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A convenient category of topological partial groups
Journal of the Egyptian Mathematical Society volume 27, Article number: 8 (2019)
Abstract
In this paper, the concept of ℘-continuous map is introduced and some of their basic properties are discussed. Also, the category \(\mathbf {K}\acute {}\), of topological partial groups, as objects and the ℘-morphisms of topological partial groups, as arrows, is introduced, which is alternative to the category K, of topological spaces, as objects and k-continuous maps,
as arrows, and satisfies the same nice properties of the category kpg, of \(\underline {k}\)-partial groups, as objects, and the morphisms of \(\underline {k}\)-partial groups, as arrows (Abd- Allah et al., J. Egyption Math. Soc 25:276-278, 2017).
Introduction
In [1], A.M. Abd- Allah et al. introduced the concept of topological partial groups and discussed some of their basic properties. Also, they introduced the category Tpg of topological partial groups, as objects and the homorphisms of topological partial groups, as arrows. So, the category Tpg has the following deficiencies:
-
(i)
If a∈S, then the right transformation ra:S→S,x↦xa and the left transformation la:S→S,x↦ax, may not be open.
-
(ii)
The quotient map ρN:S→S,x↦xN, N≤S, may not be open, in general, where S/N has the identification topology with respect to the quotient map.
-
(iii)
Let S be a topological partial group and \(N\unlhd S\). Then, the partial group S/N may not be a topological partial group, since the cartesian product of two identification maps may not be identification.
In [2], A.M. Abd- Allah et al. introduced the concept of \(\underline {k}\)-partial groups and discussed some of their basic properties. Also, they introduced the category kpg, of \(\underline {k}\)-partial groups, as objects, and the morphisms of \(\underline {k}\)-partial groups, as arrows which is modified the above deficiencies. In this paper, the concept of ℘-continuous maps is introduced and some of their basic properties are discussed. Also, the category \(\mathbf {K}\acute {}\) of topological partial groups, as objects, and the ℘-morphisms of topological partial groups, as arrows, is introduced, which is alternative to the category K, of topological spaces, as objects, and k-continuous maps, as arrows. The category \(\mathbf {K}\acute {}\) satisfied the same nice properties of the category kpg. The idea of ℘-continuous maps was taken from the definition of k-continuous map [3].
Preliminaries
We collect for sake of reference the needed definitions and results appeared in the given references.
Definition 1
[4] Let S be a semigroup. Then, x∈S is called an idempotent element if x·x=x. The set of all idempotent elements in S is denoted by E(S).
Definition 2
[5] Let S be a semigroup and x∈S. Then, an element e∈S is called a partial identity of x if:
-
(i)
ex=xe=x,
-
(ii)
If e′x=xe′=x, for some e′∈S, then ee′=e′e=e.
Theorem 1
[5] Let S be a semigroup. Then,
-
(i)
If x∈S has a partial identity, then it is unique
-
(ii)
E(S) is the set of all partial identities of the elements of S.
We will denote by ex the partial identity of the element x∈S.
Definition 3
[5] Let S be a semigroup and x∈S has a partial identity element ex. Then, y∈S is called a partial inverse of x if:
-
(i)
xy=yx=ex,
-
(ii)
exy=yex=y.
We will denote the partial inverse y of x∈S by x−1.
Definition 4
[5] A semigroup S is called a partial group if:
-
(i)
Every x∈S has a partial identity ex
-
(ii)
Every x∈S has a partial inverse x−1
-
(iii)
The map eS:S→S,x↦ex is a semigroup homomorphism
-
(iv)
The map γ:S→S,x↦x−1 is a semigroup antihomomorphism.
So, every group is a partial group.
Definition 5
[6] Let S be a partial group and x∈S. Then, we define Sx = {y ∈ S:ex=ey}.
Theorem 2
[5] Let S be a partial group and x∈S. Then,
-
(i)
Sx is a maximal subgroup of S which has identity ex
-
(ii)
\(S=\bigcup \{S_{x}:x \in S\}\).
Corollary 1
[5] Every partial group is a disjoint union of a family of groups.
Definition 6
[3] Let X be a topological space. Then, the map α:C→X is called a test map if α is continuous and C is a compact Hausdorff space.
Definition 7
[3] Let X and Y be topological spaces. Then, the map f:X→Y is called k-continuous if fα:C→Y is continuous, for each test map α:C→X.
