1 Introduction
The classical mathematical tools used for modeling, reasoning and computation have precise and accurate characteristics. This is because the classical mathematics uses two-valued logic in which some kind of certainty is involved. On the other hand, there are complex problems in economics, physics, engineering, biology, sociology, medicine, etc., which include uncertainties and incomplete information. Classical mathematical methods are not adequate for the solution of such problems. To overcome this deficiency, it is common to use such disciplines as fuzzy set theory, probability theory and rough set theory. Of course, each of the aforementioned approaches has its own shortcomings. This is why Modoltsov developed a new theory, namely soft set theory, in his 1999 paper (Molodtsov 1999).
Recently, the theory has been studied and used extensively. For example, the basic aspects of soft set theory were developed in Pie and Miao (2005), Maji et al. (2003) and Chen et al. (2005), algebraic structures of soft sets were studied in Acar et al. (2010), Aktas and Cagman (2007), Jun (2008) and Sun et al. (2008), and applications of soft set theory in game theory and measure theory were given in Molodtsov et al. (2006). The study of soft topological spaces was initiated by Shabir and Naz (2011). Many researchers defined and discussed some properties and basic notions of soft topological spaces (see Zorlutana et al. 2012; Georgiou and Megaritis 2014; Shabir and Ahmad 2015; Öztürk and Bayramo 2017; Terepeta 2019; Al-shami et al. 2018 for example). In the recent decade, algebraic structures endowed with soft topology have been studied. For example, Nazmul and Samanta (2010) and Nazmul and Samanta (2015) introduced and investigated soft topological groups. The notion of soft topological soft ring has been studied by Tahat et al. (2018).
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One important theme in mathematical research is the study of algebraic structures endowed with topology. Topological groups and topological vector spaces are among the most well-known examples of such mathematical objects. It is very important to study the continuity of the algebraic operations, because many results in the theory of these spaces rely on this. In this paper, after endowing soft semigroups with topology, the continuity of the semigroup operation will be examined. Based on the nature of soft semigroups, the concepts of e-right, e-left, e-semi and \(\varepsilon \)-soft continuity are defined. To define the last concept, we first need to introduce the \(\Delta \)-soft topology. Also, k-topology is defined to relate the notions of e-right and e-left soft continuity to \(\varepsilon \)-soft continuity. Next, the space \((Sf_p(X, X, e^\prime ), A)\) will be introduced to clarify the distinction of the aforementioned various kinds of soft continuity.
In this regard, in Sect. 3, we define e-right (e-left) soft topological semigroups and \(\varepsilon \)-soft topological semigroups and establish several theorems on their properties and interrelations. Also, we define and study the concepts of \(\bigtriangleup \)-soft topology and soft k-topology. With these tools, we give theorems, results and important classes of examples. Moreover, it is one of our main results that with extra assumptions, an e-right (e-left) soft topological semigroup is made into an \(\varepsilon \)-soft topological semigroup. In Sect. 4, we define the point open soft topology and study some of its properties. In Theorem 4.4 and Proposition 4.6, we will show that an e-right soft topological semigroup is not necessarily an e-left or e-semisoft topological semigroup. Theorem 4.10 indicates the difference between \(\varepsilon \)-soft topological semigroups and e-semisoft topological semigroups. In addition, we define soft T-spaces as a new soft separation axiom on soft topological spaces. In spaces with this property, and under a certain condition, the e-right (e-left) soft topological semigroup \((Sf_p(X, X, e^\prime ), A)\) is made into an \(\varepsilon \)-soft topological semigroup. Theorem 4.12 gives an important class of \(\varepsilon \)-soft topological semigroups.
2 Preliminaries
In this section, we present some definitions and results of the theory of soft sets and soft topological spaces which will be used later in the paper. The contents can be found in Aktas and Cagman (2007), Chen et al. (2005), Georgiou and Megaritis (2014), Molodtsov et al. (2006), Shabir and Ahmad (2015) and Zorlutana et al. (2012).
