Some results on pseudomonoids

In this paper we follow the work of López and Masa (The problem 31.10) given in [1]. The purpose of this work is to study the new concepts of pseudomonoids. We also obtain some interesting results.


Preliminaries
In mathematics, a pseudogroup is an extension of the group.But one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra (quasigroup, for example).A theory of pseudogroups developed by Élie Cartan in the early 1900's.
Recall that a transformation group G on X is a subgroup in a group Hom(X), so each g ∈ G is a diffeomorphism of X.A local homeomorphism is a homeomorphism f : U −→ V, where U and V are open subsets in X. Certainly, the set of local homeomorphisms is not a group of transformations because the composition is not well-defined.In studying a geometrical structure, it is fruitful to study its, group of automorphisms.Usually, these automorphisms are not globally defined.They therefore do not form a group in the present sense of the word but rather a pseudogroup, or the more general case, pseudomonoid.
In general, pseudogroups were studied as a possible theory of infinite-dimensional Lie groups.The concept of a local Lie group, namely a pseudogroup of functions defined in neighborhoods of the origin of E, is actually closer to Lie's original concepts of Lie group, in the case where the transformations involved depend on a finite number of parameters, than the contemporary definition via manifolds.
Examples of infinite-dimensional pseudogroups abound, beginning with the pseudogroup of all diffeomorphism of E. The interest in mainly in sub-pseudogroup of the diffeomorphisms, and therefore with objects that have a Lie algebra analogue of vector fields.The methods proposed by Lie and Cartan for studying these objects have become more practical given the progress of computer algebra.A generalization of the notion of transformation group is a pseudogroup.We summarize and develop some basics of pseudomonoids in chapter two.Here, we introduce the notion of pseudogroup, following chapter 3 of W. Thurston [12] and [8,9].Definition 1.1.(Pseudogroup) Let X be a topological space.Let Γ be a local homeomorphisms set f : U f −→ V f where U f , V f are open subsets of X.The set Γ is called a pseudogroup on X if satisfying the following conditions: (1) The identity belong to Γ. (2 Example 1.2.[5,6] The natural examples of pseudogroups are: (1) Let G be a Lie group acting on a manifold X.Then we define the pseudogroup as the set of all pairs (G|U, U ) where U is the set of all open subsets of X, (2) Pseudogroup of automorphisms of a tensor field, Next we are going to define the notion of a pseudomonoid is a collection of transformations which is closed under certain properties.

Pseudomonoid
We now change our focus from pseudogroup to pseudomonoid.We want to introduce a generalization of pseudogroups.In the new concept, we change the homeomorphism maps to continuous maps.In this case, the elements of Γ aren't necessarily invertible.So part (3) of the Definition 1.1 can't be defined in the new concept.Thus we define the pseudomonoid following.Definition 2.1.Let X be a topological space and let Γ be a set of locally continuous functions f : U f −→ X where U f is an open subset of X.The set Γ is called a pseudomonoid on X if satisfying the following conditions: (1) The identity belong to Γ. ( The pair (X, Γ) is called a pseudomonoid.
Remark 2.2.Any pseudogroup on X is a pseudomonoid on X also.

Example 2.3. We can construct many examples:
(1) For any topology space X all locally continuous functions on X form a pseudomonoid on its.This pseudomonoid is largest pseudomonoid on X.In the other words, any pseudomonoid on X is a subset of its.
(2) All linear maps on a Banach space E and their restriction to open subsets.

