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  • Original research
  • Open Access

A convenient category of topological partial groups

Journal of the Egyptian Mathematical Society201927:8

https://doi.org/10.1186/s42787-019-0010-4

  • Received: 15 September 2018
  • Accepted: 28 November 2018
  • Published:

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).

Keywords

  • Partial group
  • Partial group homomorphism
  • Topological group
  • Topological partial group
  • \(\underline {k}\)-partial group

MSC

  • 22A05
  • 22A10
  • 22A20
  • 54H11

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:
  1. (i)

    If aS, then the right transformation ra:SS,xxa and the left transformation la:SS,xax, may not be open.

     
  2. (ii)

    The quotient map ρN:SS,xxN, NS, 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].

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, xS 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 xS. Then, an element eS is called a partial identity of x if:
  1. (i)

    ex=xe=x,

     
  2. (ii)

    If ex=xe=x, for some eS, then ee=ee=e.

     

Theorem 1

[5] Let S be a semigroup. Then,
  1. (i)

    If xS 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 ex the partial identity of the element xS.

Definition 3

[5] Let S be a semigroup and xS has a partial identity element ex. Then, yS is called a partial inverse of x if:
  1. (i)

    xy=yx=ex,

     
  2. (ii)

    exy=yex=y.

     

We will denote the partial inverse y of xS by x−1.

Definition 4

[5] A semigroup S is called a partial group if:
  1. (i)

    Every xS has a partial identity ex

     
  2. (ii)

    Every xS has a partial inverse x−1

     
  3. (iii)

    The map eS:SS,xex is a semigroup homomorphism

     
  4. (iv)

    The map γ:SS,xx−1 is a semigroup antihomomorphism.

     

So, every group is a partial group.

Definition 5

[6] Let S be a partial group and xS. Then, we define Sx = {y S:ex=ey}.

Theorem 2

[5] Let S be a partial group and xS. Then,

  1. (i)

    Sx is a maximal subgroup of S which has identity ex

     
  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:CX 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:XY is called k-continuous if fα:CY is continuous, for each test map α:CX.

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×SS,(x,y)xy

     
  2. (ii)

    The partial identity map: eS:SS,xex

     
  3. (iii)

    The partial inverse map: γ:SS,xx−1.

     

Definition 9

[1] Let S be a topological partial group and aS. Then, the map ra:SS,xxa is called a right transformation and the map la:SS,xax 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]:
  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:YX×XY,(f,x)f(x) and xX, is continuous, where YX 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:CS is called a -test map if h is continuous and \(h^{-1}(S_{e_{x}})\) is open in C for each exE(S), where Cobj().

-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:ST is called -continuous if fh:CT is continuous, for each a -test map h:CS.

We note that every continuous map of topological partial group is -continuous. So, the following maps are -continuous:
  1. (i)

    The identity map I:SS

     
  2. (ii)

    The partial identity map: eS:SS,xex

     
  3. (iii)

    The partial inverse map: γ:SS,xx−1

     
  4. (iv)

    The maps ra and la.

     

Definition 12

Let f:ST 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:ST and g:TF are -morphisms, then gf:SF is also a -morphism.

Proof

It is clear that gf is a partial group homomorphism. Let h:CS be a −test map. Since f is -continuous, then fh:CT is continuous. Now, (fh)−1(Te)=h−1(f−1(Te)), for each eE(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 eE(T). That means that fh is a -test map. Since g is -continuous, then g(fh)=(gf)h:CF 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:CS

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)λL be a subfamily of -open sets. Then, h−1[Uλ] are open in C, for each λL. 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 xS if there exists a -open set U in S such that xUA.

