Schreier split extensions of preordered monoids

Properties of preordered monoids are investigated and important subclasses of such structures are studied. The corresponding full subcategories of the category of preordered monoids are functorially related between them as well as with the categories of preordered sets and monoids. Schreier split extensions are described in the full subcategory of preordered monoids whose preorder is determined by the corresponding positive cone.


Introduction
Preordered monoids are monoids equipped with a preorder compatible with the monoid operation.They are relevant tools in many areas, for instance, in computer science where they are used in the theory of language recognition (see [24]), as well as in non-classical logics, namely in fuzzy logics (see [8] and [11]).
Many fundamental results have been obtained by switching from categories of monoids to categories of preordered or ordered monoids, and the same for semigroups.Examples of this fact are new proofs of two remarkable results that we mention next.
A celebrated result of I. Simon [25] on the classification of recognizable languages in terms of J -triviality of the corresponding syntactic monoids has a radically new proof in [26], where it is proved that every finite J -trivial monoid (for the Green's J -equivalence relation [6]) is a quotient of an ordered monoid satisfying the identity x ≤ 1.In [9], the authors give another proof of this result and explain its relevance in the theory of finite semigroups.A systematic use of ordered monoids in language theory, was initiated by J.-E.Pin [20] and developed in [21], [22] and other subsequent papers.
The second example is a new proof of a well-known and important result of A. Tarski that gives a criterion for the existence of a monoid homomorphism from a given commutative monoid A to the extended positive real line R + that sends a fixed element a ∈ A to 1.In [27], F. Wehrung proves that this is a Hahn-Banach type property, stating the injectivity of R + , not in the category of commutative monoids, where there are no nontrivial injectives, but in the category of commutative monoids equipped with a preorder that makes every element positive, called there "positively ordered monoids" or P.O.M., for short.Preordered monoids have a much richer diversity of features than preordered groups.In contrast with the case of preordered groups, in preordered monoids the submonoid of positive elements, called the positive cone, neither determines the preorder nor is a cancellative monoid, in general.These features of preordered groups are rescued in the new context by considering convenient subcategories of the category of preordered monoids, OrdMon, satisfying these properties or appropriate generalizations, covering a wide range of structures.
In particular, the failure of the first property gives rise to a classification of preordered monoids according to the relation between its preorder and the preorder induced by the corresponding positive cone considered here that is, for P.O.M., the opposite of Green's preorder L as explained in Section 2. Furthermore, this last preorder may or may not be compatible with the monoid operation.The characterization of the positive cones inducing compatible preorders provides a reason as to why the commutativity of the underlying monoid is often assumed in the literature.
This classification gives rise to several categories and functors between them, some of them being part of adjoint situations.
The cancellation property is often replaced by weaker conditions like the "pseudo-cancellation" introduced in [27] that plays an important role in the characterization of the injective objects presented there.
Let Ord be the category of preordered sets and monotone maps, and Mon the category of monoids and monoid homomorphisms.We recall that the forgetful functor Ord → Set is a topological functor [10] (like the one from the category of topological spaces to the category of sets it has initial and final structures) and that the forgetful functor Mon → Set (such as the underlying functor of any variety of algebras) is monadic [17, p. 156].We prove that the forgetful functors from OrdMon to Mon and to Ord are topological and monadic functors, respectively, and derive some consequences of these facts.
Due to the fact that OrdMon is the category Mon(Ord) of internal monoids in Ord (which fails to be so in OrdGrp), we show that the construction of the left adjoint to U 1 : OrdMon = Mon(Ord) → Ord as well as its monadicity can be derived from general results for the forgetful functor Mon(C) → C, when C is a symmetric monoidal category satisfying some additional conditions [12,13,23].
It is well known that in the category of groups there is an equivalence between group actions and split extensions (which are, in this case, nothing but split epimorphisms), obtained via the semidirect product construction.
Schreier split extensions of monoids, that first appeared in [18], correspond to an important class of split epimorphisms of monoids, the Schreier split epimorphisms (whose name was inspired by the Schreier internal categories in monoids introduced by Patchkoria in [19]).Indeed, they are exactly those split epimorphisms that correspond to monoid actions: an action of a monoid B on a monoid X being a monoid homomorphism ϕ : B → End( X) from B to the monoid of endomorphisms of X .Also this class of split epimorphisms has essentially all homological and algebraic properties of the split homomorphisms in groups (see [2] and [3]).
Schreier split extensions have already been defined in categories of monoids with operations [18] and in the categories of cancellative conjugation monoids [5].
In this paper we describe Schreier split extensions in the full subcategory OrdMon * of OrdMon with objects all preordered monoids whose preorder is induced by the corresponding positive cone.
In [4] the structure of the split extensions in the category of preordered groups is studied and the case where the restriction to the positive cones gives a Schreier split epimorphism in Mon is analysed.Also the behaviour of the category Mon(Ord) and, more generally, the one Mon(C) where C satisfies suitable conditions, is considered in the last section.
This paper is organized as follows.In Section 2 we give several examples of preordered monoids and characterize the submonoids of a monoid A that induce a compatible preorder in A. Some full subcategories of the category OrdMon are defined and an isomorphism is established between the full subcategory OrdMon * and the category RNMono(Mon) of right normal monomorphisms in monoids, in a sense introduced there, which plays a central role to obtain the main result of the last section.In Section 3 we study in detail functorial relations between the main categories involved in this paper that, being quite simple, give much information about these categories.We also include a brief but complete account of the general categorical results from which they can be derived.In Section 4 we introduce the notion of Schreier split extensions in the category OrdMon * and show how they are related with what we call preordered actions via an appropriate concept of semidirect product.Finally, we point out some special cases and present an example that helps to show the real character of the notions introduced.Throughout we will denote preordered monoids additively, say by (A, +, 0, ≤) where the monoid (A, +, 0) is not necessarily commutative and ≤ is a preorder compatible with +, that is, where For concepts of category theory that are not defined here we suggest MacLane's book [17].