Let τ be the category of topological spaces, as objects and continuous maps, as arrows. Also, let K be the category of topological spaces, as objects and k-continuous maps, as arrows. It is clear that the category τ is a wide subcategory of K.
Definition 8
[1] Let S be a partial group and τ be a topology on S. Then, S is called a topological partial group if the following maps are continuous:
-
(i)
The product map: μ:S×S→S,(x,y)↦xy
-
(ii)
The partial identity map: eS:S→S,x↦ex
-
(iii)
The partial inverse map: γ:S→S,x↦x−1.
Definition 9
[1] Let S be a topological partial group and a∈S. Then, the map ra:S→S,x↦xa is called a right transformation and the map la:S→S,x↦ax is called a left transformation.
Theorem 3
[1] The maps ra and la are continuous.
Let ℘ be a non-empty full subcategory of τ which satisfies the following conditions [7]:
-
(i)
If A is a closed subspace of an object B of ℘, then A is a k℘-space.
-
(ii)
If B and C are objects in ℘, then B×C is also object in ℘.
-
(iii)
For objects X in ℘ and Y in τ, the evaluation map e:YX×X→Y,(f,x)↦f(x) and x∈X, is continuous, where YX has the compact open topology.
-
(iv)
If A and B are objects in ℘, then the topological sum \(A\bigsqcup B\) is also an object in ℘.
Definition 10
[2] Let S be a topological partial group. Then, the map h:C→S is called a ℘-test map if h is continuous and \(h^{-1}(S_{e_{x}})\) is open in C for each ex∈E(S), where C∈obj(℘).
℘-continuous maps
In this section, the notion of ℘-continuous map is introduced and some of their basic properties are discussed. Also, the category \(\mathbf {K}\acute {}\) of ℘-continuous maps, as objects and the morphisms of ℘-continuous maps as arrows, is introduced.
Definition 11
Let S and T be topological partial groups. Then, the map f:S→T is called ℘-continuous if fh:C→T is continuous, for each a ℘-test map h:C→S.
We note that every continuous map of topological partial group is ℘-continuous. So, the following maps are ℘-continuous:
-
(i)
The identity map I:S→S
-
(ii)
The partial identity map: eS:S→S,x↦ex
-
(iii)
The partial inverse map: γ:S→S,x↦x−1
-
(iv)
The maps ra and la.
Definition 12
Let f:S→T be a ℘-continuous map. Then, f is called a ℘−morphism if it is a partial group homomorphism.
We note that (i) and (iii) above are ℘-morphisms.
Theorem 4
If f:S→T and g:T→F are ℘-morphisms, then gf:S→F is also a ℘-morphism.
Proof
It is clear that gf is a partial group homomorphism. Let h:C→S be a ℘−test map. Since f is ℘-continuous, then fh:C→T is continuous. Now, (fh)−1(Te)=h−1(f−1(Te)), for each e∈E(T). Since f is a partial group homomorphism, then f−1(Te) is a maximal subgroup of S. So, (fh)−1(Te) is open in C, for each e∈E(T). That means that fh is a ℘-test map. Since g is ℘-continuous, then g(fh)=(gf)h:C→F is continuous. Then, gf is ℘-continuous. Hence, gf is a ℘-morphism. □
Definition 13
A subset V of the topological partial group S is called ℘-open if h−1[V] is open in C for each a ℘-test map h:C→S
From the above definition, we have that \(S_{e_{x}}\) is ℘-open in S.
Theorem 5
The family {℘−τS} of ℘-open sets form a topology on S.
Proof
It is clear that ϕ and S are ℘-open sets, since h−1[S]=C and h−1[ϕ]=ϕ. If U and V are ℘-open sets, then h−1[U] and h−1[V] are open sets in C. But \(h^{-1}[U\bigcap V]=h^{-1}[U]\bigcap h^{-1}[V]\) is open in C. So, \(U\bigcap V\) is a ℘-open set. Similarly, let (Uλ)λ∈L be a subfamily of ℘-open sets. Then, h−1[Uλ] are open in C, for each λ∈L. Since \(h^{-1}[\bigcup _{\lambda }U_{\lambda }]=\bigcup _{\lambda }h^{-1}[U_{\lambda }]\) is open in C. Hence, \(\bigcup U_{\lambda }\) is a ℘-open set. □
Definition 14
A subset A of the topological partial group S is called a ℘-neighbourhood of x∈S if there exists a ℘-open set U in S such that x∈U⊆A.
