On categorical aspects of S -quantales

Xia Zhang; Yunyan Zhou

doi:10.1515/math-2018-0110

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On categorical aspects of S -quantales

Xia Zhang and Yunyan Zhou

Published/Copyright: November 15, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

S-quantales are characterized as injective objects in the category of S-posets with respect to certain class of homomorphisms that are order-preserving mappings. This paper is devoted to exhibitions of categorical structures on S-quantales.

Keywords: Pomonoid; S-poset; Complete lattice; S-quantale; Adjoint situation

MSC 2010: 06F05; 20M30; 20M50

1 Introduction

The term quantale was suggested by C.J. Mulvey at the Oberwolfach Category Meeting ([1]) as a “quantization” of the term locale ([2]). An important moment in the development of the theory of quantales was the realization that quantales give a semantics for propositional linear logic in the same way as Boolean algebras give a semantics for classical propositional logic ([3, 4]). Quantales arise naturally as lattices of ideals, subgroups, or other suitable substructures of algebras ([5, 6]).

Algebraic investigations on qutale-like structures, such as quantales, quantale modules, sup-algebras, S -quantales, etc. have been studied in [5], [7], [8], and [9], respectively. Some categorical considerations are also taken into account ([10], [6]). S -quantales were firstly introduced by Zhang and Laan in [11], which have been shown to play an important role in the theory of injectivity on the category of S -posets. The current paper is devoted to the study of categorical properties of S -quantales.

In this work, S is always a pomonoid, that is, a monoid S equipped with a partial order ⩽ such that ss′⩽tt′ whenever s⩽t , s′⩽t′ in S . A poset ( A , ⩽) together with a mapping A× S → A (under which a pair (a, s) maps to an element of A denoted by as ) is called an S -poset, denoted by A_S , if for any a , b ∈ A_S , s , t∈S,

a(st) = (as)t,
a1 = a,
a⩽b, s⩽t imply that as⩽bt.

S -poset morphisms are order-preserving mappings which also preserve the S -action. We denote the category of S -posets with S -poset morphisms by Pos_S . An S -subposet of an S -poset A_S is an action-closed subset of A_S whose partial order is the restriction of the order from A_S.

Clearly, S -posets are generalizations of S -acts, whose relying category is denoted by Act_S.

Recall that an S -poset A_S is an S - quantale ([11]) if

the poset A is a complete lattice;
(∨M)s =∨{ms | m∈M} for each subset M of A and each s ∈ S.

An S -quantale morphism is a mapping between S -quantales which preserves both S -actions and arbitrary joins. An S -subquantale of an S -quantale A_S is exactly the relative S -subposet of A_S which is closed under arbitrary joins.

We denote the category of S -quantales with S -quantale morphisms by Quant_S . This work is devoted to the presentation of categorical aspects in Quant_S . We explore limits and colimits, monomorphisms and epimorphisms, respectively, and exhibit adjoint situations accordingly.

Lemma 1.1The bottom of an S -quantale is a zero element.

Proof. The result follows by the fact that for the bottom ⊥AS of an S -quantale A_S , and any s∈S,

⊥ASs=(∨∅)s=∨a∈∅(as)=∨∅=⊥AS.

Since an S -quantale morphism f : A_S → B_S preserves all joins, it follows by the

adjoint functor theorem that it has a right adjoint f_* : B_S → A_S , satisfying

(1)f(a)b⟺a≤f∗(b),

for all a∈ A_S , b∈B_S.

Lemma 1.2Let f : A_S → B_S be an S -quantale morphism. Then f preserves the bottom.

Proof. Denote by ⊥AS the bottom of A_S . Then ⊥AS≤f∗(b), for every b∈B_S. By (1), we have f(⊥AS)≤b.

2 Limits and colimits in Quant_S

Products and coproducts

Proposition 2.1

The product of a family of S -quantales is their cartesian product with componentwise action, and order.