The family of all -neighbourhoods of xS is called a -neighbourhood system and is denoted by Nx

Proposition 1

A subset AS 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, xAA, for all xA. Hence, A is a -neighbourhood of x. Conversely, for each xA, there exists a -open set Ux such that xUxA. So, \(A=\bigcup _{x\in A}U_{x}\). Hence, A is a -open set. □

Theorem 6

Let S be a topological partial group and xS. Then,
  1. (i)

    xN, for all NNx

     
  2. (ii)

    If NNx and NM, then MNx

     
  3. (iii)

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

     
  4. (iv)

    If NNx, then there exists MNx such that NNy, for each yM.

     

Proof

  1. (i)

    If NNx, then there exists a -open set U in S such that xUN. Hence, xN.

     
  2. (ii)

    If NNx, then there exists a -open set U in S such that xUN. Since, NM, then xUM. Hence, MNx.

     
  3. (iii)

    If N,MNx, then there exist two -open sets U and V, respectively such that xUN and xVM. 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 NNx, then there exists a -open set M in S such that xMN. Since M is a -open set, then MNy, for all yM. Since, NM, then NNy, for each yM.

     

Definition 15

Let S be a topological partial group and AS. Then, xA 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,BS. Then,
  1. (i)

    A0A

     
  2. (ii)

    If AB, then A0B0

     
  3. (iii)

    A0 is a -open set

     
  4. (iv)

    (A0)0=A0.

     

Proof

  1. (i)

    Let xA0. Then, ANx. So, xA.

     
  2. (ii)

    Let xA0. Then, ANx. Since, AB, then BNx and so xB0. Hence, A0B0.

     
  3. (iii)

    Let xA0. Then, ANx. Thus, there exists NNx such that ANy, for all yN. That is, yA0, for all yN. Hence, NA. Thus, xNA0. So, ANx. Therefore, A0 is a -open set.

     
  4. (iv)

    Since A0A, then from (ii) (A0)0A0. It remains that A0(A0)0. This is given from xA0. That is, A0Nx. 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 SA is a -open set.

Definition 17

Let S be a topological partial group and AS. Then, xS is called a -closure point of A if \(A\bigcap N\neq \phi \), for each NNx.

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, AτA since \(\phi =\phi \bigcap A\) and \(A=A\bigcap S\). Let M,NτA. 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)λL 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:ST be -continuous. Then, fA:AT is -continuous.

Proof

Let UT 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 AS.

Definition 19

Let S and T be topological partial groups and let (x,y)S×T. The set −(S×T), where MNx in S and NNy 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×NU.

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, xU and yV. So, UNx in S and VNy in T. This implies U×V is a -basic neighbourhood of (x,y). Since (x,y)U×VA×B, then U×VN(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 SC and TD are -open sets in S and T, respectively, then (SCT and S×(TD) 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×TS and P2:S×TT.

     
  2. (ii)

    The product map μ:S×SS.

     
  3. (iii)

    The diagonal map ΔS={(x,x):xS}.

     

Theorem 9

If f:ST and f:SF are -morphisms, then (f,g):ST×F is also a -morphism.

Proof

It is clear that (f,g) is a partial group homomorphism. Let h:CS be a -test map. Since f is -continuous, then fh:CT is continuous. Also, since g is -continuous, then gh:CT is continuous. So, (fh,gh)=(f,g)h:ST×F is continuous. That is, (f,g) is -continuous. Hence, (f,g) is a -morphism. □

Theorem 10

If f1:S1T1 and f2:S2T2 are -morphisms, then f1×f2:S1×S2T1×T2 is also a -morphism.

Proof

It is clear that f1×f2:S1×S2T1×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:ST.
  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 UT be -open. So, h−1[f−1[U]]=(fh)−1[U] is open in C, for each -test map h:CT. Hence, f−1[U] is a -open set in S.

(ii) → (iii) Let U be -closed in T. So TU is -open in T. Therefore, f−1[TU]=Sf−1[U] is -open in S. Hence, f−1[U] is -closed in S.

