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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1


Volume 13 (2015)

On categorical aspects of S -quantales

Xia Zhang / Yunyan Zhou
Published Online: 2018-11-15 | DOI: https://doi.org/10.1515/math-2018-0110


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 st , s′⩽t′ in S . A poset ( A , ⩽) together with a mapping A× SA (under which a pair (a, s) maps to an element of A denoted by as ) is called an S -poset, denoted by AS , if for any a , bAS , s , tS,

  1. a(st) = (as)t,

  2. a1 = a,

  3. ab, st imply that asbt.

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 PosS . An S -subposet of an S -poset AS is an action-closed subset of AS whose partial order is the restriction of the order from AS.

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

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

  1. the poset A is a complete lattice;

  2. (∨M)s =∨{ms | mM} for each subset M of A and each sS.

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 AS is exactly the relative S -subposet of AS which is closed under arbitrary joins.

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

Lemma 1.1 The 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 AS , and any sS,


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

adjoint functor theorem that it has a right adjoint f* : BSAS , satisfying


for all aAS , bBS.

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

Proof. Denote by AS the bottom of AS . Then ASf(b), for every bBS. By (1), we have f(AS)b.

2 Limits and colimits in QuantS

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 {Xi}iI is iIXi,(μj)jI, where μj:XjiIXi,jI, is defined by


Proof. Clearly, μj is an S -quantale morphism for every jI. Let fj : XjQS, jI, be S -quantale morphisms. Define a mapping ψ:iIXiQS by


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


Moreover, by Lemma 1.2, fi(Xi)=QS, for each iI. Hence


for any jI,xXj.

Finally, suppose that there exists an S -quantale morphism ϕ:iIXiQS such that ϕμi = fi, for every iI. Then, for each (xi)iIiIXi, one gets that


and hence ϕ = ψ as needed.

Equalizers, coequalizers, pullbacks, and pushouts

Proposition 2.3

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

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:EA 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 QuantS is complete.

Proposition 2.4

The category Quant S is complete.

Let ρ be a congruence on S -quantale AS . 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


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

Proposition 2.5

Let f , g : ASBS be morphisms of S -quantales. The coequalizer of f and g is the quotient (B / θ (H))S, where H={(f(a),g(a))aAS}.

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

Now define a mapping h¯:(B/θ(H))SCS by


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 : ASCS , g : BSCS 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 PS into AS and BS.

Proof. It is known that PS is an S -quantale. For any S -quantale QS and an pair of morphisms f1: QSAS , f2 : QSBS with ff1 = gf2 , one has that (f1(q), f2(q)) ∈ PS , for any qQS . Now define a mapping φ : QSPS by


for qQS . One gets that


for each qQS , sS , and


for all qiQS , iI . If πA : PSAS and πB : PSBS are the restricted projections, then fπA=gπB. Straightforward checking shows that φ is the unique morphism satisfying πAφ = f1 and πBφ = f2.

Proposition 2.7

Let f : ASB1 , g : ASB2 be morphisms of S -quantales. The pushout of f and g is ((B1 × B2) / θ (H))S , together with πμ1 and πμ2 , where μi : Bi → (B1 × B2)S , i =1, 2 , are defined as in Proposition 2.2, π is the canonical mapping, H = {(μ1f(a), μ2g(a))| aAS}.

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)) | aAS} , 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 QuantS . It is shown that deferent from the case of S -posets (see [13]), monomorphisms in QuantS 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 QuantS , which are exactly the category-theoretic embeddings when QuantS is considered as a concrete category over Set , ActS , and PosS , respectively.

Proposition 3.1

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

  1. f is a monomorphism;

  2. f is injective;

  3. f is an order-embedding.

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

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


imply that fh1 = fh2 and hence h1 = h2 by assumption. Therefore, x = h 1(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, a2AS . Then


According to the above result of f being injective, we soon obtain that a1a 2, and thus f is an order-embedding.

Lemma 3.2

Each inclusion mapping in QuantS is a regular monomorphism.

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


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


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:AB is the equalizer of πμ1 = πμ2 . Suppose that h : ESBS is any monomorphism satisfying πμ1h = πμ2h. Then for any eES , the equalities


indicate that ((h(e),⊥), (⊥,h(e)))∈ρ . According to the definition of ρ, we deduce that (h(e),⊥) = (xa, yb) and (⊥, h(e)) = (x'a' , y'b') for some x , y , x' , y'BS , a , a' , b , b'AS . So y = b =⊥, x' = a' =⊥, and correspondingly,




Therefore, we have h(e) = xa = b′∨ aAS , i.e., h(E)AS. As a consequence, h~=h:EA is the unique morphism satisfying ιAh~=h.

Theorem 3.3

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

  1. f is a regular monomorphism;

  2. f is an extremal monomorphism;

  3. f is a monomorphism;

  4. f is a QuantS -embedding over Set ;

  5. f is a QuantS -embedding over ActS ;

  6. f is a QuantS -embedding over PosS.

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

(3 ) ⇒ ( 4 ). Suppose that f : ASBS is a monomorphism. Let g : CSAS be a mapping with fg : CSBS being an S -quantale morphism. Then g preserves arbitrary joins by the fact that for aiCS , iI,


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 : ASBS be a QuantS -embedding over PosS , g : CSAS a mapping provided that fg : CSBS is a morphism in QuantS . We are going to show that g is an S -poset morphism. This is the case since


for any aAS , sS , and


for a1a 2 in AS . Note that the monomorphisms in PosS 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 QuantS . However, the characterization of epimorphisms in QuantS 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 QuantS are exactly onto morphisms.