Proposition 2.2

The coproduct of a family of S -quantales {X_i}_i_∈Iis∏i∈IXi,(μj)j∈I, whereμj:Xj→∏i∈IXi,j∈I,is defined by

(2)μj(x)=(xi~)i∈I,wherexi~=xi=j,⊥Xii≠j.

Proof. Clearly, μ_j is an S -quantale morphism for every j∈I. Let f_j : X_j →Q_S, j∈I, be S -quantale morphisms. Define a mapping ψ:∏i∈IXi→QS by

ψ((xi)i∈I)=⋁i∈Ifi(xi),

for any (xi)i∈I∈∏i∈IXi. It is easy to see that ψ preserves S -actions. For arbitrary indexed set K, we have

ψ⋁k∈K(xik)i∈I=ψ⋁k∈Kxiki∈I=⋁i∈Ifi⋁k∈Kxik=⋁k∈Kψ((xik)i∈I).

Moreover, by Lemma 1.2, fi(⊥Xi)=⊥QS, for each i∈I. Hence

ψ(μj(x))=ψ((xi~)i∈I)=⋁i∈Ifi(xi~)=fj(x),

for any j∈I,x∈Xj.

Finally, suppose that there exists an S -quantale morphism ϕ:∏i∈IXi→QS such that ϕμ_i = f_i, for every i∈I. Then, for each (xi)i∈I∈∏i∈IXi, one gets that

ϕ((xi)i∈I)=ϕ⋁i∈Iμi(xi)=⋁i∈Iϕμi(xi)=⋁i∈Ifi(xi)=ψ((xi)i∈I),

and hence ϕ = ψ as needed.

Equalizers, coequalizers, pullbacks, and pushouts

Proposition 2.3

Let f , g : A_S → B_S be morphisms of S -quantales. The equalizer of f and g is given by E ={ a∈ A_S | f(a)= g( a)} , with action and order inherited from A_S.

Proof. Clearly, E is an S -poset, and a complete lattice. So it is an S -quantale by the fact that f and g preserve arbitrary joins. Let ι:E↪A be the inclusion mapping. For any morphism e : E^' → A with fe = ge , since e(E^') ⊆ E , it follows that e¯, which is the codomain restriction of e , is the unique morphism fulfilling ιe¯=e.

By [12] Theorem 12.3, we immediately get that Quant_S is complete.

Proposition 2.4

The category Quant _S is complete.

Let ρ be a congruence on S -quantale A_S . In a natural way, the quotient A / ρ constitutes an S -quantale equipped with the order defined by a ρ -chain, where the joins in A / ρ are

(3)⋁i∈I[ai]ρ=⋁i∈Iaiρ,

and the canonical mapping π : A_S → (A / ρ)_S becomes an S -quantale morphism, provided that ρ = kerπ ([9]). For H ⊆ A_S × A_S, the corresponding S -quantale congruence generated by H, will be denoted by θ(H).

Proposition 2.5

Let f , g : A_S → B_S be morphisms of S -quantales. The coequalizer of f and g is the quotient (B / θ (H))_S, whereH={(f(a),g(a))∣a∈AS}.

Proof. Let f , g : A_S → B_S be morphisms of S -quantales, H={(f(a),g(a))∣a∈AS}, π be the canonical mapping from B_S to (B / θ(H))_S . Clearly, π f = π g. For any S -quantale morphism h : B_S → C_S satisfying hf = hg, we obtain that kerπ ⊆ kerh , since (f(a), g(a))∈kerh, for a∈ A_S.

Now define a mapping h¯:(B/θ(H))S→CS by

h¯[b]θ(H)=h(b),

for [b]θ(H)∈(B/θ(H))S. Clearly ̅h is an S -act morphism and preserves arbitrary joins by (3). It is quite routine to check that ̅h is the unique morphism satisfying h¯π=h..

