A convenient category of topological partial groups

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).

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:

1. (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.

2. (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.

3. (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].

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:

1. (i)

   ex=xe=x,

2. (ii)

   If e^′x=xe^′=x, for some e^′∈S, then ee^′=e^′e=e.

Theorem 1

[5] Let S be a semigroup. Then,

1. (i)

   If x∈S has a partial identity, then it is unique

2. (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:

1. (i)

   xy=yx=e_x,

2. (ii)

   e_xy=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:

1. (i)

   Every x∈S has a partial identity e_x

2. (ii)

   Every x∈S has a partial inverse x⁻¹

3. (iii)

   The map e_S:S→S,x↦e_x is a semigroup homomorphism

4. (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,

1. (i)

   S_x is a maximal subgroup of S which has identity e_x

2. (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:

1. (i)

   The product map: μ:S×S→S,(x,y)↦xy

2. (ii)

   The partial identity map: e_S:S→S,x↦e_x

3. (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 non-empty full subcategory of τ which satisfies the following conditions [7]:

1. (i)

   If A is a closed subspace of an object B of ℘, then A is a k℘-space.

2. (ii)

   If B and C are objects in ℘, then B×C is also object in ℘.

3. (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.

4. (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(℘).

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:

1. (i)

   The identity map I:S→S

2. (ii)

   The partial identity map: e_S:S→S,x↦e_x

3. (iii)

   The partial inverse map: γ:S→S,x↦x⁻¹

4. (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,

1. (i)

   x∈N, for all N∈N_x

2. (ii)

   If N∈N_x and N⊆M, then M∈N_x

3. (iii)

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

4. (iv)

   If N∈N_x, then there exists M∈N_x such that N∈N_y, for each y∈M.

Proof

1. (i)

   If N∈N_x, then there exists a ℘-open set U in S such that x∈U⊆N. Hence, x∈N.

2. (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.

3. (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 \(N\bigcap M\in N_{x}\).

4. (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 ℘−A⁰.

Proposition 2

Let S be a topological partial group and A,B⊆S. Then,

1. (i)

   ℘−A⁰⊆A

2. (ii)

   If A⊆B, then ℘−A⁰⊆℘−B⁰

3. (iii)

   ℘−A⁰ is a ℘-open set

4. (iv)

   (℘−A⁰)⁰=℘−A⁰.

Proof

1. (i)

   Let x∈℘−A⁰. Then, A∈N_x. So, x∈A.

2. (ii)

   Let x∈℘−A⁰. Then, A∈N_x. Since, A⊆B, then B∈N_x and so x∈℘−B⁰. Hence, ℘−A⁰⊆℘−B⁰.

3. (iii)

   Let x∈℘−A⁰. Then, A∈N_x. Thus, there exists N∈N_x such that A∈N_y, for all y∈N. That is, y∈℘−A⁰, for all y∈N. Hence, N⊆A. Thus, x∈N⊆℘−A⁰. So, A∈N_x. Therefore, ℘−A⁰ is a ℘-open set.

4. (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∈N_x. 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∈℘−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

1. (i)

   If A and B are ℘-open sets in S and T, respectively, then A×B is also ℘-open in S×T

2. (ii)

   If C and D are ℘-closed sets in S and T, respectively, then C×D is also ℘-closed in S×T.

Proof

1. (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.

2. (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:

1. (i)

   The projection maps P₁:S×T→S and P₂:S×T→T.

2. (ii)

   The product map μ:S×S→S.

3. (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 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.

1. (i)

   f is ℘-continuous

2. (ii)

   f⁻¹[U] is a ℘-open set in S for each ℘-open set U in T.

3. (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 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:

1. (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_ah: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.

2. (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 \(S-A=\bigcup _{x\neq A}xA\), 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]∈℘−N_x. Therefore, P₁ is ℘-open. Similarly, we can prove that P₂ 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_iP_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. □

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×_kρ_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

1. (i)

   \(N/M \unlhd S/M\)

2. (ii)

   There exists a unique bijective ℘-morphism α:(S/M)/(N/M) such that ρ_N=αρ_N/Mρ_M

Proof

1. (i)

   See [4]

2. (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/Mρ_M.

□

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:

1. (i)

   If x,y∈S_a, then S_a is a T₂-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

2. (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.

\(\mathbf {K}\acute {}\) :

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

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.

Funding

Not applicable.

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Affiliations

1. Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, 11884, Cairo, Egypt

   • A. Fathy

Contributions

The author contributed equally to this work. The author read and approved the final manuscript.

Corresponding author

Correspondence to A. Fathy.

Competing interests

The author declares that he/she has no competing interests

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