# Injective hulls for posemigroups/Jarjestatud poolruhmade injektiivsed katted.

On toestatud, et injektiivsed objektid teatud sisestuste klassi suhtes kategoorias, mille objektideks on jarjestatud poolruhmad ja morfismideks submultiplikatiivsed kujutused, on kvantaalid. Samuti on naidatud, kuidas teatud poolruhmade klassi jaoks saab konstrueerida injektiivseid katteid vaadeldava sisestuste klassi suhtes.

1. INTRODUCTION

Bruns and Lakser in their paper [2] characterized injective hulls in the category of semilattices. In a recent article [6], Lambek et al. considered injective hulls in the category of pomonoids and submultiplicative identity and order-preserving mappings.

Inspired by these results, we construct injective hulls in the category of posemigroups and submultiplicative order-preserving mappings with respect to certain class [E.sub.[less than or equal to]] of morphisms in this work. In fact, Theorem 5.8 of [6] becomes a consequence of our main theorem (Theorem 7).

As usual, a posemigroup (S, *, [less than or equal to]) is a semigroup (S, *) equipped with a partial ordering [less than or equal to] which is compatible with the semigroup multiplication, that is, [a.sub.1][a.sub.2] [less than or equal to] [b.sub.1][b.sub.2] whenever [a.sub.1] [less than or equal to] [b.sub.1] and [a.sub.2] [less than or equal to] [b.sub.2], for any [a.sub.1], [a.sub.2], [b.sub.1], [b.sub.2] [member of] S. Posemigroup homomorphisms are monotone (i.e. order-preserving) semigroup homomorphisms between two posemigroups. A subsemigroup R of S equipped with the partial order (R x R) [intersection] [less than or equal to] is called a subposemigroup of S. An order embedding from a poset (A, [[less than or equal to].sub.A]) to a poset (B, [[less than or equal to].sub.B]) is a mapping h: A [right arrow] B such that a [[less than or equal to].sub.A] a' iff h(a) [[less than or equal to].sub.B] h(a'), for all a, a' [member of] A. Every order embedding is necessarily an injective mapping.

Let C be a category and let M be a class of morphisms in C. We recall that an object S from C is M-injective in C provided that for any morphism h: A [right arrow] B in M and any morphism f: A [right arrow] S in C there exists a morphism g: B [right arrow] S such that gh = f.

A morphism [eta]: A [right arrow] B in M is called M-essential (cf. [1]) if every morphism [psi]: B [right arrow] C in C, for which the composite [psi][eta] is in M, is itself in M. An object H [member of] C is called an M-injective hull of an object S if H is M-injective and there exists an M-essential morphism S [right arrow] H.

It is natural to consider injectivity in the category of posemigroups with respect to posemigroup homomorphisms that are order embeddings. However, injective objects in this sense are only one-element posemigroups. Indeed, if a posemigroup (S, *, [less than or equal to]) is injective, then the underlying semigroup (S, *) is injective in the category of all semigroups, because every semigroup can be considered as a discretely ordered posemigroup, and homomorphisms of discretely ordered posemigroups are just the homomorphisms of underlying semigroups. But injective semigroups are only the trivial ones (see [10]).

Allowing more morphisms between posemigroups, it is still possible to obtain nontrivial injectives. This approach is taken in [6]. Namely, one can consider order-preserving submultiplicative mappings f: A [right arrow] B between posemigroups A and B, i.e. mappings with f (a) f (a') [less than or equal to] f (aa') for all a, a' [member of] A. We denote by [PoSgr.sub.[less than or equal to]] the category where objects are posemigroups and morphisms are submultiplicative order-preserving mappings.

A quantale (cf. [8]) is a posemigroup (S, *, [less than or equal to]) such that

(1) the poset (S, [less than or equal to]) is a complete lattice;

(2) s(VM) = V{sm|m [member of] M} and (VM)s = V{ms|m [member of] M} for each subset M of S and each s [member of] S.

We note that compatibility of multiplication and order actually follows from condition (2). Indeed, if s, a, b [member of] S and a [less than or equal to] b, then sb = s(V{a, b}) = V{sa, sb}, and so sa [less than or equal to] sb. Similarly as [less than or equal to] bs.

