 Original research
 Open Access
 Published:
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 kcontinuous 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 25:276278, 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 r_{a}:S→S,x↦xa and the left transformation l_{a}:S→S,x↦ax, may not be open.

(ii)
The quotient map ρ_{N}: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 kcontinuous 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 kcontinuous 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 e_{x} the partial identity of the element x∈S.
Definition 3
[5] Let S be a semigroup and x∈S has a partial identity element e_{x}. Then, y∈S is called a partial inverse of x if:

(i)
xy=yx=e_{x},

(ii)
e_{x}y=ye_{x}=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 e_{x}

(ii)
Every x∈S has a partial inverse x^{−1}

(iii)
The map e_{S}:S→S,x↦e_{x} 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 S_{x} = {y ∈ S:e_{x}=e_{y}}.
Theorem 2
[5] Let S be a partial group and x∈S. Then,

(i)
S_{x} is a maximal subgroup of S which has identity e_{x}

(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 kcontinuous 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 kcontinuous 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: e_{S}:S→S,x↦e_{x}

(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 r_{a}:S→S,x↦xa is called a right transformation and the map l_{a}:S→S,x↦ax is called a left transformation.
Theorem 3
[1] The maps r_{a} and l_{a} are continuous.
Let ℘ be a nonempty 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:Y^{X}×X→Y,(f,x)↦f(x) and x∈X, is continuous, where Y^{X} 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 e_{x}∈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: e_{S}:S→S,x↦e_{x}

(iii)
The partial inverse map: γ:S→S,x↦x^{−1}

(iv)
The maps r_{a} and l_{a}.
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}(T_{e})=h^{−1}(f^{−1}(T_{e})), for each e∈E(T). Since f is a partial group homomorphism, then f^{−1}(T_{e}) is a maximal subgroup of S. So, (fh)^{−1}(T_{e}) 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 {℘−τ_{S}} 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 (U_{λ})_{λ∈L} be a subfamily of ℘open sets. Then, h^{−1}[U_{λ}] 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 ℘−N_{x}
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 U_{x} such that x∈U_{x}⊆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∈N_{x}

(ii)
If N∈N_{x} and N⊆M, then M∈N_{x}

(iii)
If N,M∈N_{x}, then \(N\bigcap M\in N_{x}\)

(iv)
If N∈N_{x}, then there exists M∈N_{x} such that N∈N_{y}, for each y∈M.
Proof

(i)
If N∈N_{x}, then there exists a ℘open set U in S such that x∈U⊆N. Hence, x∈N.

(ii)
If N∈N_{x}, then there exists a ℘open set U in S such that x∈U⊆N. Since, N⊆M, then x∈U⊆M. Hence, M∈N_{x}.

(iii)
If N,M∈N_{x}, 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∈N_{x}, then there exists a ℘open set M in S such that x∈M⊆N. Since M is a ℘open set, then M∈N_{y}, for all y∈M. Since, N⊆M, then N∈N_{y}, 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 ℘−A^{0}.
Proposition 2
Let S be a topological partial group and A,B⊆S. Then,

(i)
℘−A^{0}⊆A

(ii)
If A⊆B, then ℘−A^{0}⊆℘−B^{0}

(iii)
℘−A^{0} is a ℘open set

(iv)
(℘−A^{0})^{0}=℘−A^{0}.
Proof

(i)
Let x∈℘−A^{0}. Then, A∈N_{x}. So, x∈A.

(ii)
Let x∈℘−A^{0}. Then, A∈N_{x}. Since, A⊆B, then B∈N_{x} and so x∈℘−B^{0}. Hence, ℘−A^{0}⊆℘−B^{0}.

(iii)
Let x∈℘−A^{0}. Then, A∈N_{x}. Thus, there exists N∈N_{x} such that A∈N_{y}, for all y∈N. That is, y∈℘−A^{0}, for all y∈N. Hence, N⊆A. Thus, x∈N⊆℘−A^{0}. So, A∈N_{x}. Therefore, ℘−A^{0} is a ℘open set.

(iv)
Since ℘−A^{0}⊆A, then from (ii) (℘−A^{0})^{0}⊆℘−A^{0}. It remains that ℘−A^{0}⊆(℘−A^{0})^{0}. This is given from x∈℘−A^{0}. That is, ℘−A^{0}∈N_{x}. Hence, x∈(℘−A^{0})^{0}.
□
Corollary 2
A subset A of the topological partial group S is ℘open if and only if ℘−A^{0}=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∈℘−N_{x}.
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∈τ_{A} since \(\phi =\phi \bigcap A\) and \(A=A\bigcap S\). Let M,N∈τ_{A}. 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=(V_{λ})_{λ∈L} be a subfamily of τ_{A}. Then, for each λ, there are ℘open sets U_{λ} 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∈N_{x} in S and N∈N_{y} 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∈℘−N_{x} in S and V∈℘−N_{y} 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 \((S\times T)(C\times D)=(SC)\times T\bigcup S\times (TD)\). 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 P_{1}:S×T→S and P_{2}:S×T→T.

(ii)
The product map μ:S×S→S.

