Injective hulls of many-sorted ordered algebras

Lemma 1

For a class of ordered Σ-algebras, σ ⊆ σ^⩽.

Proof

It is clear that a morphism h from σ is a lax morphism. Assume that tBs (h_s1(a_s1), …, h_sm(a_sm)) ⩽ h_s(a). Then h_s( tAs (a_s1, …, a_sm)) ⩽ h_s(a) since h is a homomorphism, and hence tAs (a_s1, …, a_sm) ⩽ a by the fact that h is an order-embedding.□

Theorem 2

Every sup-Σ-algebra is σ^⩽-injective and therefore σ-injective in the category OALΣ⩽ .

Proof

Let Q be a sup-Σ-algebra. It is enough to show that Q is σ^⩽-injective in OALΣ⩽ . Suppose that h : A → B is a lax order-embedding and f : A → Q is a morphism in OALΣ⩽ . We have to find a lax morphism g: B → Q such that the following diagram commutes.

Let g : B → Q be an S-indexed family of mappings g_s : B_s → Q_s, s ∈ S, which is defined by

for any b_s ∈ B_s. Let us write the above join shortly as

It is clear that g_s is monotone. Let us show that g is a lax morphism, that is, the inequality

holds for any o ∈ O with rank s₁ ⋯ s_n → s, n ∈ ℕ, b_si ∈ B_si, i = 1, …, n. Since Q is a sup-Σ-algebra whose operations are sup-lattice homomorphisms in each variable separately, it turns out that

Note that the inequality holds because the inequalities

imply that

and

is a Σ-term of sort s.

To complete the proof, let us show that g_sh_s = f_s for any s ∈ S. Take a_s ∈ A_s, by the fact that

and

we obtain that tAs (a_s1, …, a_sm) ⩽ a_s since h ∈ σ^⩽. This indicates that

Therefore g_sh_s(a_s) ⩽ f_s(a_s). On the other hand, using the term t^s = x_s, we see that f_s(a_s) is in the join that defines g_sh_s(a_s), consequently g_sh_s = f_s as required.□

Lemma 3

Let A be an ordered Σ-algebra. Then for any s ∈ S, n ∈ ℕ, o ∈ O with rank s₁ ⋯ s_n → s, and a_si ∈ A_si, i = 1, …, n, we have

and therefore

for any t^s ∈ TΣS .

Proposition 4

Let A be an ordered Σ-algebra. Then 𝒫(A) is σ^⩽-injective and σ-injective in the category OALΣ⩽ , which A can be embedded in.

Proof

Given an ordered Σ-algebra A, the injectivity of 𝒫(A) follows by the fact that 𝒫(A) is a sup-Σ-algebra. Let us consider η : A → 𝒫(A), which is an S-indexed family of mappings η_s : A_s → 𝒫(A)_s, s ∈ S, where η_s(a_s) = a_s↓, for any a_s ∈ A_s. It is clear that for each sort s, η_s is an order-embedding of the poset (A_s, ⩽_As) into the poset (𝒫(A)_s, ⊆). Moreover, for any s ∈ S, o ∈ O with rank s₁ ⋯ s_n → s, n ∈ ℕ and a_si ∈ A_si, i = 1, …, n, Lemma 3 gives that

So η : A → 𝒫(A) is a homomorphism of ordered Σ-algebras, thereby η belongs to both σ and σ^⩽.□

Lemma 5

In the category OALΣ⩽ , every retract of a sup-Σ-algebra is a sup-Σ-algebra.

Proof

Let Q be a sup-Σ-algebra and let A be a retract of Q. Then there exist lax morphisms ι : A → Q and f : Q → A, where ι and f are relative S-indexed family of mappings ι_s : A_s → Q_s and f_s : Q_s → A_s respectively, fulfilling f_sg_s = id_As for all s ∈ S, where id_As is the identity mapping on A_s. Clearly, A_s is a sup-lattice for all s ∈ S.

For any o ∈ O with rank s₁ ⋯ s_n → s, n ∈ ℕ and M ⊆ A_si, i ∈ {1, …, n}, let us show that

for a_sk ∈ A_sk, k = {1, …, n} ∖ {i}. For this purpose, assume that u is an upper bound of the set of all elements o_A(a_s1, …, m, …, a_sn), m ∈ M. Then ι_s(u) ⩾ ⋁m∈M ι_s(o_A(a_s1, …, m, …, a_sn)). Hence we have

□

Theorem 6

For an ordered Σ-algebra A, the following statements are equivalent:

1. A is σ^⩽-injective in OALΣ⩽ ,

2. A is σ-injective in OALΣ⩽ ,

3. A is a sup-Σ-algebra.

Proof

(1) ⇒ (2) follows from Lemma 1, and (3) ⇒ (1) follows from Theorem 2.