The category of preordered monoids
We start by recalling that if (A, +, 0, ≤) is a preordered group, i.e. (A, +, 0) is a (not necessarily abelian) group and the preorder ≤ is compatible with the group operation In this case, defining we have that ≤ coincides with ≤ P and P + a = a + P , because P is closed under conjugation: If P is the submonoid of positive elements in a preorder monoid, we define the relation ≤ P by and get a preorder ≤ P which is contained in the original preorder.
Proof.We have that 0 ∈ P and if a, b ∈ P then a ≥ 0 and b ≥ 0 implies that a + b ≥ 0 and so P is a submonoid of A.
The converse of this result is false, in general, as the following example shows. .
In the previous example one can easily check that ≤ P is compatible with + and so (A, +, 0, ≤ P ) is also a preordered monoid.The following example shows that this is not always the case.
Example 2. We consider the monoid (A, +, 0) with addition table + 0 1 2 3 4 0 0 1 2 3 is not compatible with the monoid operation.Indeed, 2 ≥ P 0 and 1 The following is an example of a preordered monoid where the two preorders coincide.which is exactly ≤ P , i.e. ≤ is the same as ≤ P .
Now we characterize the submonoids of a preordered monoid which induce a compatible preorder.
Definition 1.Given a monoid A and a submonoid M of A we say that M is -normal if it is both right and left normal.
Proposition 2. Let P be the positive cone of a preordered monoid (A, +, 0, ≤).Then the monoid operation is monotone with respect to ≤ P if and only if P is right normal.
Proof.If ≤ P is compatible with + and b = a + x with x ∈ P then x ≥ P 0 and a ≥ P a =⇒ b = a + x ≥ P a and so there exists an y ∈ P such that a + x = y + a, i.e. a + P ⊆ P + a. Conversely, if a ≤ P b and c ≤ P d then b = x + a and d = y + c, for some x, y ∈ P and so, because P is right normal, we can find z ∈ P for which a In Example 1 we have P = A, the so-called positively preordered monoid, and the left and right cosets are the following {3,4} 4 {4} {4} Since P is right normal (for all a ∈ A, a + A ⊆ A + a) then ≤ P is compatible with +.
In Example 2, again P = A but P is not right normal and so ≤ P is not compatible with +.
We remark that, in this case, A is not right normal in itself but it is left normal ( A + a ⊆ a + A, for every a ∈ A) and so if we consider the preorder a ≤ P b ⇐⇒ b ∈ a + P then, using a result similar to the one of Proposition 2, we conclude that (A, +, 0, ≤ P ) ∈ OrdMon.
and the same for ≤ R .
The positive cone of a commutative preordered monoid need not determine the preorder: for + 0 1 2 0 0 1 2 1 1 1 1 2 2 1 1 with P = A and ≤ as sketched below Proof.To each preorder commutative monoid we can associate a special one with the preorder induced by its positive cone.This is expressed by saying that the subcategory is coreflective.Indeed, if (A, +, 0, ≤) is a preordered commutative monoid and P is its positive cone then, by Corollary 1, (A, +, 0, ≤ P ) ∈ OrdCMon * .Furthermore, the identity morphism on A defines a monotone one c (A,≤) : (A, ≤ P ) → (A, ≤), by Proposition 1.
We prove that it is the coreflection of (A, +, 0, ≤) in OrdMon * .Indeed, given a morphism f : Hence, f (a 1 ) ≤ P f (a 2 ) and so f (a 1 ) ≤ P f (a 2 ) for all a 1 ≤ P a 2 in A .Thus OrdCMon * being a full coreflective subcategory is closed under colimits in OrdCMon.
Definition 2. We say that a monomorphism m : S → A of monoids is right normal if its image m(S) is a right normal submonoid of A and denote by RNMono(Mon) the corresponding full subcategory of the category of monomorphisms of monoids, Mono(Mon).
Example 2 shows that the identity morphism may not be a right normal monomorphism.
Theorem 1.The category OrdMon * is isomorphic to the one of right normal monomorphisms in Mon, RNMono(Mon).
(1) The set of all R-submodules of a module A over a ring R, equipped with the "Minkowski sum" and the order defined by the inclusion.Indeed, in this case every element is positive (i.e. the positive cone is the set of all R-submodules) and U ⊆ V if and only if V = V + U .
(2) All injective objects in OrdMon with respect to embeddings (not to monomorphisms) are objects in OrdMon * .In fact, let M be the submonoid of the monoid N × N, generated by (1, 0) and (1, 1) with the order induced by the product order and i : M → N × N the embedding.If a ≤ b in an injective object A then there exists a (unique) morphism in OrdMon, u : M → A such that u(1, 0) = a and u(1, 1) = b, defined by u(n + m, m) = na + mb, for every n, m ∈ N. By injectivity of A, there exists a morphism v : Then taking c = v(0, 1) we have that b = c + a ∈ P + a and so the preorder in A coincides with the one induced by its positive cone.Indeed, since (0, 0) ≤ (0, 1) and v preserves the order then 0 ≤ c.
Let OrdMon be the full subcategory of OrdMon with objects all preordered monoids whose positive cone is a right normal monoid.Note that Example 1 describes an object of OrdMon that does not belong to OrdMon * Proposition 4. The category OrdMon * is coreflective in OrdMon .
Proof.Essentially the same as the one of Proposition 3.

