A note on nuclei of quantale modules

JAN PASEKA^{∗}

Department of Mathematics, Masaryk University Janáčkovo nám. 2a 66295 Brno, Czech Republic

            e-mail: paseka@math.muni.cz

Résumé. Le but de cet article est d’établir des théorèmes utiles de factorisation pour des Q-modules de la même manière que pour des locales. On montre qu’un noyau d’un module associé à un prénoyau d’un module est l’intersection de noyaux de modules d’une forme particulière.

Our motivating source was the paper [12] where the investigation of I-simple involutive quantales Q(M) and representations Q → Q(M) given by a left action is developed. Such involutive quantales could play a similar role as do points in topological spaces or irreducible representations in C^{∗}-algebras (see [9]). Since the representations Q → Q(M) coincide with quantale modules we will be interested in studying factorization on quantale modules. This note is closely related to the papers [10] and [11] where the interested reader can find unexplained terms and notation concerning the subject. For facts concerning quantales and quantale modules in general we refer to [14] and [15]. For motivating examples concerning quantale modules we recommend the paper [13].

∗Financial Support of the Grant Agency of the Czech Republic under the grant No. 201/96/0119 is gratefully acknowledged.

The paper is organized as follows. Section 1 introduces the notion of a left Q-module and the motivation for studying such a structure is given. Section 2 contains the basic theorems for factorization on Qmodules and the structure of the complete lattice of all module nuclei is investigated. It is shown that any module nucleus associated to a module prenucleus is a meet of module nuclei of a special form.

A quantale is a complete lattice Q with an associative binary multiplication satisfying

x · ⋁_{i∈I} x_{i} = ⋁_{i∈I} x · x_{i} and (⋁_{i∈I} x_{i}) · x = ⋁_{i∈I} x_{i} · x

for all x, x_{i} ∈ Q, i ∈ I (I is a set). 1 denotes the greatest element of Q, 0 is the smallest element of Q. A quantale Q is said to be unital if there is an element e ∈ Q such that e · a = a = a · e for all a ∈ Q. A subquantale Q′ of a quantale Q is a subset of Q closed under ⋁ and ·.

By a morphism of quantales will be meant a ⋁- and · -preserving mapping f : Q → Q′. If a morphism preserves the top element we say that it is strong.

A non-trivial (0 ≠ 1 and 1 · 1 = 1) quantale Q is said to be simple if any surjective homomorphism Q → Q′ is either an isomorphism or a constant morphism.

By the quantale Q(M) of endomorphisms of the sup-lattice M will be meant the simple unital quantale of sup-preserving mappings from M to itself, with the supremum given by the pointwise ordering of mappings, with the multiplication corresponding to the composition of mappings, and with the unit given by the identity mapping.

1 Quantale modules

Definition 1.1 [1], [15] Let Q be a quantale. A left module over Q (shortly a left Q-module) is a sup-lattice M, together with a module action • : Q × M → M satisfying

\begin{aligned}(a · b) • m & = & a • (b • m) & (M1) \\ (⋁ S) • m & = & ⋁\{s • m : s ∈ S\} & (M2) \\ a • ⋁ X & = & ⋁\{a • x : x ∈ X\} & (M3) \end{aligned}

for all a, b ∈ Q, m ∈ M, S ⊆ Q, X ⊆ M. So we have two maps −→ →_{l} − : M × M → Q, −→ →_{r} − : Q × M → M such that, for all a ∈ Q, m, n ∈ M, a • m ≤ n iff a ≤ m→ →_{l} n iff m ≤ a→ →_{r} n. M is called a unital Q-module if Q is a unital quantale with the unit e and e • m = m for all m ∈ M. M is said to be a strong Q-module if 1 • m = 1 for all m ∈ M, m ≠ 0. Let a ≰ b. Then M separates a from b if there is an element m ∈ M such that a • m ≠ b • m.

Let M and N be modules over Q and let f : M → N be a sup-lattice homomorphism. f is a module homomorphism if f(a • m) = a • f(m) for all a ∈ Q, m ∈ M.

Note first that if M is a sup-lattice then M is a left Q(M)-module such that f • m = f(m) for all f ∈ Q(M) and all m ∈ M. Secondly, we may dually define the notion of a (strong, separating, unital) right Qmodule. Moreover, all propositions stated for left Q-modules are valid in a dualized form for right Q-modules.