(iii) → (ii) Let U be -open in T. So, TU is -closed in T. Therefore, f−1[TU]=Sf−1[U] is -closed in S. Hence, f−1[U] is -open in S.

(iii) → (i) Let h:CS be a -test map and UT 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:ST is called -open if f(U) is -open in T for each -open set U in S. Also, the map f:ST is called -closed if f(U) is -closed in T for each -closed set U in S.

Theorem 12

If f1:S1T1 and f2:S2T2 are -open maps, then f1×f2:S1×S2T1×T2 is also a -open map.

Proof

Let US1×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[Mf2[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 US 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:CS be a -test map. Then, rah:CS 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 a is -open. □

Theorem 14

Let S be a topological partial group and A,BS. 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 xS. Since \(S-A=\bigcup _{x\neq A}xA\), then SA is -open. Therefore, A is -closed. □

Theorem 16

The projection maps P1:S×TS and P2:S×TT are -open maps.

Proof

we only prove that P1 is -open, as follows: let WS×T be -open and xP1[W]. Then, there exists yT 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×NW. 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〉:xiSi, 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:ST if, for any topological partial group F and all maps g:TF, we have that g is -continuous if gf:SF is -continuous.

Theorem 18

The τ final topology on T with respect to the function f:ST exists and is characterized by the following condition: If UT, 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)λL 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:ST is called -identification if f is surjective and T has the -final topology with respect to f.

Theorem 19

Let f:ST be a -continuous surjection. If f is a -open (closed) map. Then, f is a -identification map.

Proof

Let UT 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 NS, then S/N with the -identification topology, with respect to the quotient map ρN:SS/N, is called the -coset space.

Theorem 20

Let S be a topological partial group and NS. Then, the quotient map ρN:SS/N is -open.

Proof

Let US be open. Then,
$$\begin{array}{l} {\rho_{N}^{-1} (\rho_{N} (U))= \left\{x\in S:\rho_{N} (x)\in \rho_{N} (U)\right\}\, \, \, \, \, \, \, \, \, \,} \\ {\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =\left\{x\in S:xN\in U/N\right\}} \\ {\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =\left\{x\in S:x\in aN \,\,\,\,\textrm{for some}\,\,\, a\in U\right\}} \\ {\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =\bigcup_{a\in U}aN} \\ {\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =UN.} \end{array}$$

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/NS/N is continuous, since μ (ρN×kρN)=ρN μ, where μ:S×k SS is the product map. The partial inverse map γ:S/NS/Nand the partial identity map eS/N:S/NS/N are continuous, since γ ρN=ρN γ and eS/N ρN=ρN eS are -continuous and ρN is an identification map, where γ:SS, xx−1 and eS:SS, xex are -continuous. □

Theorem 22

Let φ:ST be an idempotent separating surjective -morphism and K=kerφ. Then, there exists a unique bijective -morphism α:S/KT 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 MN, then
  1. (i)

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

     
  2. (ii)

    There exists a unique bijective -morphism α:(S/M)/(N/M) such that ρN=αρN/MρM

     

Proof

  1. (i)

    See [4]

     
  2. (ii)

    Let ρN:SS/N and ρN:SS/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/MS/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,yS, there exist -open sets U and V such that xU,yV, 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,yS,xy:
  1. (i)

    If x,ySa, 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 xU,yV and

     
  2. (ii)

    If xSa and ySb, 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:ST are -morphisms of topological partial group, then the difference kernel A={xS:f(x)=g(x)} is a -closed subpartial group.

Proof

A is closed (see [3]). Let x,yA. Now,
$$\begin{array}{l} {f(x\, y^{-1})=f(x)\, f(y^{-1})} \\ {\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =f(x)\, f(y)^{-1}} \\ {\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =g(x)\, g(y)^{-1} =g(xy^{-1})}. \end{array}$$
Therefore, xy−1A. 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

Declarations

Acknowledgments

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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Authors’ contributions

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

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Authors’ Affiliations

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

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