Proposition 2.6

Let f : A_S → C_S , g : B_S → C_S be morphisms of S -quantales. The pullback of f and g is the S -subposet P = {(a,b)∈(A×B)_S | f(a) = g (b)} of (A×B)_S , together with the restricted projections of P_S into A_S and B_S.

Proof. It is known that P_S is an S -quantale. For any S -quantale Q_S and an pair of morphisms f₁: Q_S → A_S , f₂ : Q_S → B_S with ff₁ = gf₂ , one has that (f₁(q), f₂(q)) ∈ P_S , for any q∈Q_S . Now define a mapping φ : Q_S → P_S by

φ(q)=(f1(q),f2(q)),

for q ∈ Q_S . One gets that

φ(q)s=(f1(q),f2(q))s=f1(q)s,f2(q)s=(f1(qs),f2(qs))=φ(qs),

for each q∈Q_S , s ∈ S , and

φ⋁i∈Iqi=f1⋁i∈Iqi,f2⋁i∈Iqi=⋁i∈If1(qi),⋁i∈If2(qi)=⋁i∈I((f1(qi),f2(qi))=⋁i∈Iφ(qi),

for all q_i ∈ Q_S , i∈I . If π_A : P_S → A_S and π_B : P_S → B_S are the restricted projections, then fπA=gπB. Straightforward checking shows that φ is the unique morphism satisfying π_Aφ = f₁ and π_Bφ = f₂.

Proposition 2.7

Let f : A_S → B₁ , g : A_S → B₂be morphisms of S -quantales. The pushout of f and g is ((B₁ × B₂) / θ (H))_S , together with πμ₁and πμ₂, where μ_i : B_i → (B₁ × B₂)_S , i =1, 2 , are defined as in Proposition 2.2, π is the canonical mapping, H = {(μ₁f(a), μ₂g(a))| a∈ A_S}.

Proof. Since ((B₁ × B₂)_S, (μ₁, μ₂)) is the coproduct of (B₁, B₂) by Proposition 2.2, the coequalizer of μ₁f and μ₂g is the quotient ((B₁×B₂)/θ(H))_S , where H = {(μ₁f(a), μ₂g(a)) | a∈ A_S} , by Proposition 2.5. The result follows immediately by [12] Remark 11.31.

3 Monomorphisms

This section contributes to the presentation of several kinds of monomorphisms in the category Quant_S . It is shown that deferent from the case of S -posets (see [13]), monomorphisms in Quant_S coincide with order-embeddings, which are precisely injective morphisms. It thus leads to the strengthening results that these classes of monomorphisms are also in accordance with those labeled regular and extremal in Quant_S , which are exactly the category-theoretic embeddings when Quant_S is considered as a concrete category over Set , Act_S , and Pos_S , respectively.

Proposition 3.1

Let f : A_S → B_S be a morphism of S -quantales. Then the following statements are equivalent:

f is a monomorphism;
f is injective;
f is an order-embedding.

Proof. It is enough to show the implications (1) ⇒ (2) and (1) ⇒ (3) hold.

Let f : A_S → B_S be a monomorphism of S -quantales. Consider S -subquantale kerf of the product (A× A)_S , and the restricted projection mappings h_i : kerf → A, i =1, 2 . For any (x, y) ∈ kerf , equalities

fh1(x,y)=f(x)=f(y)=fh2(x,y)

imply that fh₁ = fh₂ and hence h₁ = h₂ by assumption. Therefore, x = h₁(x, y) = h₂ (x, y) = y , and hence f is injective as needed.

It remains to prove that f is an order-embedding whenever it is a monomorphism. Suppose that f(a₁)⩽f(a₂) for a₁, a₂ ∈ A_S . Then

f(a2)=f(a1)⋁f(a2)=fa1⋁a2.

According to the above result of f being injective, we soon obtain that a₁ ⩽a₂, and thus f is an order-embedding.

Lemma 3.2

Each inclusion mapping in Quant_S is a regular monomorphism.