We shall denote by Q-Mod (resp. Mod-Q) the category of left Q-modules (resp. right Q-modules).

Let us make the following elementary observation. Evidently, for a quantale Q, any left Q-module M is a unital left Q[e]-module with Q[e] defined as in [11] by

Q[e] = {a ∨ ε : a ∈ Q, ε ∈ {0, e}} ,

e arbitrary such that e ∉ Q, and the multiplication •_{e} defined by

(a ∨ ε) •_{e}m = \{ \begin{matrix}a • m & if ε = 0 \\ a • m ∨ m & if ε = e. \end{matrix}

So we may always assume that any left quantale module is unital over a unital quantale.

Note that, for any quantale Q, any left Q-module M and all n ∈ M, the antitone maps −→ →_{l} n : M → Q and −→ →_{r} n : Q → M form a Galois connection between Q and M since, for all a ∈ Q and all M ∈ M, a ≤ (a→ →_{r} n)→ →_{l} n and m ≤ (m→ →_{l} n)→ →_{r} n. So we have that m→ →_{l} n = ((m→ →_{l} n)→ →_{r} n)→ →_{l} n and a→ →_{r} n = ((a→ →_{r} n)→ →_{l} n)→ →_{r} n.

A simple calculation gives us the following two lemmas.

Lemma 1.2 Let Q be a quantale. If M is a (strong) Q-module then we have a (strong) homomorphism f_{M} : Q → Q(M), f_{M}(a)(x) = a • x. Conversely, let f : Q → Q(M) be a (strong) homomorphism of quantales. Then M is a (strong) Q-module with the module action a •_{f} x = f(a)(x).

Lemma 1.3 Let Q be a quantale, a ≰ b. If M is a Q-module separating a from b then we have a separating homomorphism f_{M} : Q → Q(M). Conversely, let f : Q → Q(M) be a homomorphism separating a from b. Then M is a separating Q-module.

The following theorem (see [6]) is our main motivation for studying quantale modules.

Theorem 1.4 Let Q be a quantale. Then the following conditions are equivalent:

1. Q is spatial i.e. Q is a strong subquantale of a product of simple quantales.

2. Each element of Q is an intersection of primes.

3. Q has enough strong separating left Q-modules.

Proof. (1) ⇔(2) by [6]. (1) ⇔(3) by lemma 1.2 and by lemma 1.3.

2 Factorization on Q-modules

In the following, we shall assume that Q is an arbitrary, fixed quantale.

Definition 2.1 Let M be a left Q-module, j : M → M an operator on M satisfying:

(i)  m ≤ j(m)
(ii)  m ≤ n implies j(m) ≤ j(n)
(iii)  a • j(m) ≤ j(a • m)

for all a ∈ Q, m, n ∈ M. We say that j is a module prenucleus on M. We put M_{j} = {m ∈ M : m = j(m)}. Evidently, M_{j} is a closure system in M, that is, closed under arbitrary meets in M. Note that ⋁_{j} = j ∘ ⋁. We let ν(j) be the associated closure operator, so that

ν(j)(n) = ⋀{m ∈ M_{j} : m ≥ n}.

We say that ν(j) is generated by j. If j satisfies

(iv)  j(m) = j(j(m))

for all m ∈ M we shall say that j is a module nucleus on M. We shall denote by N(M) the lattice of all module nuclei on M.

Note that any module prenucleus (nucleus) on a Q-module M is a module prenucleus (nucleus) on a Q[e]-module M and conversely.

Lemma 2.2 Let j be a module nucleus on a left Q-module M. Then M_{j} is a left Q-module with the join ⋁_{j} and the action •_{j} : Q×M_{j} → M_{j} defined as follows

a •_{j} m = j(a • m)

for all a ∈ Q and all m ∈ M_{j}. Furthermore, M_{j} is unital (strong) if M is.

Proof. Let a, b ∈ Q, m ∈ M_{j}. Then (a · b) •_{j} m = j((a · b) • m) = j(a • (b • m)) = j(a • j(b • m)) = a •_{j} (b •_{j} m). Similarly, let S ⊆ Q, m ∈ M_{j}. Then (⋁ S) •_{j} m = j((⋁ S) • m) = j(⋁{s • m : s ∈ S}) = j(⋁{j(s • m) : s ∈ S}) = ⋁_{j}{s •_{j} m : s ∈ S}. Now, let a ∈ Q, N ⊆ M. Then a •_{j} (⋁_{j} N) = j(a • j(⋁ N)) = j(a • ⋁ N) = j(⋁{a • n : n ∈ N}) = j(⋁{j(a • n) : n ∈ N}) = ⋁_{j}{a •_{j} n : n ∈ N}).