The family of all ℘-neighbourhoods of x∈S is called a ℘-neighbourhood system and is denoted by ℘−Nx
Proposition 1
A subset A⊆S of the topological partial group S is a ℘-open set if and only if it is a ℘-neighbourhood of each of its points.
Proof
Let A be a ℘-open set. Then, x∈A⊆A, for all x∈A. Hence, A is a ℘-neighbourhood of x. Conversely, for each x∈A, there exists a ℘-open set Ux such that x∈Ux⊆A. So, \(A=\bigcup _{x\in A}U_{x}\). Hence, A is a ℘-open set. □
Theorem 6
Let S be a topological partial group and x∈S. Then,
-
(i)
x∈N, for all N∈Nx
-
(ii)
If N∈Nx and N⊆M, then M∈Nx
-
(iii)
If N,M∈Nx, then \(N\bigcap M\in N_{x}\)
-
(iv)
If N∈Nx, then there exists M∈Nx such that N∈Ny, for each y∈M.
Proof
-
(i)
If N∈Nx, then there exists a ℘-open set U in S such that x∈U⊆N. Hence, x∈N.
-
(ii)
If N∈Nx, then there exists a ℘-open set U in S such that x∈U⊆N. Since, N⊆M, then x∈U⊆M. Hence, M∈Nx.
-
(iii)
If N,M∈Nx, then there exist two ℘-open sets U and V, respectively such that x∈U⊆N and x∈V⊆M. So, we have that \(x\in U\bigcap V\subseteq N\bigcap M\). Since \(N\bigcap M\) is a ℘-open set, then \(N\bigcap M\in N_{x}\).
-
(iv)
If N∈Nx, then there exists a ℘-open set M in S such that x∈M⊆N. Since M is a ℘-open set, then M∈Ny, for all y∈M. Since, N⊆M, then N∈Ny, for each y∈M.
□
Definition 15
Let S be a topological partial group and A⊆S. Then, x∈A is called a ℘-interior point of A if A is a ℘-neighbourhood of x.
The set of all ℘-interior points of A is called ℘-interior set and is denoted by ℘−A0.
Proposition 2
Let S be a topological partial group and A,B⊆S. Then,
-
(i)
℘−A0⊆A
-
(ii)
If A⊆B, then ℘−A0⊆℘−B0
-
(iii)
℘−A0 is a ℘-open set
-
(iv)
(℘−A0)0=℘−A0.
Proof
-
(i)
Let x∈℘−A0. Then, A∈Nx. So, x∈A.
-
(ii)
Let x∈℘−A0. Then, A∈Nx. Since, A⊆B, then B∈Nx and so x∈℘−B0. Hence, ℘−A0⊆℘−B0.
-
(iii)
Let x∈℘−A0. Then, A∈Nx. Thus, there exists N∈Nx such that A∈Ny, for all y∈N. That is, y∈℘−A0, for all y∈N. Hence, N⊆A. Thus, x∈N⊆℘−A0. So, A∈Nx. Therefore, ℘−A0 is a ℘-open set.
-
(iv)
Since ℘−A0⊆A, then from (ii) (℘−A0)0⊆℘−A0. It remains that ℘−A0⊆(℘−A0)0. This is given from x∈℘−A0. That is, ℘−A0∈Nx. Hence, x∈(℘−A0)0.
□
Corollary 2
A subset A of the topological partial group S is ℘-open if and only if ℘−A0=A.
Proof
It is obvious. □
Definition 16
A subset A of the topological partial group S is called ℘-closed if S−A is a ℘-open set.
Definition 17
Let S be a topological partial group and A⊆S. Then, x∈S∈ is called a ℘-closure point of A if \(A\bigcap N\neq \phi \), for each N∈℘−Nx.
The set of all ℘-closure points of A is called the ℘-closure of A and is written by \(\wp -\overline {A}\).
Proposition 3
Let A be a subset of the topological partial group S. Then, the family \(\tau _{A}= \{U\bigcap A: U\ {is} \ \wp -open\ in\ \textit {S}\}\) is a topology on A, which is called ℘-relative topology.