2. INJECTIVE POSEMIGROUPS

Let [E.sub.[less than or equal to]] denote the class of all morphisms h: A [right arrow] B in the category [PoSgr.sub.[less than or equal to]] which are order-preserving, submultiplicative, and satisfy the following condition: h([a.sub.1]) ... h([a.sub.n]) [less than or equal to] h(a) implies [a.sub.1] ... [a.sub.n] [less than or equal to] a for all [a.sub.1], ..., [a.sub.n], a [member of] A. Each such morphism is necessarily an order-embedding. In this section we show that [E.sub.[less than or equal to]]-injective objects in the category [PoSgr.sub.[less than or equal to]] are precisely the quantales. This is largely a restatement of arguments from [6] for posemigroups.

Proposition 1. Quantales are [E.sub.[less than or equal to]]-injective objects in the category [PoSgr.sub.<].

Proof. Let S be a quantale, h: A [right arrow] B be a morphism in [E.sub.[less than or equal to]], and let f: A [right arrow] S be a morphism in [PoSgr.sub.[less than or equal to]]. Define a mapping g: B [right arrow] S by

g(b) = V {f ([a.sub.1]) ... f([a.sub.n])|h([a.sub.1]) ... h([a.sub.n]) [less than or equal to] b, [a.sub.1], ..., [a.sub.n] [member of] A},

for any b [member of] B. Then g is clearly an order-preserving mapping. The fact that g is submultiplicative and satisfies gh = f follows from the proof of [6] Theorem 4.1.

Proposition 2. In the category [PoSgr.sub.[less than or equal to]], every retract of a quantale is a quantale.

Proof. Let (E, [omicron], [[less than or equal to].sub.E]) be a quantale and let (S, *, [[less than or equal to].sub.S]) be a retract of E. Then there exist submultiplicative order-preserving mappings i: S [right arrow] E and g: E [right arrow] S such that gi = i[d.sub.S], where i[d.sub.S] is the identity mapping on S. It is obvious that (S, [[less than or equal to].sub.S]) is complete.

Let s [member of] S and M [subset or equal to] S. Obviously, s(VM) is an upper bound of the set {sm|m [member of] M}. Suppose that u is an upper bound of {sm|m [member of] M} in S. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

So s(VM) is the least upper bound of {sm|m [member of] M}, that is,

s (VM) = V{sm|m [member of] M}.

Similarly one can prove the equality

(VM)s = V{ms|m [member of] M}. []

A subset A of a poset (S, [less than or equal to]) is said to be a down-set if s [member of] A whenever s [less than or equal to] a for s [member of] S, a [member of] A. For any I [subset or equal to] S, we denote by 11 the down-set {x [member of] S|x [less than or equal to] i for some i [member of] I} and by a[down arrow] the down-set {s [member of] S|s [less than or equal to] a} for a [member of] S.

Now one can construct an [E.sub.[less than or equal to]]-injective posemigroup starting from any posemigroup.

Let (S, *, [less than or equal to]) be a posemigroup, and let P(S) be the set of all down-sets of S. Define a multiplication * on P(S) by

I x J = (IJ)[down arrow] = {x [member of] S|x [less than or equal to] ij for some i [member of] I, j [member of] J}. (1)

As in [6], (P(S), *, [subset or equal to]) is a quantale. Hence, by Proposition 1 we have the following result.

Proposition 3. Let (S, *, [less than or equal to]) be a posemigroup. Then (P(S), *, [subset or equal to]) is [E.sub.[less than or equal to]]-injective in the category [PoSgr.sub.[less than or equal to]].

Theorem 4. Let (S, *, [less than or equal to]) be a posemigroup. Then S is [E.sub.[less than or equal to]]-injective in [PoSgr.sub.[less than or equal to]] if and only if S is a quantale.

Proof. Sufficiency follows by Proposition 1.