Similarly as for frames (see [2]) we have

Proposition 2.3 Let j be a module prenucleus on a left Q-module M. Then ν(j) is the smallest module nucleus greater than j. Moreover, the module homomorphism ν(j) : M → M_{j} is universal among all module homomorphisms f : M → N for which f(m) = f(j(m)) for all m ∈ M.

Proof. Evidently, ν(j) is inflationary, idempotent and order-preserving. We have to show that, for all a ∈ Q and m ∈ M, a • ν(j)(m) ≤ ν(j)(a • m). We put E_{a,m} = {m′ ∈ M : m ≤ m′ ≤ ν(j)(m), a • m′ ≤ ν(j)(a • m)}. Then m ∈ E_{a,m}, m′ ∈ E_{a,m} implies j(m′) ∈ E_{a,m} – m ≤ m′ ≤ ν(j)(m) implies m ≤ j(m) ≤ j(m′) ≤ j(ν(j)(m)) = ν(j)(m) and a • m′ ≤ ν(j)(a • m) implies a • j(m′) ≤ j(a • m′) ≤ j(ν(j)(a • m)) = ν(j)(a • m). Moreover, for any non-void M′ ⊆ E_{a,m}, ⋁ M′ ∈ E_{a,m} by the left distribution law of modules. Hence m_{0} = ⋁ E_{a,m} ∈ E_{a,m}. This gives us j(m_{0}) ≤ m_{0} ≤ j(m_{0}) i.e. m ≤ m_{0} ≤ ν(j)(m) ≤ m_{0}. So we have a • ν(j)(m) ≤ a • m_{0} ≤ ν(j)(a • m).

Let us prove the second part of the lemma. Note that ν(j)(j(m)) = ν(j)(m) for all m ∈ M. Let f : M → N be a module homomorphism such that f(m) = f(j(m)) for all m ∈ M. We have to show that f(ν(j)(m)) = f(m) for all m ∈ M. We put F_{m} = {m′ ∈ M : m ≤ m′ ≤ ν(j)(m), f(m′) = f(m)}. Then m ∈ F_{m}, m′ ∈ F_{m} implies j(m′) ∈ F_{m} by the condition on f and, for all non-void F′ ⊆ F_{m}, ⋁ F′ ∈ F_{m} since f is a join-preserving. We put m_{1} = ⋁ F_{m}. Then j(m_{1}) ≤ m_{1} ≤ j(m_{1}) i.e. m ≤ m_{1} ≤ ν(j)(m) ≤ ν(j)(m_{1}) = m_{1} i.e. f(m) ≤ f(ν(j)(m)) ≤ f(m_{1}) = f(m). We then have a uniquely determined map f̂ : M_{j} → N defined by f̂(m) = f(m) for all m ∈ M_{j} i.e. f̂ ∘ j = f. Evidently, for all a ∈ Q, m ∈ M_{j} and S ⊆ M_{j}, we have f̂(⋁_{M_{j}} S) = f̂(j(⋁ S)) = f(⋁ S) = ⋁_{s∈S} f(s) = ⋁_{s∈S} f̂(j(s)) = ⋁_{s∈S} f̂(s) and f̂(a •_{M_{j}}m) = f̂(j(a • m)) = f(a • m) = a • f(m) = a • f̂(j(m)) = a • f̂(m) i.e. f̂ is a module homomorphism.

Another way of obtaining ν(j) is to iterate the prenucleus j until it converges by the following procedure:

\begin{aligned}j^{0}(m) & = & m, \\ j^{σ+1}(m) & = & j(j^{σ}(m)) \\ j^{λ}(m) & = & ⋁\{j^{σ}(m) : σ < λ\} \end{aligned}

for all m ∈ M, all ordinals σ and all limit ordinals λ. Then each j^{σ} is a module prenucleus with the same fixpoints as j and, for sufficiently large ordinal σ, j^{σ} has to be idempotent.