Proof. Suppose that A_S is an S -subquantale of B_S . Let ((B × B)_S , (μ₁, μ₂)) be the coproduct of (B_S , B_S) , described as in Proposition 2.2. Write

R={(a,⊥),(⊥,a)∣a∈AS},

where ⊥ is the bottom element of B_S . Then the relation ρ , which is defined by

ρ={(x∨a,y∨b),(x′∨a′,y′∨b′)|x,y,x′,y′∈BS,a,b,a′,b′∈AS,x∨b=x′∨b′,y∨a=y′∨a′}

is the smallest congruence relation on B×B containing R . So (( B× B) / ρ)_S becomes an S -quantale equipped with a suitable order defined by a ρ -chain, and the canonical mapping π:(B× B)_S → ((B × B)/ ρ)_S given by π(x, y) = [(x, y)]_ρ , for each (x , y) ∈ (B × B)_S , is a morphism.

Next we show that the inclusion mapping ιA:A↪B is the equalizer of πμ₁ = πμ₂ . Suppose that h : E_S → B_S is any monomorphism satisfying πμ₁h = πμ₂h. Then for any e ∈ E_S , the equalities

[(h(e),⊥)]ρ=π(h(e),⊥)=πμ1h(e)=πμ2h(e)=π(⊥,h(e))=[(⊥,h(e))]ρ

indicate that ((h(e),⊥), (⊥,h(e)))∈ρ . According to the definition of ρ, we deduce that (h(e),⊥) = (x ∨ a, y ∨ b) and (⊥, h(e)) = (x^' ∨ a^' , y^' ∨ b^') for some x , y , x^' , y^' ∈ B_S , a , a^' , b , b^' ∈ A_S . So y = b =⊥, x^' = a^' =⊥, and correspondingly,

a=⊥∨a=y∨a=y′∨a′=y′∨⊥=y′,

and

x=x∨⊥=⊥∨b′=b′.

Therefore, we have h(e) = x ∨ a = b′∨ a∈ A_S , i.e., h(E)⊆AS. As a consequence, h~=h:E→A is the unique morphism satisfying ιAh~=h.

Theorem 3.3

Let f : A_S → B_S be a morphism of S -quantales. Then the following assertions are equivalent:

f is a regular monomorphism;
f is an extremal monomorphism;
f is a monomorphism;
f is a Quant_S -embedding over Set ;
f is a Quant_S -embedding over Act_S ;
f is a Quant_S -embedding over Pos_S.

Proof. (1) ⇒ ( 2 ) ⇒ ( 3 ) are general category-theoretic results.

(3 ) ⇒ ( 4 ). Suppose that f : A_S → B_S is a monomorphism. Let g : C_S → A_S be a mapping with fg : C_S → B_S being an S -quantale morphism. Then g preserves arbitrary joins by the fact that for a_i ∈ C_S , i∈I,

fg⋁i∈Iai=⋁i∈Ifg(ai)=f⋁i∈Ig(ai),

and f being injective by Proposition 3.1. Similarly, we get that g preserves S -actions. Thus f is initial and then an S -quantale embedding over Set.

( 4 ) ⇒ ( 3 ), ( 4 ) ⇒ (5 ) ⇒ ( 6 ) are clear.

( 6 ) ⇒ ( 4 ). Let f : A_S → B_S be a Quant_S -embedding over Pos_S , g : C_S → A_S a mapping provided that fg : C_S → B_S is a morphism in Quant_S . We are going to show that g is an S -poset morphism. This is the case since

fg(as)=fg(a)s=f(g(a)s),

for any a∈ A_S , s∈S , and

fg(a2)=fga1⋁a2=fg(a1)⋁fg(a2)=fg(a1)⋁g(a2),

for a₁ ⩽a₂ in A_S . Note that the monomorphisms in Pos_S are just the S -poset morphisms with injective underlying mappings, we immediately achieve that g(as) = g(a)s and g(a₁)⩽g(a₂) . Therefore, g is an S -poset morphism as required.