Necessity. The mapping [eta]: (S, *, [less than or equal to]) [right arrow] (P(S), *, [subset or equal to]), given by [eta](a) = a[down arrow] for each a [member of] S, is clearly an order-embedding of the poset (S, [less than or equal to]) into the poset (P(S), [subset or equal to]). It is routine to check that [eta] preserves multiplication and hence [eta] is also submultiplicative. Being a multiplicative order-embedding, [eta] belongs to [E.sub.[less than or equal to]].

Since S is [E.sub.[less than or equal to]]-injective by assumption, there exists g: P(S) [right arrow] S such that g[eta] = i[d.sub.S], so S is a retract of P(S). Consequently, (S, *, [less than or equal to]) is a quantale by Proposition 2.

3. ON INJECTIVE HULLS OF POSEMIGROUPS

In this section we show that, for a certain class of posemigroups, [E.sub.[less than or equal to]]-injective hulls exist. This class will include all pomonoids, but not only those. Similarly to Proposition 2.1 in [6] it can be shown that [E.sub.[less than or equal to]]-injective hulls are unique up to isomorphism.

For any down-set I of a posemigroup S we define its closure by

cl(I) := {x [member of] S|aIc [subset or equal to] b[down arrow] implies axc [less than or equal to] b for all a, b, c [member of] S}.

Let I be a down-set and s [less than or equal to] x [member of] cl (I). Suppose that aIc [subset or equal to] b[down arrow]. Since x [member of] cl (I), axc [less than or equal to] b. But then also asc [less than or equal to] axc [less than or equal to] b, which means that s [member of] cl (I). Thus cl (I) is a down-set and we may consider the mapping cl: P(S) [right arrow] P(S).

Recall (see [8], Definition 3.1.1) that a quantic nucleus on a quantale Q is a submultiplicative closure operator on Q.

Lemma 5. The mapping cl is a quantic nucleus on the quantale P(S).

Proof. First, let us show that cl is a closure operator.

If alc [subset or equal to] b[down arrow], then clearly axc [less than or equal to] b for every x [member of] I. Hence I [subset or equal to] cl(I) and cl is extensive.

Let I [subset or equal to] J, x [member of] cl (I), and aJc [subset or equal to] b[down arrow]. Then we have aIc [subset or equal to] aJc [subset or equal to] b[down arrow] and hence axc [less than or equal to] b. So x [member of] cl(J) and we have proven that cl is order-preserving.

The inclusion cl (I) [subset or equal to] cl (cl (I)) holds because cl is extensive and order-preserving. Conversely, suppose that aIc [subset or equal to] b[down arrow] and y [member of] cl (cl (I)). Then axc [less than or equal to] b for any x [member of] cl (I). This means acl(I)c [subset or equal to] b[down arrow]. So ayc [less than or equal to] b by the definition of cl, and y [member of] cl(I). Thus cl is also idempotent, and therefore a closure operator.

It remains to prove that cl is a submultiplicative mapping. To this end, let us first prove that cl (I) x J [subset or equal to] cl (I * J) for all I, J [member of] P (S). Take z [member of] cl (I) x J and suppose that a(I x J)c [subset or equal to] b[down arrow]. Then for any j [member of] J the inclusion Ij [subset or equal to] I x J implies that aI(jc) [member of] b[down arrow]. We have z [less than or equal to] mj for some m [member of] cl(I) and j [member of] J. So am(jc) [less than or equal to] b. It follows that azc [less than or equal to] amjc [less than or equal to] b, which results in z [member of] cl(I x J), as needed.

Similarly, I x cl(J) [subset or equal to] cl (I x J) holds. Consequently, we obtain that

cl(I) x cl(J) [subset or equal to] cl (I x cl(J)) [subset or equal to] cl(cl (I x J)) = cl (I x J). []

One can immediately get the following corollary.

Corollary 6. For a posemigroup S and I, J [member of] P (S), we have

cl(cl(I) x cl(J)) = cl(I x J).

We put

Q(S) := {I [member of] P(S)|I = cl(I)}

and define a multiplication [omicron] on Q(S) by

I [omicron] J := cl(I x J). (2)

By Theorem 3.1.1 of [8] we immediately have that, for every posemigroup S, (Q(S), [omicron], C) is a quantale which is the image of the quantic nucleus cl. From Theorem 4 we conclude that Q(S) is [E.sub.[less than or equal to]]-injective in the category [PoSgr.sub.[less than or equal to]].