Following Macnab (see [8]) for Heyting algebras we have

Theorem 2.4 Let Q be a unital quantale, M a unital left Q-module, j : M → M an operator on M. The following conditions are equivalent:

1. j is a module nucleus on M.

2. For all m, n in M, m→ →_{l} j(n) = j(m)→ →_{l} j(n).

Proof. 1⟹2. We have, for all a ∈ Q, a ≤ m→ →_{l} j(n) iff a • m ≤ j(n) iff j(a • m) ≤ j(n) iff a • j(m) ≤ j(n) iff a ≤ j(m)→ →_{l} j(n) i.e. the condition 2 holds.

2⟹1. We have, for all m ∈ M, j(m) ≤ j(m) i.e. e • j(m) ≤ j(m) i.e. e ≤ j(m)→ →_{l} j(m) i.e. e ≤ m→ →_{l} j(m) i.e. e • m ≤ j(m) i.e. m ≤ j(m) i.e. j is inflationary.

Now, let m, n ∈ M, m ≤ n. Then m ≤ n ≤ j(n) i.e. e • m ≤ j(n) i.e. e ≤ m→ →_{l} j(n) = j(m)→ →_{l} j(n) i.e. e • j(m) ≤ j(n) i.e. j(m) ≤ j(n).

Let a ∈ Q, m ∈ M. We have a • m ≤ j(a • m) i.e. a ≤ m→ →_{l} j(a • m) i.e. a ≤ j(m)→ →_{l} j(a • m) i.e. a • j(m) ≤ j(a • m).

Let us show that j is idempotent. Assume m ∈ M. We have j(m) ≤ j(m) i.e. e • j(m) ≤ j(m) i.e. e ≤ j(m)→ →_{l} j(m) i.e. e ≤ j(j(m))→ →_{l} j(m) i.e. e • j(j(m)) ≤ j(m) i.e. j(j(m)) ≤ j(m) ≤ j(j(m)) i.e. j(j(m)) = j(m). So it follows that j is a module nucleus on M.

Proposition 2.5 Let M be a left Q-module, n ∈ M. Then the map w_{n} : M → M defined by w_{n}(m) = (m→ →_{l} n)→ →_{r} n is a module nucleus. Similarly, the map W_{n} : Q → Q defined by W_{n}(a) = (a→ →_{r} n)→ →_{l} n is a right module nucleus on Q, seen as a right module over itself. Moreover, for all a ∈ Q and m ∈ M, we have a • m ≤ n iff W_{n}(a) • w_{n}(m) ≤ n.

Proof. Clearly, the Galois connection makes m ≤ w_{n}(m) = w_{n}(w_{n}(m)) and m_{1} ≤ m_{2} implies w_{n}(m_{1}) ≤ w_{n}(m_{2}) for all m, m_{1}, m_{2} ∈ M. Let a ∈ Q, m ∈ M. Then y ≤ a • m→ →_{l} n iff (y · a) • m = y • (a • m) ≤ n iff (y · a) ≤ m→ →_{l} n iff (y · a)→ →_{r} n ≥ w_{n}(m) iff (y · a) • w_{n}(m) ≤ n iff y • (a • w_{n}(m)) ≤ n iff y ≤ (a • w_{n}(m))→ →_{l} n. Hence a • m→ →_{l} n = (a • w_{n}(m))→ →_{l} n, and thus w_{n}(a • m) = w_{n}(a • w_{n}(m)), i.e. w_{n}(a • m) ≥ a • w_{n}(m). To prove the second part of the lemma note that Q is a right module over itself. By similar arguments as above, we have that W_{n} is a right module nucleus.

Finally, we show that a • m ≤ n ⟺ W_{n}(a) • w_{n}(m) ≤ n: a • m ≤ n iff a ≤ m→ →_{l} n iff, by the Galois connection, a ≤ w_{n}(m)→ →_{l} n iff a • w_{n}(m) ≤ n iff, by similar steps using → →_{r}, W_{n}(a) • w_{n}(m) ≤ n.

Lemma 2.6 Let Q be a quantale, j_{α}, α ∈ Λ module prenuclei (nuclei) on a left Q-module M. Then the operator ⋀_{α∈Λ} j_{α} is a module prenucleus (nucleus).