RNMono(Mon)
where OrdMon * is coreflective in OrdMon but OrdMon is not coreflective in OrdMon as we prove in the following section.

The forgetful functors
Let us consider the following commutative diagram of forgetful functors

Set
where V 2 is topological and V 1 is a monadic functor.We are going to prove that also U 2 is a topological functor and U 1 is a monadic one.
We recall that U : C → D is a topological functor if every family ( f i : D → U (C i )) i∈I , where I may be a proper class, has a unique U -initial lift: (i) there exists a family ( for each i in I , there exists a unique morphism The uniqueness, up to isomorphism, of the U -lift comes from the uniqueness of h. If U : C → D is a topological functor then the same holds for its dual, U op : C op → D op , which means that every family ( f i : U (C i ) → D) i∈I has a unique U -final lift, a generalization of the well-known fact that each meet-complete partially ordered set is also join-complete.This implies that a topological functor has a left adjoint (the discrete object functor) and a right adjoint (the indiscrete object functor).
In this case we say that the category C is topological over D which is a powerful condition with nice consequences.It is easy to prove that the functor V 2 is topological.However, if we replace Ord by the category Pos of partially ordered sets then it is no longer the case.This is a reason why the category of preordered sets is better behaved than the one of partially ordered sets, for our purposes.Proposition 5.The functor U 2 : OrdMon → Mon is a topological functor.
Proof.Given a family of monoid homomorphisms we obtain a preorder which, in addition, is compatible with the monoid operation: It is easy to check that condition (ii) above holds and so that the family has a unique U 2 -initial lift.
From this we conclude that: (1) U 2 has a left and a right adjoint defined by equipping each monoid with the discrete and the total preorder, respectively; (2) OrdMon is complete and cocomplete, since Mon is complete and cocomplete, and U 2 preserves limits and colimits.Proposition 6.The functor U 1 : OrdMon → Ord has a left adjoint.
Proof.For (X, ≤), let F 1 (X, ≤) = (X * , •, , ≤), where X * is the set of all words in the alphabet X with the operation of concatenation, having the empty word as identity (i.e., X * is the free monoid on the set X ), equipped with the preorder if and only if n = m and w i ≤ w i for i = 1, 2, . . ., n.In this way we define a preorder compatible with concatenation.
The morphism Consequently, this defines a functor that is left adjoint of U 1 with unit η. • U 1 reflects isomorphisms; • OrdMon has and U 1 preserves coequalizers of all parallel pairs ( f , g) such that its image under U 1 , (U 1 ( f ), U 1 (g)), has a contractible coequalizer in Ord.
Given a morphism f : (A, +, 0, ≤) → (B, +, 0, ≤) in OrdMon such that U 1 ( f ) is an isomorphism in Ord then, being also a bijective homomorphism of monoids, it is an isomorphism of monoids and so it is also an isomorphism in OrdMon.Hence U 1 reflects isomorphisms.
Let us assume that (U 1 ( f ), U 1 (g)) has a contractible coequalizer (U 1 ( f ), U 1 (g), h; i, j) in Ord.We have to prove that the unique morphism t ∈ Ord such that t Indeed, by definition of the preorder in C , there exists a zig-zag in Proposition 8.The subcategory OrdMon is not coreflective in the category OrdMon.