Now we can prove our main result.

Theorem 7. (cf. Theorem 5.8 in [6]). Let S be a posemigroup such that cl(s[down arrow]) = s[down arrow] for every s [member of] S. Then Q(S) is an [E.sub.[less than or equal to]]-injective hull of S in [PoSgr.sub.[less than or equal to]].

Proof. Since cl(s[down arrow]) = s[down arrow], we can consider the mapping [eta]: S [right arrow] Q(S), a [??] a[down arrow]. We shall prove that [eta] is an [E.sub.[less than or equal to]]-essential morphism in [PoSgr.sub.[less than or equal to]].

Let us show that [eta] is a posemigroup homomorphism. Take a, b [member of] S. It is easy to see (see also the proof of Proposition 3.3 in [6]) that (ab)[down arrow] = ((a[down arrow])(b[down arrow]))[down arrow]. Hence, using (2) and (1), we have

[eta](a) [omicron] [eta](b) = cl(a[down arrow] x b[down arrow]) = cl(((a[down arrow])(b[down arrow]))[down arrow]) = cl((ab)[down arrow]) = (ab)[down arrow] = [eta](ab),

i.e. [eta] is a semigroup homomorphism. For every a, b [member of] S, a [less than or equal to] b if and only if a[down arrow] [subset or equal to] b[down arrow]. This means that [eta] is both monotone and an order-embedding. If now [eta]([a.sub.1]) [omicron] ... [omicron] [eta]([a.sub.n]) [subset or equal to] [eta](a), then [eta]([a.sub.1] ... [a.sub.n]) [subset or equal to] [eta](a), which implies [a.sub.1] ... [a.sub.n] [less than or equal to] a. Thus [eta] belongs to [E.sub.[less than or equal to]].

Finally, let [psi]: Q(S) [right arrow] B be a morphism in [PoSgr.sub.[less than or equal to]] such that [psi][eta] [member of] [E.sub.[less than or equal to]]. We have to show that [psi] [member of] [E.sub.[less than or equal to]]. Suppose that [psi]([I.sub.1]) ... [psi]([I.sub.n]) [less than or equal to] [psi](J) in B, where [I.sub.1], ..., [I.sub.n], J [member of] Q(S). First we prove that

([for all]a, b, c [member of] S)(aJc [subset or equal to] b[down arrow] [??] a([I.sub.1] [omicron] ... [omicron] [I.sub.n])c [subset or equal to] b[down arrow]). (3)

Suppose that aJc [subset or equal to] b[down arrow], a, b, c [member of] S. Then also a[down arrow] x J x c[down arrow] [subset or equal to] b[down arrow]. Let us show that

a([I.sub.1] ... [I.sub.n])c [subset or equal to] b[down arrow]. (4)

Take [i.sub.1] [member of] [I.sub.1], ..., [i.sub.n] [member of] [I.sub.n]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [psi][eta] [member of] [E.sub.[less than or equal to]], we conclude that [a.sub.i1] ... [i.sub.n]c [less than or equal to] b. Consequently, a([I.sub.1] ... [I.sub.n])c [subset or equal to] b[down arrow].

Now (4) implies a(([I.sub.1] ... [I.sub.n])l)c [subset or equal to] b[down arrow]. If x [member of] cl(([I.sub.1] ... [I.sub.n])[down arrow]) = cl([I.sub.1] ... [I.sub.n]) = [I.sub.1] [omicron] ... [omicron] In, then a(([I.sub.1] ... [I.sub.n])[down arrow])c [subset or equal to] b[down arrow] implies axc [less than or equal to] b by the definition of closure. Thus we have proven (3).