Proof. Evidently, for all m, n ∈ M, a ∈ Q, we have m ≤ ⋀_{α∈Λ} j_{α}(m) = (⋀_{α∈Λ} j_{α}) (m), m ≤ n implies j_{α}(m) ≤ j_{α}(n) i.e. (⋀_{α∈Λ} j_{α}) (m) ≤ (⋀_{α∈Λ} j_{α}) (n) and a • (⋀_{α∈Λ} j_{α}) (m) ≤ a • j_{β}(m) ≤ j_{β}(a • m) for all β ∈ Λ i.e. a • (⋀_{α∈Λ} j_{α}) (m) ≤ (⋀_{α∈Λ} j_{α}) (a • m). Let j_{α}, α ∈ Λ be module nuclei. Then ⋀_{α∈Λ} j_{α}(⋀_{β∈Λ} j_{β}(m)) ≤ j_{γ}(j_{γ}(m)) = j_{γ}(m) for all γ ∈ Λ and all m ∈ M i.e. ⋀_{α∈Λ} j_{α} is idempotent.

We have then that, for any set Λ, ⋀{j_{λ} ∈ N(M) : λ ∈ Λ} (with ⋀ defined pointwise), exists and it is a module nucleus i.e. N(M) is a complete lattice. Because the meet in N(M) is pointwise so also is the order on it. We shall denote by ⨆ the join in N(M) and by ⋁ the pointwise join, which in general is only a prenucleus. Let j_{1}, j_{2} ∈ N(M).

Then j_{1} ∘ j_{2}, j_{2} ∘ j_{1} ≤ j_{1} ⊔ j_{2}. Note that j_{1} ∘ j_{2} is a module prenucleus. So it follows that if j_{1} ∘ j_{2} is idempotent then j_{1} ∘ j_{2} = j_{1} ⊔ j_{2}. If j_{1} ∘ j_{2} ≤ j_{2} ∘ j_{1} then j_{2} ∘ j_{1} is idempotent i.e. a module nucleus.

Recall the following notions from [3]. Let Q be a semigroup, a ∈ Q. We then put Q⟨a⟩ = {a^{k} : k ∈ N}. We say that Q⟨a⟩ is a cyclic subsemigroup of Q generated by the element a. For any a ∈ Q, we have either that Q⟨a⟩ is infinite or that Q⟨a⟩ is finite i.e. there exist k_{1}, k_{2} ∈ N minimal such that a^{k_{1}+k_{2}} = a^{k_{2}}; here k_{1} = p(a) is the period of a and k_{2} = i(a) the index of a. We say that an element a ∈ Q is periodic if Q⟨a⟩ is finite. We shall denote by Q_{P} = {a ∈ Q : a is periodic} and Q_{E} = {a ∈ Q : a is idempotent}. If Q = Q_{P} (Q = Q_{E}) we say that Q is periodic (idempotent).

Now we generalise the nuclei w_{n} of Proposition 2.5.

Lemma 2.7 Let Q be a unital quantale, j a module prenucleus on a unital left Q-module M, k ∈ N, n ∈ M, z ≤ j(n)→ →_{l} n. Then the operator ν_{z,n}^{k} : M → M defined by the prescription

ν_{z,n}^{k}(m) = (z^{k}(m→ →_{l} n)) → →_{r} n

is a module prenucleus on M greater than j. Especially, the operator ν_{n}^{k}(j) = ν_{j(n)→ →_{l} n,n}^{k} is a module prenucleus on M greater than j. Moreover, if z ∈ Q_{P} and k ≥ i(z) we have that ν_{z,n}^{k} is a module nucleus greater than j.

Proof. We first show that, for all m ∈ M, j(m) ≤ ν_{z,n}^{k}(m) – we have then m ≤ ν_{z,n}^{k}(m). It suffices to show that

(z^{k}(m→ →_{l} n)) • j(m) ≤ n.

We have

(z^{k}(m→ →_{l} n)) • j(m) = z^{k} • ((m→ →_{l} n) • j(m))
≤ z^{k} • j ((m→ →_{l} n) • m)
≤ z^{k} • j(n)
≤ n,

where the last inequality follows by induction on k:

- z • j(n) ≤ (j(n)→ →_{l} n) • j(n) ≤ n,

- z^{k+1} • j(n) = z • (z^{k} • j(n)) ≤ z • n ≤ z • j(n) ≤ n.