OrdMon
where U 1 is the restriction of U 1 to OrdMon , F 1 is the corestriction of F 1 giving a left adjoint to U 1 , and T is the monad that both adjunctions induce in Ord.
From the above we conclude that OrdMon cannot be coreflective in OrdMon otherwise, being closed under coequalizers, U 1 would be monadic and so OrdMon ∼ = Ord T ∼ = OrdMon which is false as Example 2 shows.
Direct proofs presented in this section are simple and informative about the categories involved.However, since OrdMon is the category Mon(Ord) of internal monoids in the category of preordered sets (which is not true for preordered groups) these results can be derived from more general ones relative to categories of models of the theory of monoids in monoidal categories.In our case, since Ord is a cartesian closed category which, furthermore, is locally finitely presentable (see [1]), the construction of the left adjoint of U 1 : OrdMon = Mon(Ord) → Ord is a special case of the construction of the left adjoint of the forgetful functor of Mon(C) → C, when C is a symmetric monoidal category, satisfying some additional conditions, presented by G. M. Kelly in [12], see also [13].Also the monadicity of U 1 was proved by H. Porst [23,Cor. 2.6].
In more detail, S. Lack [13] proves that the forgetful functor of Mon(C) → C has a left adjoint when C is a symmetric monoidal category with countable coproducts that are preserved by tensoring on either side, with the free monoid over an object X ∈ C given by 1 + X + X 2 + • • • where X n means the n-fold tensor product of X .
H. Porst [23] deals with "admissible monoidal categories" which are locally presentable categories that, in addition, are symmetric monoidal with the property that tensoring by a fixed object defines a finitary functor (i.e., a functor preserving directed colimits).
(2) the right normal submonoid of A, P A = P ξ , is defined by This gives a Schreier split extension in RNMono(Mon).Indeed: (a) P ξ is a submonoid of X ϕ B by (A3) and the fact that P B is a monoid.(b) The right normality of P A comes from (A4).
Then H G ∼ = 1 S : in the diagram It is easy to check that also G H = 1 A , thus giving the desired equivalence of categories.
Finally, we point out two interesting special cases: • When q is a monotone map then it restricts to q : P A → P X and ξ is trivial, in the sense that ξ(x, b) = x when x ∈ P X and b ∈ P B and it is zero otherwise.In this case, the upper row of the diagram (2) is a Schreier split epimorphism of monoids and hence P A is isomorphic to the semidirect product P X × φ P B .
• When q is an homomorphism then the monoid action ϕ is trivial, i.e. ϕ b (x) = x, for all b ∈ B. However, we may still have a non trivial ξ in this case, as the following example shows.
In the diagram (2) if q is a monoid homomorphism then A ∼ = X × B but the upper row need not be a Schreier split epimorphism.
which is an example of a Schreier split epimorphism in the category RNMono(Mon).The left Z has the discrete order because its positive cone is {0}, while the one on the right has the usual order since its positive cone is N.The positive cone N × N and the corresponding order on Z × Z will be described below.
In this case we have a non trivial ξ : Z × N → Z, defined by ξ(u, v) = u if u ∈ N and u ≤ v 0 otherwise giving a preordered action (Z, Z, {0}, N, ϕ, ξ) where ϕ is trivial, which induces a Schreier split extension in RNMono(Mon)

Example 1 .
Let (A, +, 0) be the monoid with the following addition table + with the preorder ≤ with P = A and generated by the following diagram (where the arrows from zero have been omitted)

Example 3 .
Let (A, +, 0) be the monoid of Example 1 now with a different positive cone, P = {0, 1}, and the preorder

Remark 1 .
For a submonoid M of a monoid A, we can define two preorders on A a ≤ M b ⇔ b ∈ M + a and a≤ M b ⇔ b ∈ a + M, whose positive cones are precisely M. When M = A we have that ≤ M = ≤ op L and ≤ M = ≤ op R , where L and R are the Green's relations defined, in additive notation, by

B
since β(x, b) = k(x) + s(b), by definition of P ξ , we conclude that β : P ξ → P A is an isomorphism.

Example 4 .
Let us consider the following diagram {0} ≤ 0, and P = A, that is, 0 ≤ x, for all x ∈ A. It is easy to check that (A, +, 0, ≤) is a preordered monoid.However, ≤ P being the following preorder, plus 3 ≤ P 0, 3