To prove that [I.sub.1] [omicron] ... [omicron] In [subset or equal to] J, let x [member of] [I.sub.1] [omicron] ... [omicron] In. Suppose that a, b, c [member of] S and aJc [subset or equal to] b[down arrow]. By (3), a([I.sub.1] [omicron] ... [omicron] In)c [subset or equal to] b[down arrow]. Since x [member of] [I.sub.1] [omicron] ... [omicron] [I.sub.n] = cl([I.sub.1] [omicron] ... [omicron] [I.sub.n]), we have axc [less than or equal to] b. Hence x [member of] cl(J) = J. []

It turns out that the assumptions of Theorem 7 are satisfied for several natural classes of posemigroups.

A posemigroup S is negatively ordered (cf. [9]) if st [less than or equal to] s and st [less than or equal to] t for all s, t [member of] S. Negatively ordered semigroups and monoids arise naturally in various semigroup theoretic contexts; see, for example, [3,4,11].

Example 8.

(1) Every lower semilattice with respect to its natural order is negatively ordered.

(2) If S is any semigroup, then the set Id(S) of all its ideals is a negatively ordered posemigroup with respect to inclusion and the product IJ = {ij|i [member of] I, j [member of] J}.

(3) The real interval [0,1] is negatively ordered with respect to the usual multiplication and order of real numbers.

(4) In [7], negatively ordered semigroups with respect to natural partial order in many classes of semigroups are determined.

A semigroup S has weak local units (see, e.g., [5]) if for every s [member of] S there exist u, v [member of] S such that s = su = vs.

Corollary 9. The posemigroup Q(S) is an [E.sub.[less than or equal to]]-injective hull of S in [PoSgr.sub.[less than or equal to]] in any of the following four cases:

(1) S is a pomonoid;

(2) S is a negatively ordered posemigroup with weak local units;

(3) S is a linearly ordered cancellative posemigroup;

(4) S is an upper semilattice with natural order.

Proof. We shall show that the assumption of Theorem 7 is fulfilled in all these cases. Since s[down arrow] [subset or equal to] cl(s[down arrow]) holds always, we have to prove that cl(s[down arrow]) [subset or equal to] s[down arrow] for every s [member of] S.

(1) Suppose that x [member of] cl(s[down arrow]). Since 1 (s[down arrow])1 [subset or equal to] s[down arrow], we have that x = 1 x 1 [less than or equal to] s, that is, x [member of] s[down arrow].

(2) Take x [member of] cl(s[down arrow]). By assumption there exist u, v [member of] S such that x = ux = xv. Since S is negatively ordered, we have usv [less than or equal to] s. This implies u(s[down arrow])v [subset or equal to] s[down arrow], and hence, by the definition of cl(s[down arrow]), x = uxv [less than or equal to] s. Thus, cl(s[down arrow]) [subset or equal to] s[down arrow], as needed.

(3) To prove that cl(s[down arrow]) [subset or equal to] s[down arrow] for every s [member of] S, we show that x [not member of] s[down arrow] implies x [not member of] cl(s[down arrow]) for every x [member of] S. So let x [member of] s[down arrow], i.e. s < x. Suppose that x [member of] cl(s[down arrow]). Choose arbitrary a, c [member of] S, and put b := asc. Then a(s[down arrow])c [subset or equal to] b[down arrow], and hence axc [less than or equal to] b, because x [member of] cl(s[down arrow]). Consequently, b = asc [less than or equal to] axc [less than or equal to] b, which gives asc = axc. Cancelling a and c, we obtain s = x, contradicting inequality s < x. Thus x [not member of] cl(s[down arrow]).

(4) Let (S, V, [less than or equal to]) be an upper semilattice with its natural order. Assume x [member of] cl(s[down arrow]). Since s [disjunction] (s[down arrow]) [disjunction] s [subset or equal to] s[down arrow], it follows that s [disjunction] x [disjunction] s [less than or equal to] s. Hence x [member of] s[down arrow].

Example 10. Both additive and multiplicative posemigroups of natural numbers are linearly ordered and cancellative. Note that neither of them is a pomonoid or negatively ordered.

There exist semigroups S for which Q(S) is not an [E.sub.[less than or equal to]]-injective hull of S in [PoSgr.sub.[less than or equal to]].