(3 ) ⇒ (1). This follows by [12] Proposition 7.53 (2) and Lemma 3.2.

4 Epimorphisms

Dual to discussions on monomorphisms studied in Section 3, this section is intended to motivate our investigation on relationships between various type of epimorphisms in Quant_S . However, the characterization of epimorphisms in Quant_S is quite complicated. So we merely cite the result and the reader is suggested to find complete illustrations in [14].

Proposition 4.1

(Th. 4.2) Epimorphisms in Quant_S are exactly onto morphisms.

Theorem 4.2

For a morphism f : A_S → B_S of S -quantales, the following statements are equivalent:

f is a regular epimorphism;
f is an extremal epimorphism;
f is an epimorphism;
f is a Quant_S -quotient morphism over Set ;
f is a Quant_S -quotient morphism over Act_S ;
f is a Quant_S -quotient morphism over Pos_S.

Proof. (1) ⇒ ( 2 ) ⇒ ( 3 ) are clear.

(3 ) ⇒ (1) follows by [14] Corollary 14.

(3 ) ⇒ ( 4 ). Let g : B_S → C_S be a mapping between S -quantales such that gf is an S -quantale morphism. Let us verify that g is an S -quantale morphism, as well. It is easy to see that g is an S -poset morphism. Since f is an epimorphism, it is onto by Proposition 4.1. Hence we may assume that for any M ⊆ B_S , ∨M = f (a) for some a∈ A_S . By the reason that f preserves arbitrary joins, we have

f(a)=⋁M=⋁x∈f−1(M)f(x)=f⋁x∈f−1(M)x.

Consequently,

g⋁M=gf(a)=gf⋁x∈f−1(M)x=⋁x∈f−1(M)gf(x)=⋁m∈Mg(m).

( 4 ) ⇒ ( 3 ), ( 4 ) ⇒ (5 ) ⇒ ( 6 ) are clear.

( 6 ) ⇒ ( 2 ). Let f : A_S → B_S be a Quant_S -quotient morphism over Pos_S . Suppose that g : A_S → C_S and h : C_S → B_S are S -quantale morphisms such that f = hg and h is a monomorphism. Then h is injective by Proposition 3.1. Note that f is a Pos_S -epimorphism by hypotheses, and hence is surjective. So h is surjective, as well, and thus bijective. Now, considering the inverse mapping h⁻¹ with g = h⁻¹f , we remain to show that h⁻1 is an S -poset morphism. In fact, f bing onto indicates that h⁻¹ is action-preserving. Observe that

hh−1(b)⋁h−1(b′)=hh−1(b)⋁hh−1(b′)=b⋁b′=hh−1(b′),

for any b⩽b′ in B_S . Thus h⁻¹(b)∨h⁻¹(b′) = h⁻¹(b′) , which expresses that h⁻¹ is an S -poset morphism, and hereby an S -quantale morphism by assumption.

5 Adjoint situations

The final part is devoted to observation on the adjoint situation between Pos and Quant_S . By a free S -quantale on a poset P we mean an S -quantale Q _S together with a monotone mapping ψ : P→Q _S with the universal property that given any S -quantale A_S and a monotone mapping f : P→ A_S , there exists a unique S -quantale morphism f¯:QS→AS such that f can be factored through.

Lemma 5.1

(Th.10) For a given poset P and a pomonoid S , the free S -poset on P is given by P × S , with componentwise order and the action (x,s)t = (x, st), for every x∈P, s,t∈S.

Let (P×S)_S be the free S -poset presented in Lemma 5.1. Write

Q(P×S)={D⊆P×S∣D=D↓},

where D ↓ is the down-set of D for D ⊆ P×S , more precisely,

D↓={(p,s)∈P×S∣(p,s)≤(p1,s1) for some (p1,s1)∈D}.