Note that the function (z^{k}(−→ →_{l} n)) → →_{r} n is order-preserving. Now, let us show that a • ν_{z,n}^{k}(m) ≤ ν_{z,n}^{k}(a • m) for all a ∈ Q and m ∈ M, that is, we check that a • [(z^{k}(m→ →_{l} n)) → →_{r} n] ≤ (z^{k}((a • m)→ →_{l} n)) → →_{r} n. Evidently, (a • m→ →_{l} n) · a ≤ m→ →_{l} n. We have

\begin{aligned}[(z^{k}((a • m)→ →_{l} n)) · a] • [(z^{k}(m→ →_{l} n)) → →_{r} n] & ≤ \\ (z^{k}(m→ →_{l} n)) • [(z^{k}(m→ →_{l} n)) → →_{r} n] & ≤ & n. \end{aligned}

So we have shown that ν_{z,n}^{k} is a module prenucleus.

Now assume that z ∈ Q_{P} and k ≥ i(z). We show that ν_{z,n}^{k}(ν_{z,n}^{k}(m)) ≤ ν_{z,n}^{k}(m). This is equivalent to the following inequality (⋆)

(z^{k}(m→ →_{l} n)) • (z^{k}((z^{k}(m→ →_{l} n)) → →_{r} n→ →_{l} n)) → →_{r} n ≤ n.

Since −→ →_{l} n and −→ →_{r} n form a Galois connection and z is periodic the inequality (⋆) follows from the inequality (⋆⋆_{i}), i ∈ N

\begin{matrix}[z^{ip(z)} · ((z^{k}(m→ →_{l} n)) → →_{r} n→ →_{l} n)] & • \\ • (z^{k}((z^{k}(m→ →_{l} n)) → →_{r} n→ →_{l} n)) → →_{r} n ≤ n. \end{matrix}

Let us prove (⋆⋆_{i}) implies (⋆). We have

(z^{k}(m→ →_{l} n)) • (z^{k}((z^{k}(m→ →_{l} n)) → →_{r} n→ →_{l} n)) → →_{r} n =
[z^{ip(z)} · (z^{k}(m→ →_{l} n))] • (z^{k}((z^{k}(m→ →_{l} n)) → →_{r} n→ →_{l} n)) → →_{r} n =
z^{ip(z)} • ((z^{k}(m→ →_{l} n)) • (z^{k}((z^{k}(m→ →_{l} n)) → →_{r} n→ →_{l} n)) → →_{r} n) ≤
z^{ip(z)} • (→ →_{l} n) • (z^{k}((z^{k}(m→ →_{l} n)) → →_{r} n→ →_{l} n)) → →_{r} n) ≤
z^{ip(z)} • n ≤ n.

(in fact, we also have (⋆) implies (⋆⋆_{i}) since we obtain by (⋆) and the Galois connection that

((z^{k}(m→ →_{l} n)) → →_{r} n→ →_{l} n) • (z^{k}((z^{k}(m→ →_{l} n)) → →_{r} n→ →_{l} n)) → →_{r} n ≤ n.

Then

[z^{ip(z)} · ((z^{k}(m→ →_{l} n)) → →_{r} n→ →_{l} n)] •
• (z^{k}((z^{k}(m→ →_{l} n)) → →_{r} n→ →_{l} n)) → →_{r} n =
= z^{ip(z)} • (((z^{k}(m→ →_{l} n)) → →_{r} n→ →_{l} n) •
• (z^{k}((z^{k}(m→ →_{l} n)) → →_{r} n→ →_{l} n)) → →_{r} n ) ≤ z^{ip(z)} • n ≤ n.)

Now, for a large enough i, such that ip(z) = i_{0} + k for some i_{0} ∈ N, the condition (⋆⋆_{i}) becomes equivalent to

\begin{matrix}z^{i_{0}} • [z^{k} ((z^{k}(m→ →_{l} n)) → →_{r} n→ →_{l} n) • \\ • (z^{k}((z^{k}(m→ →_{l} n)) → →_{r} n→ →_{l} n)) → →_{r} n] ≤ n. \end{matrix}

The last inequality evidently holds since z^{i_{0}} • n ≤ n and

z^{k} ((z^{k}(m→ →_{l} n)) → →_{r} n→ →_{l} n) • (z^{k}((z^{k}(m→ →_{l} n)) → →_{r} n→ →_{l} n)) → →_{r} n ≤ n.

We have then that ν_{z,n}^{k} is a module nucleus.