Proof
It is clear that ϕ, A∈τA since \(\phi =\phi \bigcap A\) and \(A=A\bigcap S\). Let M,N∈τA. Then, there exist two ℘-open sets U and V such that \(M=U\bigcap A\) and \(N=V\bigcap A\). So, \(M\bigcap N \in \tau _{A}\). Also, let V=(Vλ)λ∈L be a subfamily of τA. Then, for each λ, there are ℘-open sets Uλ such that \(V=U_{\lambda }\bigcap A\). Then, \(V=\bigcup _{\lambda \in L}V_{\lambda } =\bigcup _{\lambda \in L}(U_{\lambda } \bigcap A)= (\bigcup _{\lambda \in L}U_{\lambda })\bigcap A\). □
Theorem 7
Let f:S→T be ℘-continuous. Then, f∣A:A→T is ℘-continuous.
Proof
Let U⊆T be ℘-open. Now, \((f \mid A)^{-1}(U)=f^{-1}(U)\bigcap A\). Since f−1(U) is a ℘−open set in S, then f−1(U) is a ℘-open in A. □
Definition 18
Let S be a topological partial group and A be a subpartial group of S. Then, A with the ℘-relative topology is a topological partial group, called a topological subpartial group, denoted by A≤S.
Definition 19
Let S and T be topological partial groups and let (x,y)∈S×T. The set ℘−(S×T), where M∈Nx in S and N∈Ny in T is called a ℘-basic neighbourhood of (x,y).
Definition 20
A subset U of M×N is called a ℘-neighbourhood if there exists a ℘-basic neighbourhood M×N of (x,y) such that (x,y)∈M×N⊆U.
We note that if M and N are ℘-open sets in the topological partial groups S and T, respectively, then M×N is a ℘-basic neighbourhood of any (x,y)∈M×N.
Theorem 8
-
(i)
If A and B are ℘-open sets in S and T, respectively, then A×B is also ℘-open in S×T
-
(ii)
If C and D are ℘-closed sets in S and T, respectively, then C×D is also ℘-closed in S×T.
Proof
-
(i)
Let (x,y)∈U×V. Then, x∈U and y∈V. So, U∈℘−Nx in S and V∈℘−Ny in T. This implies U×V is a ℘-basic neighbourhood of (x,y). Since (x,y)∈U×V⊆A×B, then U×V∈N(x,y). Hence, A×B is also ℘-open in S×T.
-
(ii)
We have \((S\times T)-(C\times D)=(S-C)\times T\bigcup S\times (T-D)\). Since S−C and T−D are ℘-open sets in S and T, respectively, then (S−C)×T and S×(T−D) are ℘-open sets in S×T and so (S×T)−(C×D) is ℘-open set in S×T. That is, C×D is ℘-closed in S×T.
□
We note that the following maps are ℘-continuous, for each topological partial group S:
-
(i)
The projection maps P1:S×T→S and P2:S×T→T.
-
(ii)
The product map μ:S×S→S.
-
(iii)
The diagonal map ΔS={(x,x):x∈S}.
Theorem 9
If f:S→T and f:S→F are ℘-morphisms, then (f,g):S→T×F is also a ℘-morphism.
Proof
It is clear that (f,g) is a partial group homomorphism. Let h:C→S be a ℘-test map. Since f is ℘-continuous, then fh:C→T is continuous. Also, since g is ℘-continuous, then gh:C→T is continuous. So, (fh,gh)=(f,g)h:S→T×F is continuous. That is, (f,g) is ℘-continuous. Hence, (f,g) is a ℘-morphism. □
Theorem 10
If f1:S1→T1 and f2:S2→T2 are ℘-morphisms, then f1×f2:S1×S2→T1×T2 is also a ℘-morphism.
Proof
It is clear that f1×f2:S1×S2→T1×T2 is a partial group homomorphism. Since f1×f2=(f1 P1, f2 P2), then from the last theorem, we have that f1×f2 is ℘-continuous. Hence, f1×f2 is a ℘-morphism. □
Theorem 11
Let S and T be topological partial groups. Then, the following conditions are equivalent for any map f:S→T.
-
(i)
f is ℘-continuous
-
(ii)
f−1[U] is a ℘-open set in S for each ℘-open set U in T.
-
(iii)
f−1[U] is a ℘-closed set in S for each ℘-closed set U in T.
Proof
(i) → (ii) Let f be ℘-continuous and let U⊆T be ℘-open. So, h−1[f−1[U]]=(fh)−1[U] is open in C, for each ℘-test map h:C→T. Hence, f−1[U] is a ℘-open set in S.