Basic definitions and notations
Before we define some concepts of pseudomonoid for a topological space, we need to introduce the following definitions.See below for more details.A C r -pseudomonoid, is a pseudomonoid Γ on a C r -manifold that Γ contains C r -maps (with 0 ≤ r ≤ ∞).
We say S ⊂ Γ generates Γ and write Γ =< S >, if composition and restriction to open subsets in S obtain Γ (expect identity).The orbit of a point x ∈ X is the set Γ(x) := {f (x) : f ∈ Γ and x ∈ U f }.
We will denote by X the set of all orbits of (X, Γ).The stability or isotropy of x, show by Γ x , contain all f ∈ Γ leaving x fixed: We will define on X the following equivalence relation: "xRy if and only if there exists a path of x to y".The equivalence classes of R are said leaves of (X, Γ).We will denote by L x the leaf contain of x ∈ X.The notation X/Γ means the quotient space of R with quotient topology.
The union of leaves that meet X 0 ⊆ X is called the saturation of X 0 and denoted by Γ(X 0 ).We say the set X 0 invariant or saturated, if Γ is called transitive if for any x, y ∈ X, there exists f ∈ Γ that f (x) = y.A sub-pseudomonoid of Γ is a subset Σ ⊆ Γ which is a pseudomonoid on X also.
Example 2.5.(1) Let Γ be the set of all locally continuous maps of X.Then any pseudomonoid on X is a sub-pseudomonoid of Γ. (2) Let Γ be contain of identity and all identity restrictions to open subsets of X.Then Γ is a subpseudomonoid of any pseudomonoid on X. (3) Let M be a smooth manifold and let C r (M ) be all locally C r -maps on M.
Proof.Λ stands for the set.We have divided the proof into two parts: First, we show that Λ is a pseudomonoid, next we will prove that Λ =< Γ x >.Clearly, identity belongs to Λ and (2) of pseudomonoid definition is valid by the definition of Λ.Let 3) of pseudomonoid definition is valid.Finally, we conclude that Λ is a pseudomonoid.Obviously, Γ x is a subset of Λ.Since Λ is a pseudomonoid and Γ x is a subset of Λ, we have < Γ x >⊆ Λ.By definition, Λ ⊆< Γ x >, So Λ =< Γ x > and this finishes the proof.
Lemma 2.7.Let S be a subset of Γ and S generates Γ.Then Proof.Since X 0 is invariant.Consequently, X 0 is the union of some leaves of Γ.Thus X c 0 is union some leaves in Γ that do not intersect with X 0 , therefore X c 0 is invariant.
Example 2.10.In following, we give several examples of pseudomonoids.
(1) Consider a pseudomonoid (X, Γ) where X = R n with Γ =< f >, which f is defined by Then the isotropy of a ∈ S n is Γ a = {id| U : a ∈ U ⊆ S n } , and we have X/Γ ∼ = RP n , because L a = Γ(a) = {a, −a}.
(3) Suppose that X = R n \{0} and consider the pseudomonoid < f > on X, where Then the isotropy set Γ a = {g ∈ Γ : a ∈ U g } if a = 1, and other wise Γ a = {id| U : a ∈ U } .We have Then by using the Lemma 2.7 we have: Example 2.11.(a) The nth symmetric product of topological space Y [13].Let (Y, * ) be a pointed topological space.Suppose that X = Π n i=1 Y and Γ is a pseudomonoid such that it is generated by the maps where S n is the symmetric group.Consequently, we have X/Γ = SP n Y.
(b) The infinite symmetric product of topological space Y [13].Let Y is a topological space.Consider the pseudomonoid (X, Γ) such that X = Π m∈N Y .Here the set Γ is a pseudomonoid such that it is generated by the following maps on X.
for all i, j ∈ N and i = j.Therefore, we have Proof.Because any Γ i contain the identity, therefore the this finishes the proof.
Proposition 2.13.Let S is a family of locally continuous maps on X.Then the intersection all pseudomonoids on X that contain S is the pseudomonoid such that it is generated by S.
Proof.Suppose Γ 1 is the pseudomonoid is generated by S and Γ 2 is the intersection all pseudomonoids on X and contain S. By the Lemma 2.12, the set Γ 2 is a pseudomonoid.Because Γ 1 contain S, thus by to define Γ 2 ⊆ Γ 1 .From any pseudomonoid contain S is a superset of Γ 1 .We have Γ 1 ⊆ Γ 2 .Thus, we showed that Γ 1 = Γ 2 .