Johnstone gives in ([4]) an explicit formula for ν(j) for frames. The following proposition is a slight modification of it for Q-modules.

Proposition 2.8 Let Q be a unital periodic quantale, j a module prenucleus on a unital left Q-module M. Then the operator ν_{p}(j) : M → M defined by the prescription

ν_{p}(j)(m) = ⋀{((j(n)→ →_{l} n)^{i(j(n)→ →_{l} n)}(m→ →_{l} n)) → →_{r} n : n ∈ M}

is the associated module nucleus ν(j).

Proof. Note that ν_{p}(j) = ⋀_{n∈M}{ν_{n}^{i(j(n)→ →_{l} n)}(j)} i.e. ν_{p}(j) is a module nucleus greater than j. Let j(m) = m. We put

m′ = ((m→ →_{l} m)^{i(m→ →_{l} m)}(m→ →_{l} m)) → →_{r} m, k = i(m→ →_{l} m).

We then have ν_{p}(j)(m) ≤ m′ and

m′ = (e · e) • m′ ≤ (e · (m→ →_{l} m)) • m′
≤ ((m→ →_{l} m)^{k}(m→ →_{l} m)) • m′
≤ m

i.e. ν_{p}(j)(m) = j(m). Hence, ν_{p}(j) has the same fixed points as j i.e. ν_{p}(j) is the least module nucleus greater than j.

Corollary 2.9 Let Q be a unital idempotent quantale, j a module prenucleus on a unital left Q-module M. Then the operator ν_{e}(j) : M → M defined by the prescription

ν_{e}(j)(m) = ⋀{((j(n)→ →_{l} n)(m→ →_{l} n)) → →_{r} n : n ∈ M}

is the associated module nucleus ν(j).

Theorem 2.10 Let Q be a unital periodic quantale, M a unital left Qmodule. Then, for any set Λ, ⨆{j_{λ} ∈ N(M) : λ ∈ Λ} = ^{⨆↑}{ν_{p}(j_{λ_{1}} ∘ … ∘ j_{λ_{s}}) : {λ_{1}, …, λ_{s}} ⊆ Λ} = ν_{p} (⋁{j_{λ} ∈ N(M) : λ ∈ Λ}).

Proof. The first equality follows from the proposition 2.8 and from the fact that a directed join of module nuclei is computed pointwise. The second one follows from an easy observation that the pointwise join of module prenuclei is a module prenucleus.

Corollary 2.11 Let Q be a unital idempotent quantale, M a unital left Q-module. Then, for any set Λ, ⨆{j_{λ} ∈ N(M) : λ ∈ Λ} = ^{⨆↑}{ν_{e}(j_{λ_{1}} ∘ … ∘ j_{λ_{s}}) : {λ_{1}, …, λ_{s}} ⊆ Λ} = ν_{e} (⋁{j_{λ} ∈ N(M) : λ ∈ Λ}).

Proposition 2.12 Let Q be a unital quantale, j a module prenucleus on a unital left Q-module M. Then the operators ν_{P}(j), ν_{E}(j) : M → M defined by the prescriptions

\begin{aligned}ν_{P}(j) & = & ⋀\{ν_{i(z)}^{z,n}(j) : n ∈ M, z ∈ Q_{P}, z ≤ j(n)→ →_{l} n\} \\ ν_{E}(j) & = & ⋀\{ν_{1}^{z,n}(j) : n ∈ M, z ∈ Q_{E}, z ≤ j(n)→ →_{l} n\} \end{aligned}

coincide with the associated nucleus ν(j).

Proof. By 2.6 and 2.7 we know that both operators are module nuclei on M greater than j.

Now, let j(m) = m. Then e ∈ Q_{E} ⊆ Q_{P} and e ≤ m→ →_{l} m. We then have

j(m) ≤ ν_{P}(j)(m) ≤ ν_{E}(j)(m) ≤ (e · (m→ →_{l} m)) → →_{r} m
= e • [(m→ →_{l} m) → →_{r} m]
≤ (m→ →_{l} m) • [(m→ →_{l} m) → →_{r} m]
≤ m = j(m)

i.e. ν_{P}(j)(m) = ν_{E}(j)(m) = j(m).

Since both ν_{E}(j) and ν_{E}(j) have the same fixed points as j we have

ν_{E}(j) = ν(j) = ν_{P}(j).