(ii) → (iii) Let U be ℘-closed in T. So T−U is ℘-open in T. Therefore, f−1[T−U]=S−f−1[U] is ℘-open in S. Hence, f−1[U] is ℘-closed in S.
(iii) → (ii) Let U be ℘-open in T. So, T−U is ℘-closed in T. Therefore, f−1[T−U]=S−f−1[U] is ℘-closed in S. Hence, f−1[U] is ℘-open in S.
(iii) → (i) Let h:C→S be a ℘-test map and U⊆T be open. So, f−1[U] is ℘-open in S. Therefore, h−1[f−1[U]]=(fh)−1[U] is open in C. Hence, f is ℘-continuous. □
Definition 21
Let S and T be topological partial groups. Then, the map f:S→T is called ℘-open if f(U) is ℘-open in T for each ℘-open set U in S. Also, the map f:S→T is called ℘-closed if f(U) is ℘-closed in T for each ℘-closed set U in S.
Theorem 12
If f1:S1→T1 and f2:S2→T2 are ℘-open maps, then f1×f2:S1×S2→T1×T2 is also a ℘-open map.
Proof
Let U⊆S1×T1 be ℘-open and (x,y)∈U. Then, there exists a ℘-basic neighbourhood M×N of (x,y) such that (x,y)∈℘−(M×N)⊆U. So, (f1×f2)[M×N]⊆(f1×f2)[U]. Therefore, f1[M]×f2[N]⊆(f1×f2)[U]. Since f1 and f2 are ℘-open maps, then f1[M] and f2[N] are ℘-open sets in T1 and T2, respectively. Hence, f1×f2 is ℘-open. □
Theorem 13
The maps ra and la are ℘-open maps.
Proof
We only prove that ra is ℘-open as follows: Let U⊆S be ℘-open. Then, \(U\bigcap S_{e_{x}} \) is open in the maximal topological subgroup \(S_{e_{x}} \) and so is open in S. Now, we have two cases:
-
(i)
Let \(\phantom {\dot {i}\!}r_{a} \, |_{_{_{S_{e_{x}} }} } :S_{e_{x}} \to S_{e_{y} }\). So, \(\phantom {\dot {i}\!}r_{a} \, |_{_{_{S_{e_{x}} }} } (U \bigcap S_{e_{x} })=Ua \bigcap S_{e_{y} }\). We show that \(Ua \bigcap S_{e_{y} }\) is open in S as follows: Let h:C→S be a ℘-test map. Then, rah:C→S is a ℘-test map. Now, \((r_{a} h)^{-1}(Ua \bigcap S_{e_{y} })=h^{-1}((r_{a})^{-1}(Ua \bigcap S_{e_{y} }))=h^{-1}((r_{a})^{-1}(Ua) \bigcap (r_{a})^{-1}(S_{e_{y}}))=h^{-1}(U \bigcap S_{e_{x}})\phantom {\dot {i}\!}\). Since \(U \bigcap S_{e_{x}}\) is open in S, then \(\phantom {\dot {i}\!}h^{-1}(U \bigcap S_{e_{x}})\) is open in C. Hence, \(Ua \bigcap S_{e_{y} }\) is ℘-open in S.
-
(ii)
Let \(\phantom {\dot {i}\!}r_{a} \, |_{_{_{S_{e_{x}} }} } :S_{e_{x}} \to S_{e_{x}} \). Since, the right transformation \(\phantom {\dot {i}\!}r_{a} \, |_{_{_{S_{e_{x}} }} }\) is a homeomorphism of the topological maximal subgroups \(S_{e_{x} }\), then \(\phantom {\dot {i}\!}r_{a} \, |_{_{_{S_{e_{x}} }} } (U\bigcap S_{e_{x}})\) is open in \(S_{e_{x}} \). Since \(S_{e_{x}} \) is open in S, then \(\phantom {\dot {i}\!}r_{a} \, |_{_{_{S_{e_{x}} }} } (U\bigcap S_{e_{x}})=Ua\bigcap S_{e_{x}} \) is open in S. That means \(\phantom {\dot {i}\!}r_{a} (U)=\bigcup _{e_{x} \in E(S)}\, r_{a} |_{_{_{S_{e_{x}} }} } (U\bigcap S_{e_{x}})\) is ℘-open in S.
Similarly, we can prove that ℓa is ℘-open. □
Theorem 14
Let S be a topological partial group and A,B⊆S. Then, if A is ℘-open in S, then AB and BA are also ℘-open in S.