Proposition 2.14.If Γ is a pseudomonoid on X and where U an open subset of X, then Γ| U is a pseudomonoid on U.
The restriction of Γ to a subspace X 0 ⊆ X, Γ| X 0 , contains continuous maps between open subsets of X 0 that can be locally extended to maps in Γ.If X 0 is open in X, then Γ| X 0 consists of maps in Γ whose domains and images are subsets of X 0 (the Proposition 2.14).
Lemma 2.15.If X 0 ⊆ X is a subspace of X, then Proof.Let y ∈ Γ| X 0 (x).Thus there is a f ∈ Γ| X 0 (x) that y = f (x).It follows that f can be locally extended to f ∈ Γ, therefore Proof.We can prove Γ| U (x) ⊆ Γ(x) ∩ U as the Lemma 2.15.Now, assume y ∈ Γ(x) ∩ U , there is a continuous map f ∈ Γ that y = f (x).Because y and x are in U .We can restrict f to an open subset U ⊆ U and contain x, where f (U ) ⊆ U.So f = f | U is a member of Γ| U .Thus, we have y = f (x) ∈ Γ| U (x).
Note 2.17.In the Lemma 2.15 may not be equal (see the following example).
for all x ∈ X and for every ψ ∈ Φ that x ∈ dom ψ.
Proof.Fix ψ ∈ Φ.Let y ∈ Γ 2 ψ(x) is an arbitrary element.Then for some . By the property (3) of morphism definition, there is The proof is now complete.
Proof.Suppose that Φ : (X 1 , Γ 1 ) → (X 2 , Γ 2 ) is a morphism.Let Ψ contain all possible combinations of maps in Φ.Then Ψ satisfies (1)-(3) of Definition 2.24 and Ψ contains Φ, by the maximality of Φ, Φ = Ψ.Now, Suppose that Φ is closed under map combinations of and let us show the maximality of Definition 2.24.Assume Φ is contained in another collection Ψ of continuous maps of open subsets of X 1 to X 2 satisfying the (1)-(3) of Definition 2.24.Take any ψ ∈ Ψ.By property (2) for Φ, for every x ∈ dom ψ there is some φ ∈ Φ with x ∈ dom φ.From property (3) of Ψ, there is some (1).Therefore, every germ of Ψ is some member germ of Φ, yielding that ψ is a combination of members of Φ.Thus ψ ∈ Ψ as desired because Φ is closed under combinations of maps.
Proof.Let Φ be the family of continuous maps from open subset X to X that satisfying the following conditions: • All composites h • φ • h with φ ∈ Φ 0 , h ∈ Γ 1 and h ∈ Γ 2 , wherever defined.
• All possible combinations of composites of the above type are in Φ.
The composition of two morphisms, is the morphism Ψ • Φ : (X 1 , Γ 1 ) → (X 3 , Γ 3 ) be generated by all compositions of maps in Φ with maps in Ψ.With this operation, the pseudomonoids morphisms form a category PsMo.The identity morphism id (X,Γ) of PsMo at X is the morphism be generated by id X , note that Γ ⊆ id (X,Γ) .
The restriction of a morphism Φ : (X, Γ) → (X , Γ ) to a subspace X 0 ⊂ X is the morphism Φ| X 0 : (X 0 , Γ| X 0 ) → (X , Γ ) consisting of all maps of open subsets of X 0 to X that can be locally extended to maps in Φ.
If X 0 is open in X, then Γ| X 0 consists of all maps in Φ whose domain is contained in X 0 .The inclusion map X 0 → X generates a morphism (X 0 , Γ| X 0 ) → (X, Γ), whose composition with Φ is Φ| X 0 .
Given X = ∪ i X i and suppose that Φ i : (X i , Γ| X i ) → (X , Γ ) for each i.Let Φ be the family of continuous maps φ : U → X , where U is an open subset of X, such that φ| U ∩X i ∈ Φ i for all i.If Φ is a morphism and the maps in each Φ i can be locally extended to maps in Φ, then Φ| X i = Φ i for all i, and Φ is called the combination of the morphisms Φ i .
When every X i is open in X, then the combination of morphisms Φ i is defined just when Φ i | X i ∩X j = Φ j | X i ∩X j for all i and j.We define the image of a morphism Φ : (X, Γ) → (X , Γ ) be the set im Φ = ∪ φ∈Φ im φ.If Γ is a pseudomonoid on X and X 0 ⊂ X, define direct image We say Φ is constant if im Φ is one orbit (Φ orb is constant).If Γ is a pseudomonoid on X and X 0 ⊂ X , define the inverse image Proposition 2.32.Let X 0 is a subspace of X and im Φ ⊂ X 0 .Then the restrictions φ : dom φ → X 0 , for any φ ∈ Φ, form a morphism that we denote by Φ : (X, Γ) → (X 0 , Γ | X 0 ).The morphism is called the restriction of Φ too.