Note that ( p ↓ ×s ↓) ↓= p ↓ ×s ↓ provides that p↓×s↓∈Q(P×S) for every element p∈P, s∈S. Define an action ∗ on Q(P×S) by

D∗t:={(p,s)∈P×S∣(p,s)≤(p1,s1t) for some (p1,s1)∈D},

for t∈S. Then it is clear that D∗t = (Dt) ↓ . We claim that (Q(P×S)S,∗,⊆) is the free object in Quant_S.

Proposition 5.2

Let S be a pomonoid, P be a poset. Then(Q(P×S)S,∗,⊆)is an S -quantale.

Proof. Observe first that

(D∗t1)∗t2={(p,s)∈P×S∣(p,s)≤(p1,s1t2) for some (p1,s1)∈D∗t1}={(p,s)∈P×S∣(p,s)≤(p1,s1t2),(p1,s1)≤(p2,s2t1) for some (p2,s2)∈D}={(p,s)∈P×S∣(p,s)≤(p2,s2t1t2), for some (p2,s2)∈D}=D∗(t1t2)

for any t₁, t₂ ∈ S, D∈Q(P×S), and D∗1 = (D1) ↓= D. This shows that (Q(P×S),∗) is an S -act. Clearly, D₁ ∗ s ⊆ D₂ ∗t, whenever D₁ ⊆ D₂ in Q(P×S), and s⩽t in S . So Q(P×S) is an S -poset. It is straightforward to check that (⋃i∈IDi)∗t=⋃i∈I(Di∗t) for every Di∈Q(P×S),i∈I,t∈S.

Lemma 5.3 comes true directly by the definition of Q(P×S).

Lemma 5.3

Let S be a pomonoid, P be a poset. ThenD=⋃(p,s)∈D(p↓×s↓)for everyD∈Q(P×S).

Lemma 5.4

Let S be a pomonoid, P be a poset. Then p ↓ ×t ↓= (p ↓ ×1↓)∗t holds in Q(P×S)Sfor every p ∈ P, t ∈ S.

Proof. It is clear that (q, s) ∈ (p ↓ ×1↓)∗t for every (q, s) ∈ p ↓ ×t ↓ , since (q, s)⩽(p,t) . On the other hand, for any (q,s)∈(p↓×1↓)∗t,(q,s)≤(p1,s1t)=(p1,s1)t for some (p₁, s₁) ∈ p ↓ ×1↓ , it follows that (q, s)⩽( p,1)t = (p,t). Hence (q, s) ∈ p ↓ ×t ↓.

Theorem 5.5

Let S be a pomonoid, P be a poset. Then the free S -quantale on P is given by the S -quantale Q(P×S)S.

Proof. Define a mapping τ:P→Q(P×S)S by τ(p) = p ↓ ×1↓ for every p∈P. Obviously, τ is order-preserving. Let Q_S be an S -quantale, f : P→Q_S be any monotone mapping. Define a mapping f¯:Q(P×S)S→QS by

f¯(D)=⋁{f(p)s∣(p,s)∈D},

for every D∈Q(P×S)S. We claim that f¯ is the unique S -quantale morphism with the property that f¯τ=f.

It is clear that f¯ preserves S -actions. Take Di∈Q(P×S)S,i∈I, then equalities

f¯⋃i∈IDi=⋁{f(p)s|(p,s)∈⋃i∈IDi}=⋁i∈I{⋁{f(p)s|(p,s)∈Di}}=⋁i∈If¯(Di)

indicate thatf¯ preserves arbitrary joins. Evidently, for any p ∈ P,

f¯τ(p)=f¯(p↓×1↓)=⋁{f(q)s∣(q,s)∈p↓×1↓}≤f(p),

while the fact that f(p) being one of the terms in the sup that defines fτ(p) guarantees the opposite implication. Suppose that f′:Q(P×S)S→QS is an S -quantale morphism such that f′τ=f. Then by Lemma 5.3 and Lemma 5.4, we achieve that

f′(D)=f′⋃(p,s)∈D(p↓×s↓)=⋁(p,s)∈Df′(p↓×s↓)=⋁(p,s)∈Df′((p↓×1↓)∗s)=⋁(p,s)∈Df′(p↓×1↓)s=⋁(p,s)∈Df′τ(p)s=⋁(p,s)∈Df(p)s=f¯(D),

for every D∈Q(P×S)S, which finishes our proof.