(2) ⇒ (3) holds by the reason that η : A → 𝒫(A) given in Proposition 4 has a left inverse, and thus A is a retract of 𝒫(A). Therefore A is a sup-Σ-algebra by Lemma 5.□

Example 7

posemigroup (S, ⊗, ⩽) is a semigroup (S, ⊗) equipped with a partial ordering ⩽ which is compatible with the semigroup multiplication, that is, s ⊗ s′ ⩽ t ⊗ t′ whenever s ⩽ t, s′ ⩽ t′ in S.

A mapping f : S → T between posemigroups (S, ⊗, ⩽) and (T, ⊗, ⩽)is said to be submultiplicative if f(a) ⊗ f(a′) ⩽ f(a ⊗ a′) for all a, a′ ∈ S.

A quantale (cf. [17]) is a posemigroup (S, ⊗, ⩽) such that

1. the poset (S, ⩽) is a complete lattice,

2. s ⊗(⋁ M) = ⋁{s ⊗ m | m ∈ M} and (⋁ M) ⊗ s = ⋁{m ⊗ s | m ∈ M} for each subset M of S and each s ∈ S.

Then quantales are injectives in the category PoSgr_⩽ where objects are posemigroups and morphisms are submultiplicative order-preserving mappings ([7]).

Example 8

Let (S, ⊗, ⩽) be a posemigroup. A poset (A, ⩽) together with a mapping S × A → A (under which a pair (s, a) maps to an element of A denoted by s ∗ a) is called an S-poset, denoted by _SA, if for any a, b ∈ A, s, t ∈ S,

1. (s ⊗ t) ∗ a = s ∗(t∗ a),

2. a ⩽ b, s ⩽ t ⇒ s ∗ a ⩽ t ∗ b.

If S is a pomonoid, then _SA fulfils

An S-poset A_S is called an S-quantale if

1. the poset A is a complete lattice,

2. s*(⋁ M) = ⋁{s * m | m ∈ M} for each subset M of A and each s ∈ S.

A mapping f: _SA → _SB of S-posets is said to be S-submultiplicative if s * f(a) ⩽ f(s * a) for any a ∈ A, s ∈ S. Let S be a pomonoid, PosS⩽ be the category where objects are right S-posets and morphisms are S-submultiplicative order-preserving mappings, 𝓔_⩽ be the class of morphisms e : _SA → _SB in the category PosS⩽ which satisfy the following condition : s * e(a) ⩽ e(a′) implies s * a ⩽ a′ for all a, a′ ∈ A and s ∈ S. Evidently, each morphism in 𝓔_⩽ is an order-embedding. Then 𝓔_⩽-injectives in the category PosS⩽ are exactly S-quantales ([8]).

Example 9

Let (S, ⊗, ⩽) be a posemigroup. A posemigroup (A, ⋅, ⩽), which is also an S-poset, where the action is denoted by ∗, is called an S-semigroup, if for any a, b ∈ A, s, t ∈ S,

An S-semigroup quantale is an S-semigroup (_SA, ⋅, ∗) such that (A, ⋅, ⋁) is a quantale and (_SA, ∗) is an S-quantale.

An order-preserving mapping f: _SA → _SB of S-semigroups is called a subhomomorphism if it is both submultiplicative in posemigroups, i.e.,

for all a₁, a₂ ∈ A, and S-submultiplicative in S-posets, i.e.,

for all a ∈ A, s ∈ S. The category of S-semigroups with subhomomorphisms as morphisms is denoted by Ssgr_⩽.

Let 𝓔_⩽ be the class of morphisms e : _SA → _SB in the category Ssgr_⩽ which satisfy the following conditions:

and

for all a₁, a₂, …, a_n, a ∈ A, s ∈ S. Then 𝓔_⩽-injectives in the category Ssgr_⩽ are indeed S-semigroup quantales ([13]).

Example 10

Let us denote by Pos the category of posets and order-preserving mappings and Sup the category of sup-lattices and sup-preserving mappings. An ordered semicategory (category) is a locally small semicategory (category) such that hom-sets are partially ordered and composition on both sides is order-preserving.