Top, PsGr and PsMo Categories
Here we focus on Top, PsGr and PsMo Categories and want to prove the following results.
Let Top denote the category of topological spaces and continuous maps between them.There is a canonical injective covariant functor Top → PsMo which the pseudomonoid be generated by id X to each space X, and assigns the morphism be generated by f to each map.
Conclude, we can consider Top as a subcategory of PsMo.If PsGr be the pseudogroup category [1], then Top ⊂ PsGr ⊂ PsMo .
Let Y be a topological space and Γ is a pseudomonoid on X.Then any continuous map Y → X generates a morphism (Y, < id Y >) → (X, Γ).
Lemma 3.1.A continuous map f : X → Y generates a morphism (X, Γ) → (Y, < id Y >) if and only if f is constant on the Γ-orbits.
A set M equipped with a diffeology is called a diffeological space.Now, consider X = ∞ n=1 R n and Γ is all smooth maps f : U ⊆ R n → V ⊆ R m that U and V are open subsets.If M is a diffeological space be equipped by the diffeology D. Then D is a (X, Γ)-atlas for M. Consequently (M, D) is a ( ∞ n=1 R n , Γ)-structure.
Lemma 4.8.Suppose that X = ∞ n=1 R n and Γ is all smooth maps f : U ⊆ R n → V ⊆ R m that U and V are open subsets.Then a (X, Γ)-structure, on (M, A ), is a diffeology if and only if A satisfies the following conditions: (1) A contain all constant parametrizations, (2) A is closed under combination maps.Definition 4.9.Let A be a (X, Γ)-atlas on M. Then the weakest topology on M such that all parametrizations in A are continuous is called (X, Γ)-topology on M induced by A .
is a (X, Γ)-atlas on V. Therefore, (V, A | V ) is a (X, Γ)-structure.Definition 4.11.Let (M 1 , A 1 ) and (M 2 , A 2 ) are (X, Γ)-structure.Then α : M 1 → M 2 is called (X, Γ)-map if for any φ ∈ A 1 , α • φ ∈ A 2 .Definition 4.12.Let (M 1 , A 1 ) and (M 2 , A 2 ) are (X, Γ)-structure.Then α : M 1 → M 2 is called (X, Γ)equivalent if α satisfies the following properties: (1) α is one-to-one and onto, (2) α and α −1 are (X, Γ)-maps.Remark 4.13.We say (M 1 , A 1 ) is local (X, Γ)-equivalent to (M 2 , A 2 ) if for p ∈ M 1 there are open subsets U ⊆ M 1 , V ⊆ M 2 such that p ∈ U and there is a (X, Γ)-equivalent β : U → V. Lemma 4.14.Assume that Γ is all local diffeomorphism maps on R n .Suppose that A is a (R n , Γ)-atlas on M.Then, there is A ⊆ A that (M, A ) is a n-manifold if and only if (M, A ) is local (R n , Γ)-equivalent to (R n , Γ).A Frölicher space is a triple (S, C, F ) where S is a set and the couple(C, F ) is a Frölicher structure on it.Let X = R and Γ be pseudomonoid generated by C ∞ (R, R).Therefore C is a cover for S.So by Lemma 4.3 the (S, < C >) is a (X, Γ)-structure.Hence, any Frölicher spaces equipped a (X, Γ)-atlas.Example 4.17.(Sikorski spaces [11]) Suppose that S is a nonempty set.A Sikorski structure on S is a non-empty collection F = {f : S → R}, with weakest topology on S that all elements of F are continuous, satisfying the following conditions: (1) For any m ∈ N, if

( 4 )
Pseudogroup of analytical (or symplectic) diffeomorphisms, The pseudogroup PL of piecewise linear homeomorphisms between open subsets of R n .

Lemma 4 . 10 .
Suppose that (M, A ) is a (X, Γ)-structure.If V ⊆ M is open subset, then

Example 4 .
15. (Frölicher spaces[4]) A Frölicher structure on a set S is a couple (C, F ) where C ⊆ S R and F ⊆ R S such that C = D * F and F = D * C holds, withD * F = {c : R → S : f • c ∈ C ∞ (R, R) for all f ∈ F } D * C = {f : S → R : f • c ∈ C ∞ (R, R) for all c ∈ C} .