Theorem 4.2

For a morphism f : ASBS of S -quantales, the following statements are equivalent:

  1. f is a regular epimorphism;

  2. f is an extremal epimorphism;

  3. f is an epimorphism;

  4. f is a QuantS -quotient morphism over Set ;

  5. f is a QuantS -quotient morphism over ActS ;

  6. f is a QuantS -quotient morphism over PosS.

Proof. (1) ⇒ ( 2 ) ⇒ ( 3 ) are clear.

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

(3 ) ⇒ ( 4 ). Let g : BSCS 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 MBS , ∨M = f (a) for some aAS . By the reason that f preserves arbitrary joins, we have




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

( 6 ) ⇒ ( 2 ). Let f : ASBS be a QuantS -quotient morphism over PosS . Suppose that g : ASCS and h : CSBS 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 PosS -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−1 f , we remain to show that h1 is an S -poset morphism. In fact, f bing onto indicates that h−1 is action-preserving. Observe that


for any bb′ in BS . 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 QuantS . By a free S -quantale on a poset P we mean an S -quantale Q S together with a monotone mapping ψ : PQ S with the universal property that given any S -quantale AS and a monotone mapping f : PAS , there exists a unique S -quantale morphism f¯:QSAS 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 xP, s,tS.

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


where D ↓ is the down-set of D for DP×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 pP, sS. Define an action on Q(P×S) by

Dt:={(p,s)P×S(p,s)(p1,s1t) for some (p1,s1)D},

for tS. Then it is clear that Dt = (Dt) ↓ . We claim that (Q(P×S)S,,) is the free object in QuantS.

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

(Dt1)t2={(p,s)P×S(p,s)(p1,s1t2) for some (p1,s1)Dt1}={(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, t2S, DQ(P×S), and D∗1 = (D1) ↓= D. This shows that (Q(P×S),) is an S -act. Clearly, D1sD2t, whenever D1D2 in Q(P×S), and st in S . So Q(P×S) is an S -poset. It is straightforward to check that (iIDi)t=iI(Dit) for every DiQ(P×S),iI,tS.

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. Then D=(p,s)D(p×s) for every DQ(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)S for every pP, tS.

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


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

It is clear that preserves S -actions. Take DiQ(P×S)S,iI, then equalities


indicate thatf¯ preserves arbitrary joins. Evidently, for any pP,


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


for every DQ(P×S)S, which finishes our proof.

Corollary 5.6

The category QuantS has a separator.

Proof. Let f , g : ASBS be a pair of morphisms in QuantS with fg . Then there exists aAS such that f(a) ≠ g(a). Let P be a poset. Define a mapping k : PAS by k(p) = a,∀pP . We are aware that k is a morphism in Pos . Hence there is a unique S -quantale morphism k¯:Q(P×S)SAS with k¯τ=k, where τ:PQ(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 → QuantS given by

where FP=Q(P×S)S, and


for any monotone mapping f : PQ and DFP.

Theorem 5.8

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

Proof. Let us prove that η:idposF with ηP:PQ(P×S)S, where P is a Pos -object, ηP(p) = p ↓ × 1 ↓ , ∀pP , is a natural transformation. Suppose that f : PP' is a morphism in Pos . Then


for pP , and


It results in FfηP=ηPf as needed. Now, by Theorem 5.5 and [12] 19.4(2), we obtain that F is left adjoint to.


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.


  • [1]

    Mulvey C.J., § . Suppl. Rend. Circ. Mat. Palermo ser. II, 1986, 12, 99–104 Google Scholar

  • [2]

    Jonestone P.T., Stone spaces, Cambridge university press, Cambridge, 1982 Google Scholar

  • [3]

    Girard J.-Y., Linear logic, Theoretical Computer Science, 1997, 50, 1–102 Web of ScienceGoogle Scholar

  • [4]

    Belohlávek R., Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic/Plenum Press, New York, 2002 Google Scholar

  • [5]

    Rosenthal K.I., Quantales and their applications, Longman Scientific and Technical, London, 1990 Google Scholar

  • [6]

    Kruml D., Paseka J., In: Hazewinkel M. (Ed.), Handbook of Algebra, Vol. 5, Algebraic and categorical aspects of quantales, Elsevier/North-Holland, Amsterdam, 2008, 323–362 Google Scholar

  • [7]

    Russo C., Quantale modules, Lambert Academic publishing, Saarbrücken, 2009 Google Scholar

  • [8]

    Zhang X., Laan V., Quotients and subalgebras of sup-algebras, Proc. Est. Acad. Sci., 2015, 64, 311-322 CrossrefGoogle Scholar

  • [9]

    Zhang X., Li T.Y., On nuclei and conuclei of S -quantales, Quasigroups Related Systems., 2017, 25, 155-164 Google Scholar

  • [10]

    Solovyov S.A., On the category of Q-Mod. Algebra Universalis, 2008, 58, 35-58 CrossrefWeb of ScienceGoogle Scholar

  • [11]

    Zhang X., Laan V., On injective hulls of S -posets, Semigroup Forum, 2015, 91, 62-70 CrossrefWeb of ScienceGoogle Scholar

  • [12]

    Adámek J., Herrlich H., Strecker, G.E., Abstract and Concrete Categories: The Joy of Cats, John Wiley and Sons, New York, 1990 Google Scholar

  • [13]

    Bulman-Fleming S., Mahmoudi M., The category of S -posets, Semigroup Forum, 2005, 71, 443-461 CrossrefWeb of ScienceGoogle Scholar

  • [14]

    Xiao Z.J., Zhang X., On epimorphisms of S -quantales, Algebra Universalis, 2018, 79: 65 Web of ScienceGoogle Scholar

About the article

Received: 2018-07-29

Accepted: 2018-10-11

Published Online: 2018-11-15

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1326–1334, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0110.

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© 2018 Zhang and Zhou, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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