A lax semifunctor F: 𝓒 → 𝓓 of ordered categories is given by functions

for all X ∈ ob 𝓒 (with F_X,Y usually written only as F), such that

1. F_X,Y is monotone;

2. Fg ∘ Ff ⩽ F(g ∘ f),

   for all X, Y, Z ∈ ob 𝓒, f: X → Y, g: Y → Z in 𝓒. A lax semifunctor F: 𝓒 → 𝓓 is a semifunctor if (2 =) Fg ∘ Ff = F(g ∘ f),

   for all X, Y, Z ∈ ob 𝓒, f: X → Y, g: Y → Z in 𝓒.

Let 𝓢 be a small ordered semicategory. An 𝓢-module is a semifunctor A: 𝓢 → Pos of ordered semicategories.

An 𝓢-morphism is a lax natural map between 𝓢-modules A and B, that is, a family α = {α_X ∈ Pos(AX, BX) | X ∈ 𝓢} of order-preserving mappings such that for every f: X → Y in 𝓢, we have B(f) ∘ α_X ⩽ α_Y ∘ A(f).

An 𝓢-𝓠-module is an 𝓢-module A such that for every two objects X, Y ∈ 𝓢 and for every f: X → Y we have that AX and AY are sup-lattices and A(f) is a sup-preserving mapping, i.e., A yields a semifunctor of ordered semicategories into Sup.

The category 𝓢 – Mod of 𝓢-modules has 𝓢-modules as objects and lax natural maps as morphisms.

An order embedding ε between 𝓢-modules A and B is an 𝓢-morphism ε: A → B such that ε_X : AX → BX is an order embedding in posets for all X ∈ 𝓢. Let 𝓔_⩽𝓢 be the class of order embeddings ε : A ⟶ B in the category 𝓢-Mod which satisfy the following conditions:

for all a ∈ A(X), b ∈ A(Y) and all f: X → Y in 𝓢. Then 𝓢-𝓠-modules are 𝓔_⩽𝓢-injectives in the category 𝓢-Mod of 𝓢-modules ([13]).

Proposition 11

Let A be an ordered Σ-algebra, j: 𝒫(A) → 𝒫(A) be an S-indexed family of mappings j_s defined as in (4.1). Then j is a nucleus on 𝒫(A) satisfying j_s(a_s↓) = (a_s↓) for any a_s ∈ A_s, s ∈ S.

Proof

Let us first show that for any s ∈ S, j_s is a closure operator on 𝒫(A)_s. It is obvious that j_s is increasing and monotone, in particular j_s(D_s) ⊆ j_s(j_s(D_s)) for every D_s ∈ 𝒫(A)_s. Take u ∈ j_s(j_s(D_s)) and assume that p^s͠(D_s) ⊆ a↓ for some a ∈ A_s͠ and p^s͠ ∈ PAs1 , s͠ ∈ S. Since for any v ∈ j_s(D_s), we have p(v) ⩽ a, it follows that p^s͠(j_s(D_s)) ⊆ a↓. Thus p^s͠(u) ⩽ a and u ∈ j_s(D_s). Consequently, j_s(j_s(D_s)) = j_s(D_s) as needed.

We next show that j is a lax morphism on 𝒫(A), namely, we need to prove o_𝒫(A)(j_s1(D_s1), …, j_sn(D_sn)) ⊆ j_s(o_𝒫(A)(D_s1, …, D_sn)) for any s ∈ S, o ∈ O with rank s₁ ⋯ s_n → s, n ∈ ℕ and D_si ∈ 𝒫(A)_si, i = 1, …, n. It is sufficient to show that the inclusion

is satisfied, that is, the inclusion

holds. For this aim, take z ∈ A_s such that z ⩽ o_A(d_s1, …, d_si–1, d_si, d_si+1, …, d_sn), where d_sj ∈ D_sj, j ∈ {1, …, n} ∖ {i}, and d_si ∈ j_si(D_si). Then z ⩽ o_si(d_si). Suppose that for a unary polynomial function p^s͠ over A_s, p^s͠(o_A(D_s1, …, D_sn)↓) ⊆ a↓ for some a ∈ A_s͠, then we have

Since p^s͠ o_si ∈ PAsi1 , we obtain p^s͠ o_si(d_si) ⩽ a because d_si ∈ j_si(D_si). Therefore, p^s͠(z) ⩽ p^s͠(o_si(d_si)) ⩽ a. We have shown that z ∈ j_s(o_A(D_s1, …, D_sn)↓), and hence the inclusion (4.3) holds.