Proof
We only prove that AB is ℘-open in S as follows: Since \(AB=\bigcup _{b\in B}r_{b}(A)\), and rb(A) is ℘-open in S, then AB is ℘-open in S. Similarly, we can prove that BA is also ℘-open in S. □
Theorem 15
If S is a topological partial group, then every ℘-open topological subpartial group of S is ℘-closed.
Proof
Let A be a ℘-open topological subpartial group of S. Then, xA is ℘-open in S, for all x∈S. Since \(S-A=\bigcup _{x\neq A}xA\), then S−A is ℘-open. Therefore, A is ℘-closed. □
Theorem 16
The projection maps P1:S×T→S and P2:S×T→T are ℘-open maps.
Proof
we only prove that P1 is ℘-open, as follows: let W⊆S×T be ℘-open and x∈P1[W]. Then, there exists y∈T such that (x,y)∈W. Since W is ℘-open, then there exists a ℘-basic neighbourhood M×N of (x,y) such that (x,y)∈M×N⊆W. So, \(x\in M=P_{1}^{-1} \left [M\times N\right ]\subseteq P_{1} [W]\). Hence, P1[W]∈℘−Nx. Therefore, P1 is ℘-open. Similarly, we can prove that P2 is ℘-open. □
Let {Si:i=1,2,⋯,n} be a family of topological partial groups and \(S=\mathop {\otimes }\limits _{i=1}^{n} \, S_{i} \, \) be the cartesian product of topological partial groups. That is, S={x=〈xi〉:xi∈Si, ∀ i=1,2,…,n}.
Theorem 17
The partial group S with the cartesian product topology \(S=\mathop {\otimes }\limits _{i=1}^{n} \, S_{i} \,\) is a topological partial group.
Proof
The maps μ,γ and eS are ℘-continuous, since μ=〈μi (Pi×Pi)〉, γ=〈γiPi〉 and \(e_{S}=\left \langle e_{S_{i}} P_{i} \right \rangle \), respectively, where \(P_{i} :\mathop {\otimes }\limits _{i=1}^{n} \, (S_{i})\to S_{i} \), are the projection maps. □
Definition 22
Let S and T be topological partial groups. A topology ℘−τ∗ on T is called ℘−final with respect to the map f:S→T if, for any topological partial group F and all maps g:T→F, we have that g is ℘-continuous if gf:S→F is ℘-continuous.
Theorem 18
The ℘−τ∗ final topology on T with respect to the function f:S→T exists and is characterized by the following condition: If U⊆T, then U is ℘-open (℘-closed) in T if and only if f−1[U] is ℘-open (℘-closed) in S.
Proof
It is clear that ϕ and T are ℘-open sets in S. If U and V are ℘-open sets in T, then \(f^{-1}[U\bigcap V]=f^{-1}[U]\bigcap f^{-1}[V]\) is ℘-open in S. So, \(U\bigcap V\) is ℘-open in T. Similarly, let (Uλ)λ∈L be a subfamily of ℘-open sets in T. Then, \(f^{-1}[\bigcup (U_{\lambda })]\) are ℘-open sets in S. So, \(\bigcup U_{\lambda }\) is a ℘-open set in S. A similar proof applies with ℘-open replaced by ℘-closed. □
Definition 23
Let S and T be topological partial groups. Then, the map f:S→T is called ℘-identification if f is surjective and T has the ℘-final topology with respect to f.
Theorem 19
Let f:S→T be a ℘-continuous surjection. If f is a ℘-open (closed) map. Then, f is a ℘-identification map.
Proof
Let U⊆T be a ℘-open set. Then, f−1[U] is ℘-open in S. Since f is surjective, then f[f−1[U]]=U. Hence, f−1[U] is ℘-open in S if and only if U is ℘-open. A similar proof applies with open replaced by ℘-closed. □
Quotients in topological partial groups
Definition 24
If S is a topological partial group and N≤S, then S/N with the ℘-identification topology, with respect to the quotient map ρN:S→S/N, is called the ℘-coset space.
Theorem 20
Let S be a topological partial group and N≤S. Then, the quotient map ρN:S→S/N is ℘-open.
Proof
Let U⊆S be open. Then,
Since U is open in S, then UN is open in S. Since ρN is an identification map and UN is open in S, then ρN(U) is open is S/N. □
Theorem 21
If S is a topological partial group and \(N\unlhd S\), then S/N is a topological partial group.