Theorem 2.13 Let Q be a unital quantale, M a unital left Q-module. Then, for any set Λ, ⨆{j_{λ} ∈ N(M) : λ ∈ Λ} = ^{⨆↑}{ν_{P}(j_{λ_{1}} ∘ … ∘ j_{λ_{s}}) : {λ_{1}, …, λ_{s}} ⊆ Λ} = ^{⨆↑}{ν_{E}(j_{λ_{1}} ∘ … ∘ j_{λ_{s}}) : {λ_{1}, …, λ_{s}} ⊆ Λ} = ν_{P} (⋁{j_{λ} ∈ N(M) : λ ∈ Λ}) = ν_{E} (⋁{j_{λ} ∈ N(M) : λ ∈ Λ}).

Proof. Similar to the proof of theorem 2.10, with ν_{p} replaced by ν_{P} and ν_{E} and using proposition 2.12 instead of proposition 2.8.

Another way how to construct new module nuclei is the method of R-saturated elements introduced for frames by Kříž (see [7]).

Definition 2.14 Let M be a left Q-module, R ⊆ M × M a relation on M. We shall say that an element m ∈ M is R-saturated if

a • n_{1} ≤ m iff a • n_{2} ≤ m and n_{1} ≤ m iff n_{2} ≤ m

holds for all a ∈ Q, n_{1}, n_{2} ∈ M such that (n_{1}, n_{2}) ∈ R. We shall denote by M_{R} the set of all R-saturated elements.

Note that M_{R} coincides with the sup-lattice quotient of M by the relation R′ = R ∪ {(a • m_{1}, a • m_{2}) : a ∈ Q, (m_{1}, m_{2}) ∈ R} [5]. Equivalently the map j_{R} : M → M defined by j_{R}(m) = ⋀{n ∈ M_{R} : m ≤ n} is a closure operator on M, M_{j_{R}} = M_{R}, and the sup-lattice homomorphism j_{R} : M → M_{R} is universal amongst all sup-lattice homomorphisms f : M → N for which f(m_{1}) = f(m_{2}) for all (m_{1}, m_{2}) ∈ R′.

Lemma 2.15 Let a ∈ Q, m ∈ M and m is R-saturated. Then the element a→ →_{r} m = ⋁{n ∈ M : a • n ≤ m} is R-saturated.

Proof. Let n_{1}, n_{2} ∈ M, (n_{1}, n_{2}) ∈ R, c ∈ Q. Then c • n_{1} ≤ a→ →_{r} m iff a • (c • n_{1}) ≤ m iff (a · c) • n_{1} ≤ m iff (a · c) • n_{2} ≤ m iff a • (c • n_{2}) ≤ m iff c • n_{2} ≤ a→ →_{r} m. Similarly, n_{1} ≤ a→ →_{r} m iff n_{2} ≤ a→ →_{r} m.

Theorem 2.16 Let M be a left Q-module, R ⊆ M × M. Then the closure operator j_{R} is a module nucleus, and the module homomorphism j_{R} : M → M_{R} is universal amongst all module homomorphisms f : M → N for which f(m_{1}) = f(m_{2}) for all (m_{1}, m_{2}) ∈ R.

Proof. Assume a ∈ Q, m ∈ M. Then a • m ≤ j_{R}(a • m) i.e. m ≤ a→ →_{r} j_{R}(a • m) i.e. j_{R}(m) ≤ a→ →_{r} j_{R}(a • m) i.e. a • j_{R}(m) ≤ j_{R}(a • m). Hence, j_{R} is a module nucleus.

Now, let f : M → N be a module homomorphism such that f(m_{1}) = f(m_{2}) for all (m_{1}, m_{2}) ∈ R, m ∈ M. Then f factors uniquely through j_{R} and a sup-lattice homomorphism f′ : M_{R} → N, because (m_{1}, m_{2}) ∈ R′ implies f(m_{1}) = f(m_{2}). Let us prove that f′ preserves the action. Denoting by •_{j_{R}} the action on M_{R} and by •_{N} the action on N we have f′(a •_{j_{R}} j_{R}(m)) = f′(j_{R}(a • j_{R}(m))) = f(a • j_{R}(m)) = a •_{N} f(j_{R}(m)) = a •_{N} f′(m).

Acknowledgement

I want to thank here the anonymous referee for many valuable comments that allow to highly improve the final version of this paper.

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