Applying this fact n times and using idempotency of j_s, it follows that

which imply that j is a lax morphism and hence a nucleus on 𝒫(A). Finally, by the definition of j_s and the fact the id_As is a Σ-term of sort s, we immediately get that j_s(a_s↓) = a_s↓, for all a_s ∈ A_s.□

Lemma 12

[18, Lemma 2.2.6] Let j be a nucleus on a sup-Σ-algebra A with an S-indexed family of closure operators j_s, s ∈ S. If for n ∈ ℕ, o ∈ O with rank s₁ ⋯ s_n → s, x_si, xsi′ ∈ A_si, we have x_si ⩽ xsi′ ⩽ j_si(x_si), for any i ∈ {1, …, n}, then

Lemma 13

If j is a nucleus on a sup-Σ-algebra A, then for any o ∈ O with rank s₁ ⋯ s_n → s, s ∈ S, n ∈ ℕ, and a_si ∈ A_si, i = 1, …, n, one has that

Lemma 14

Let A be an ordered Σ-algebra, j be the nucleus on 𝒫(A) with an S-indexed family of closure operators j_s, s ∈ S, defined in (4.1). Then for any o ∈ O with rank s₁ ⋯ s_n → s, n ∈ ℕ, and D_si ∈ 𝒫(A)_si, i = 1, …, n, we have

Therefore, for every t^s ∈ TΣS , s ∈ S, D_si ∈ 𝒫(A)_si, i = 1, …, n, n ∈ ℕ, we have

Corollary 15

Let A be an ordered Σ-algebra. Then

for every t^s ∈ TΣS , s ∈ S, D_si ∈ 𝒫(A)_jsi, i = 1, …, n, n ∈ ℕ.

Theorem 16

Let A be an ordered Σ-algebra. Then 𝒫(A)_j is the σ^⩽-injective hull of A in the category OALΣ⩽ .

Proof

It is clear that 𝒫(A)_j is σ^⩽-injective. Consider the mapping η : A → 𝒫(A)_j, which is given by an S-indexed family of mappings η_s : A_s → 𝒫(A)_js, where η_s(a_s) = a_s↓, for any a_s ∈ A_s, s ∈ S. Then obviously η is an order-embedding. Moreover, η is indeed a homomorphism and thus belongs to σ and σ^⩽, respectively. In fact, for any s ∈ S, o ∈ O with rank s₁ ⋯ s_n → s, n ∈ ℕ, and a_si ∈ A_si, i = 1, …, n, by Lemma 3, we have

It remains to show that η is σ^⩽-essential in the category OALΣ⩽ . Let ψ : 𝒫(A)_j → B be a morphism of ordered Σ-algebras in OALΣ⩽ such that ψη is a lax order-embedding. We have to show that ψ is a lax order-embedding, as well, i.e., condition (2) in the definition of σ^⩽ is satisfied.

Assume that tBs (ψ_s1(D_s1), …, ψ_sm(D_sm)) ⊆ ψ_s(D_s) in B, for t^s ∈ TΣS , s ∈ S, where D_si ∈ 𝒫(A)_jsi, i = 1, …, m, m ∈ ℕ and D_s ∈ 𝒫(A)_js. Our aim is to establish the inclusion tP(A)js (D_s1, …, D_sm) ⊆ D_s = j_s(D_s). Take u ∈ tP(A)js (D_s1, …, D_sm) = j_s( tAs (D_s1, …, D_sm)↓). Suppose that p^s͠(D_s) ⊆ a↓ for some p^s͠ ∈ PAs1 , a ∈ A_s͠, s͠ ∈ S, we may further suppose that p^s͠ has the form tAs~ (a_s1, …, a_si–1, _–, a_si+1, …, a_sn), i.e., a function from A_s to A_s͠i, where tAs~ is a Σ-term of sort s, a_sj ∈ A_sj, j ∈ {1, …, n} ∖ {i}, n ∈ ℕ. If p^s͠( tAs (D_s1, …, D_sm)↓) ⊆ a↓, then we conclude that p^s͠(u) ⩽ a because u ∈ j_s( tAs (D_s1, … D_sm)↓), which means that u ∈ j_s(D_s) = D_s as needed. Therefore, to complete the proof, it remains to prove the implication

Since p^s͠(D_s) ⊆ a↓, we have p^s͠(D_s)↓ ⊆ a↓. By Lemma 3, it comes out that

We need to verify that

Take d_s1 ∈ D_s1, … d_sm ∈ D_sm, then