Corollary 5.6

The category Quant_S has a separator.

Proof. Let f , g : A_S → B_S be a pair of morphisms in Quant_S with f ≠ g . Then there exists a∈ A_S such that f(a) ≠ g(a). Let P be a poset. Define a mapping k : P → A_S by k(p) = a,∀p∈P . We are aware that k is a morphism in Pos . Hence there is a unique S -quantale morphism k¯:Q(P×S)S→AS with k¯τ=k, where τ:P→Q(P×S)S is defined as in Theorem 5.5. This yields that fk¯≠gk¯, and consequently gives that Q(P×S)S is a separator.

We thereby obtain a free functor from the category of posets into the category of S -quantales, which is shown to be left adjoint to the forgetful functor.

Proposition 5.7

There is a free functor F : Pos → Quant_S given by

where FP=Q(P×S)S, and

Ff(D)={(x,y)∈Q×S∣(x,y)≤(f(p),s)forsome(p,s)∈D},

for any monotone mapping f : P→Q and D ∈ FP.

Theorem 5.8

The free functor F : Pos → Quant_S is left adjoint to the forgetful functor:QuantS→Pos.

Proof. Let us prove that η:idpos→F with ηP:P→⌊Q(P×S)S⌋, where P is a Pos -object, η_P(p) = p ↓ × 1 ↓ , ∀p∈P , is a natural transformation. Suppose that f : P→ P^' is a morphism in Pos . Then

Ff∘ηP(p)=Ff(p↓×1↓)={(x,y)∈P′×S∣(x,y)≤(f(p~),s)forsome(p~,s)∈p↓×1↓}={(x,y)∈P′×S∣(x,y)≤(f(p~),s)≤(f(p),1),(p~,s)∈p↓×1↓},

for p ∈ P , and

(ηP′∘f)(p)=ηP′(f(p))=f(p)↓×1↓.

It results in Ff∘ηP=ηP′∘f as needed. Now, by Theorem 5.5 and [12] 19.4(2), we obtain that F is left adjoint to⌊⌋.

Dedicated to Professor Ulrich Knauer on his 75th Birthday.

Acknowledgement

This work was supported by the Natural Science Foundation of Guangdong Province, China under Grant number 2016A030313832, the Science and Technology Program of Guangzhou, China under Grant number 201607010190, the State Scholarship Fund, China under Grant number 201708440512, and the research funding of School of Mathematical Sciences, SCNU under Grant number 2016YN32.

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Received: 2018-07-29

Accepted: 2018-10-11

Published Online: 2018-11-15

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0110

Keywords for this article

Pomonoid; S-poset; Complete lattice; S-quantale; Adjoint situation

Creative Commons

BY-NC-ND 4.0

On categorical aspects of S -quantales

Article

Abstract

1 Introduction

2 Limits and colimits in QuantS

Products and coproducts

Proposition 2.1

Proposition 2.2

Equalizers, coequalizers, pullbacks, and pushouts

Proposition 2.3

Proposition 2.4

Proposition 2.5

Proposition 2.6

Proposition 2.7

3 Monomorphisms

Proposition 3.1

Lemma 3.2

Theorem 3.3

4 Epimorphisms

Proposition 4.1

Theorem 4.2

5 Adjoint situations

Lemma 5.1

Proposition 5.2

Lemma 5.3

Lemma 5.4

Theorem 5.5

Corollary 5.6

Proposition 5.7

Theorem 5.8

Acknowledgement

References

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

2 Limits and colimits in Quant_S