(iii)
The diagonal map Δ_{S}={(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 f_{1}:S_{1}→T_{1} and f_{2}:S_{2}→T_{2} are ℘morphisms, then f_{1}×f_{2}:S_{1}×S_{2}→T_{1}×T_{2} is also a ℘morphism.
Proof
It is clear that f_{1}×f_{2}:S_{1}×S_{2}→T_{1}×T_{2} is a partial group homomorphism. Since f_{1}×f_{2}=(f_{1} P_{1}, f_{2} P_{2}), then from the last theorem, we have that f_{1}×f_{2} is ℘continuous. Hence, f_{1}×f_{2} 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 f_{1}:S_{1}→T_{1} and f_{2}:S_{2}→T_{2} are ℘open maps, then f_{1}×f_{2}:S_{1}×S_{2}→T_{1}×T_{2} is also a ℘open map.
Proof
Let U⊆S_{1}×T_{1} 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, (f_{1}×f_{2})[M×N]⊆(f_{1}×f_{2})[U]. Therefore, f_{1}[M]×f_{2}[N]⊆(f_{1}×f_{2})[U]. Since f_{1} and f_{2} are ℘open maps, then f_{1}[M] and f_{2}[N] are ℘open sets in T_{1} and T_{2}, respectively. Hence, f_{1}×f_{2} is ℘open. □
Theorem 13
The maps r_{a} and l_{a} are ℘open maps.
Proof
We only prove that r_{a} 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, r_{a}h: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 ℓ_{a} 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 r_{b}(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 \(SA=\bigcup _{x\neq A}xA\), then S−A is ℘open. Therefore, A is ℘closed. □
Theorem 16
The projection maps P_{1}:S×T→S and P_{2}:S×T→T are ℘open maps.
Proof
we only prove that P_{1} is ℘open, as follows: let W⊆S×T be ℘open and x∈P_{1}[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, P_{1}[W]∈℘−N_{x}. Therefore, P_{1} is ℘open. Similarly, we can prove that P_{2} is ℘open. □
Let {S_{i}: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=〈x_{i}〉:x_{i}∈S_{i}, ∀ 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 e_{S} are ℘continuous, since μ=〈μ_{i} (P_{i}×P_{i})〉, γ=〈γ_{i}P_{i}〉 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 (U_{λ})_{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 ρ_{N}:S→S/N, is called the ℘coset space.
Theorem 20
Let S be a topological partial group and N≤S. Then, the quotient map ρ_{N}: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 ρ_{N} is an identification map and UN is open in S, then ρ_{N}(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 ρ_{N} is a ℘open identification map, then ρ_{N}×ρ_{N} is a ℘ identification map. So, the product map μ:S/N×S/N→S/N is continuous, since μ (ρ_{N}×_{k}ρ_{N})=ρ_{N} μ^{′}, where μ^{′}:S×_{k} S→S is the product map. The partial inverse map γ:S/N→S/Nand the partial identity map e_{S/N}:S/N→S/N are continuous, since γ ρ_{N}=ρ_{N} γ^{′} and e_{S/N} ρ_{N}=ρ_{N} e_{S} are ℘continuous and ρ_{N} is an identification map, where γ^{′}:S→S, x↦x^{−1} and e_{S}:S→S, x↦e_{x} 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=αρ_{K}.
Proof
It is clear that α is bijective and a partial group homomorphism. Also, α is ℘continuous since φ is ℘continuous and ρ_{K} 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 ρ_{N}=αρ_{N/M}ρ_{M}
Proof

(i)
See [4]

(ii)
Let ρ_{N}:S→S/N and ρ_{N}:S→S/M be the quotient maps. Then, ρ_{N} is an idempotent separating surjective ℘morphism and kerρ_{N}=N. So, from the last theorem, there exists a unique bijective ℘morphism φ:S/M→S/N such that φρ_{M}=ρ_{N}. 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 ρ_{N}=αρ_{N/M}ρ_{M}.
□
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 T_{0}space.
Proof
Let S be a Hausdorff partial group. Then, S is a T_{0}space. Conversely, let S be a T_{0}space. Let x,y∈S,x≠y:

(i)
If x,y∈S_{a}, then S_{a} is a T_{2}group and there exist two open sets U,V in S_{a} and also ℘open in S such that \(U\bigcap V\neq \phi \) and x∈U,y∈V and

(ii)
If x∈S_{a} and y∈S_{b}, then, we have that S_{a} and S_{b} 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 kcontinuous 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 nonempty full subcategory of τ
References
 1
Abd Allah, A. M., Aggour, A. I., Fathy, A.: Strong semilattices of topological groups. J. Egyption Math. Soc. 24, 597–602 (2016). https://doi.org/10.1016/j.joems.2016.03.003.
 2
Abd Allah, A. M., Aggour, A. I., Fathy, A.: \(\underline {k}\)partial groups. J. Egyption Math. Soc. 25, 276–278 (2017). https://doi.org/10.1016/j.joems.2017.01.008.
 3
Brown, R.: Topology and Groupoids. McGraw Hill, Oxon (2006). this updated version first published in 2006 by www.groupoids.org.uk Deganwy, United Kingdom.
 4
Howie, J. M.: An introduction to semigroup theory. Academic Press (1976).
 5
Abd Allah, A. M., Abdallah, M. E. G. M.: On Clifford Semigroup,. Pur. Math. Manuscr. 7, 1–17 (1988).
 6
Abd Allah, A. M., Abdallah, M. E. G. M.: Quotient in partial groups. Delta J. Sci. 8(2), 470–480 (1984).
 7
Vogt, R. M.: Convenient category of topological spaces for algebraic topology. Proc. Adv. Study Inst. Alg. Top. Aarhus XXII, 545–555 (1970).
Acknowledgments
We thank our colleagues from AlAzhar 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.
Funding
Not applicable.
Availability of data and materials
Not applicable
Author information
Affiliations
Contributions
The author contributed equally to this work. The author read and approved the final manuscript.
Corresponding author
Correspondence to A. Fathy.
Ethics declarations
Competing interests
The author declares that he/she has no competing interests
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Received
Accepted
Published
DOI
Keywords
 Partial group
 Partial group homomorphism
 Topological group
 Topological partial group
 \(\underline {k}\)partial group
MSC
 22A05
 22A10
 22A20
 54H11