Proof
Since ρN is a ℘-open identification map, then ρN×ρN is a ℘ identification map. So, the product map μ:S/N×S/N→S/N is continuous, since μ (ρN×kρN)=ρN μ′, where μ′:S×k S→S is the product map. The partial inverse map γ:S/N→S/Nand the partial identity map eS/N:S/N→S/N are continuous, since γ ρN=ρN γ′ and eS/N ρN=ρN eS are ℘-continuous and ρN is an identification map, where γ′:S→S, x↦x−1 and eS:S→S, x↦ex are ℘-continuous. □
Theorem 22
Let φ:S→T be an idempotent separating surjective ℘-morphism and K=kerφ. Then, there exists a unique bijective ℘-morphism α:S/K→T such that φ=αρK.
Proof
It is clear that α is bijective and a partial group homomorphism. Also, α is ℘-continuous since φ is ℘-continuous and ρK is a ℘-identification map. □
Theorem 23
Let S be a topological partial group and \(M,N\unlhd S\) such that M⊆N, then
-
(i)
\(N/M \unlhd S/M\)
-
(ii)
There exists a unique bijective ℘-morphism α:(S/M)/(N/M) such that ρN=αρN/MρM
Proof
-
(i)
See [4]
-
(ii)
Let ρN:S→S/N and ρN:S→S/M be the quotient maps. Then, ρN is an idempotent separating surjective ℘-morphism and kerρN=N. So, from the last theorem, there exists a unique bijective ℘-morphism φ:S/M→S/N such that φρM=ρN. Since kerφ=N/M is a topological partial group, then by the last theorem, there exists a unique bijective ℘-morphism α:(S/M)/(N/M), such that ρN=αρN/MρM.
□
Separation axioms.
Definition 25
Let S be a topological partial group. Then, S is called ℘-Hausdorff if, for all x,y∈S, there exist ℘-open sets U and V such that x∈U,y∈V, and \(U\bigcap V \neq \phi \).
Theorem 24
Let S be a topological partial group. Then, S is Hausdorff if and only if S is a T0-space.
Proof
Let S be a Hausdorff partial group. Then, S is a T0-space. Conversely, let S be a T0-space. Let x,y∈S,x≠y:
-
(i)
If x,y∈Sa, then Sa is a T2-group and there exist two open sets U,V in Sa and also ℘-open in S such that \(U\bigcap V\neq \phi \) and x∈U,y∈V and
-
(ii)
If x∈Sa and y∈Sb, then, we have that Sa and Sb are ℘-open and \(S_{a}\bigcap S_{b}\neq \phi \). So, S is a Hausdorff partial group.
□
Theorem 25
Let S be a Hausdorff topological partial group. If f,g:S→T are ℘-morphisms of topological partial group, then the difference kernel A={x∈S:f(x)=g(x)} is a ℘-closed subpartial group.
Proof
A is closed (see [3]). Let x,y∈A. Now,
Therefore, xy−1∈A. Hence, A is a ℘-closed subpartial group. □
Let \(\mathbf {K}\acute {}\) be the category of topological partial groups, as objects and the ℘-morphisms, as arrows.
The category \(\mathbf {K}\acute {}\) is a convenient category since this category has a product and a quotient.
Abbreviations
- \(\mathbf {K}\acute {}\) :
-
The category of topological partial groups, as objects and the ℘-morphisms of topological partial groups, as arrows
- K :
-
The category of topological spaces, as objects and k-continuous maps, as arrows
- k p g :
-
The category of \(\underline {k}\)-partial groups, as objects, and the morphisms of \(\underline {k}\)-partial groups, as arrows
- T p g :
-
The category of topological partial groups, as objects and the homorphisms of topological partial groups, as arrows
- τ :
-
The category of topological spaces, as objects and continuous maps, as arrows
- ℘ :
-
A non-empty full subcategory of τ
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We thank our colleagues from Al-Azhar University who provided insight and expertise that greatly assisted the research. Further, the authors are very grateful to the editor and referees for their comments and suggestions.
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Fathy, A. A convenient category of topological partial groups. J Egypt Math Soc 27, 8 (2019). https://doi.org/10.1186/s42787-019-0010-4
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DOI: https://doi.org/10.1186/s42787-019-0010-4
Keywords
- Partial group
- Partial group homomorphism
- Topological group
- Topological partial group
- \(\underline {k}\)-partial group