Example 11. Let S = {a, b, c} be a left zero semigroup with the ordering a [less than or equal to] c, b [less than or equal to] c. Then

P(S) = {a[down arrow], b[down arrow], c[down arrow], [empty set], {a, b}},

where a[down arrow] = {a} and cl(a[down arrow]) = S [not equal to] a[down arrow]. In fact, cl(a[down arrow]) = cl(b[down arrow]) = cl(c[down arrow]) = cl({a, b}) = S. The reason is that for any u, v, w [member of] S and nonempty I [member of] P(S), if uIv = {u} [subset or equal to] w[down arrow], then uxv = u [less than or equal to] w for any x [member of] S. Therefore, Q(S) = {S, [empty set]} and there is no [E.sub.[less than or equal to]]-essential morphism from S to Q(S), because such a morphism would have to be an order-embedding and hence injective. Consequently, Q(S) is not an [E.sub.[less than or equal to]]-injective hull of S in [PoSgr.sub.[less than or equal to]].

As the last thing we show that Theorem 5.8 of [6] follows from Theorem 7.

Let [PoMon.sup.1.sub.[less than or equal to]] be the category where objects are pomonoids and morphisms are submultiplicative order-preserving mappings which preserve identity (this is the category considered in [6]). Thus [PoMon.sup.1.sub.[less than or equal to]] is a subcategory of [PoSgr.sub.[less than or equal to]]. By [E.sup.1.sub.[less than or equal to]] we denote the class of those morphisms which belong to [PoMon.sup.1.sub.[less than or equal to]] and [E.sub.[less than or equal to]].

Corollary 12. Let S be a pomonoid. Then Q(S) is an [E.sup.1.sub.[less than or equal to]]-injective hull of S in the category [PoMon.sup.1.sub.[less than or equal to]].

Proof. From the proof of Corollary 9 we know that cl(s[down arrow]) = s[down arrow] for every s [member of] S. Observe that Q(S) is a pomonoid with the identity element 1[down arrow]. Let us show that Q(S) is [E.sup.1.sub.[less than or equal to]]-injective in the category [PoMon.sup.1.sub.[less than or equal to]]. Consider a morphism h: A [right arrow] B in [E.sup.1.sub.[less than or equal to]] and any morphism f: A [right arrow] Q(S) in [PoMon.sup.1.sub.[less than or equal to]]. Since Q(S) is [E.sub.[less than or equal to]]-injective in [PoSgr.sub.[less than or equal to]], there exists g: B [right arrow] Q(S) in [PoSgr.sub.[less than or equal to]] such that gh = f. Then 1[down arrow] = f (1) = (gh)(1) = g(1) and g is a morphism in [PoMon.sup.1.sub.[less than or equal to]]. A similar argument shows that [eta]: S [right arrow] Q(S), s [??] s[down arrow] is an [E.sup.1.sub.[less than or equal to]]-essential morphismin [PoMon.sup.1.sub.[less than or equal to]].

As the authors of [6] mention, in the category of pomonoids it would be natural to require 1 [less than or equal to] f (1) instead of 1 = f (1) from a morphism f. It is an open problem if in such a category injective hulls can be constructed in a similar way.

doi: 10.3176/proc.2014.4.02

Received 4 April 2014, accepted 13 August 2014, available online 20 November 2014

ACKNOWLEDGEMENTS

Research of the first named author was supported by the Specialized Research Fund for the Doctoral Program of Higher Education of Ministry of China (20124407120004), the National Natural Science Foundation of China (11171118), the Program on International Cooperation and Innovation, Department of Education, Guangdong Province (2012gjhz0007), and the Foundation for Distinguished Young Talents in Higher Education of Guangdong ([2008]342). Research of the second named author was supported by the Estonian Science Foundation grant No. 8394 and Estonian Institutional Research Project IUT20-57.

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Xia Zhang (a,b) * and Valdis Laan (c)

(a) School of Mathematical Sciences, South China Normal University, 510631 Guangzhou, China

(b) Department of Mathematics, Southern Illinois University Carbondale, 62901 Carbondale, USA

(c) Institute of Mathematics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia

* Corresponding author, xiazhang@scnu.edu.cn, xiazhang- 1@yahoo.com