ(a) Q S σ(b), as required.
Let a, b ∈ S such that a, b / ∈ x.As x is irreducible, there exist c ∈ x and there exists d / ∈ x such that a∧c ≤ d and b∧c ≤ d.Then a ≤ c d and b ≤ c d.If c d ∈ x, then c ∧ (c d) ≤ d ∈ x, but as −1 (x) = x, we get that d ∈ x, which is impossible.So, c d / ∈ x, and thus x ∩ S is prime.We recall that in implicative semilattices the notions of prime filter and irreducible filter coincide.Let x ∈ X(S ).We prove that −1 (x) is an irreducible filter of S. Let a, b / ∈ x.Then there exists d / ∈ x such that a ≤ d and b ≤ d.As d = d ∈ S , we get d / ∈ −1 (x).If we take c = 1, we have that a ∧ 1 ≤ d and b ∧ 1 ≤ d, and thus −1 (x) is an irreducible filter of S.

S
and by Proposition 5.23, the set x ∩ S is an irreducible filter of the implicative semilattice S .By Proposition 5.13, x ∩ S is prime.Then there exists c ∈ S such that c / ∈ x ∩ S , a ≤ c and b ≤ c.Taking into account that σ( d) c ∩ X * = σ(d) c ∩ X * , for any d ∈ S, we have that σ

Proposition 5 .
39.  Let S be a ⊕-implicative semilattice.Then R is dense, R is included in the set-theorical inclusion ⊆, and σ(a⊕ b) = R (σ(a) ∪ σ(b)), for any a, b ∈ S.Proof.As the modal operator is a nucleus, we have thatR ⊆ R • R and R is included in ⊆.Let x ∈ σ(a ⊕ b), i.e., a ⊕ b ∈ x.Suppose that (x, y) ∈ R but a, b / ∈ y.As y is irreducible and S is a distributive semilattice, there exists c / ∈ y such that a, b ≤ c.So, a ⊕ b ≤ c ⊕ c = c ∈ x, and so c ∈ y, which is a contradiction.Therefore, a ∈ y or b ∈ y.Then, σ(a ⊕ b) ⊆ R (σ(a) ∪ σ(b)).Let a, b ∈ S and we suppose that a ⊕ b / ∈ x.Consider the order ideal (a ⊕ b].Then it is easy to see that −1 (x) ∩ (a ⊕ b] = ∅.Thus there exists an irreducible filter y such that −1 (x) ⊆ y and a ⊕ b / ∈ y.As a, b ≤ a ⊕ b, we get a, b / ∈ y.So, x / ∈ R (σ(a) ∪ σ(b)), i.e, R (σ(a) ∪ σ(b)) ⊆ σ (a ⊕ b).

2 .
x ∈ Fi * (S), and for all a, b ∈ S, one has a ⊕ b ∈ x iff a ∈ x or b ∈ x.Proof.(1) ⇒ (2).Let a, b ∈ S be such that a ⊕ b ∈ x.By Proposition 5.39 we have x ∈ σ(a ⊕ b) = R (σ(a) ∪ σ(b)), and as x ∈ R (x), we get x ∈ σ(a) ∪ σ(b), i.e, a ∈ x or b ∈ x.The converse is immediate because a, b ≤ a ⊕ b.We note that this condition implies that x is irreducible.