Let X be an initial universe and A be a set of parameters, and letEvery element of SS(X, A) is called a soft set over X. If (F, A) and (G, A) are soft sets over X, then Let I be an arbitrary index set and \(\{(F_i, A) : i\in I\} \subseteq SS(X, A)\). The soft union of these soft sets is denoted by \(\sqcup \{(F_i, A): i\in I\}\) and is the soft set \((F, A) \in SS(X, A)\), where the map \(F : A \longrightarrow P(X)\) is defined by \(F(a) =\bigcup \{F_i(a) : i\in I\}\), for every \(a\in A\). Similarly, the soft intersection of the aforementioned soft sets is the soft set \((F, A) \in SS(X, A)\), where the map \(F : A \longrightarrow P(X)\) is defined by \(F(a) =\bigcap \{F_i(a) : i\in I\}\), for every \(a\in A\). The soft intersection is denoted by
$$\begin{aligned} SS(X,A)=\{(F,A): F\ \mathrm {is}\ \mathrm {a}\ \mathrm {map}\ \mathrm {from}\ A\ \mathrm {to}\ P(X)\}. \end{aligned}$$
(1)
(F, A) is said to be a soft subset of (G, A) if \(F(a) \subseteq G(a),\) for every \(a\in A\). In this situation, we write \((F ,A)\sqsubseteq (G, A)\);
(2)
(F, A) and (G, A) are soft equal if \((F, A)\sqsubseteq (G, A)\) and \((G,A)\sqsubseteq (F, A)\), in which case we write \((F, A) = (G,A)\). Also, (F, A) and (G, A) are said to be soft disjoint if \(F(a)\cap G(a)=\phi \) for each \(a\in A\);
(3)
the soft complement of (F, A) is the soft set \((F^c, A),\) where the map \(F^c : A \longrightarrow P(X)\) is defined by \(F^c(a) = X\setminus F(a)\), for every \(a\in A\);
(4)
(0, A) and (1, A) are elements of SS(X, A) such that for each \(a\in A\);
$$\begin{aligned} 0(a)=\phi \ \ \ \ \mathrm {and}\ \ \ \ 1(a)=X,\end{aligned}$$
(5)
(F, A) is said to be a soft point of (X, A) if \(F(a)\not =\phi \) for some \(a\in A\), and \(F(a^\prime )=\phi \) for every \(a^\prime \not = a\). We denote this by \(a_F\). A soft point \(a_F\) belongs to (G, A) if \(F(a)\subseteq G(a)\). If \(F(a)\cap G(a)=\phi \), then \(a_F\not = a_G\) for each \(a\in A\).
$$\begin{aligned} \sqcap \{(F_i, A): i\in I\}. \end{aligned}$$
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Definition 2.1
Georgiou and Megaritis (2014) Suppose \((F, A)\in SS(X, A)\) and \((G, B)\in SS(Y, B)\). The Cartesian product of (F, A) and (G, B) is a soft set \((H, A\times B)\), where \(H:A\times B\longrightarrow P(X\times Y)\) is defined byWe denote the Cartesian product by \((F\times G , A\times B).\)
$$\begin{aligned} H(a, b)=F(a)\times G(b)=\{(x, y): x\in F(a) , y\in G(b)\}. \end{aligned}$$
Definition 2.2
Georgiou and Megaritis (2014) Consider soft sets (X, A) and (Y, B). Let \(f : X\longrightarrow Y\) and \(e : A\longrightarrow B\) be maps. Then, by \(\varphi _{f, e}\) we denote a map from SS(X, A) to SS(Y, B) for which:
(i)
if \((F, A)\in SS(X,A) \), then the image of (F, A) under \(\varphi _{f, e},\) denoted by \(\varphi _{f, e}(F, A)\), is the soft set \((G, B) \in SS(Y, B)\) such that for every \( b \in B\);
$$\begin{aligned} G(b)=\left\{ \begin{array}{lr} \cup \{f(F(a)): a\in e^{-1}(b)\}\ \ \ \ &{} e^{-1}(b)\not =\phi \\ \phi &{}\quad \mathrm{otherwise}, \end{array}\right. \end{aligned}$$
(ii)
if \((G , B)\in SS(Y, B)\), then the inverse image of (G, B) under \(\varphi _{f, e}\), denoted by \(\varphi ^{-1}_{f, e}(G, B)\), is the soft set \((F, A) \in SS(X,A)\) such that \(F(a) = f^{-1}(G(e(a)))\), for every \(a\in A\).
Proposition 2.3
In Definition 2.2, if the map e is a bijection, then the following hold:
(i)
\(G(b)= f(F\circ e^{-1}(b))\) for each \(b\in B\).
(ii)
If f is a bijection, then \(\varphi _{f, e}(F, A)=\varphi ^{-1}_{f^{-1}, e^{-1}}(F,A)\).
Definition 2.4
Georgiou and Megaritis (2014) A family \(\tau \) of subsets of SS(X, A) is said to be a soft topology on (X, A) if \(\tau \) satisfies the following conditions: The triple \((X,\tau , A)\) is called a soft topological space.
(i)
\((0, A), (1,A)\in \tau \).
(ii)
If \((G, A), (H, A) \in \tau \), then \((G, A) \sqcap (H, A) \in \tau \).
(iii)
If \((F_i, A)\in \tau \) for every i in some index set I, then \(\sqcup _i{(F_i, A)}\in \tau \).
If \((X,\tau , A)\) is a soft topological space, then A soft topological space \((X,\tau , A)\) is called
(6)
the members of \(\tau \) are called soft open sets in X;
(7)
a soft set (F, A) is called soft closed if the complement \((F^c, A)\) belongs to \(\tau \). The family of all soft closed sets is denoted by \(\tau ^ c\). The set is called the soft closure of (F, A);
$$\begin{aligned} \quad cl(F,A)=\cap \{(H,A)\in \tau ^c:(F,A)\subseteq (H,A)\}\end{aligned}$$
(8)
the soft topology \(\tau =SS(X,A)\) is known as the discrete soft topology on X, and \((X, \tau ,A)\) is known as the discrete soft topological space;
(9)
if \(a\in A\) and \(x \in X,\) then the soft set \((F, A)\in \tau \) is called the a-soft open neighborhood of x if \(x\in F(a)\);
(10)
a subfamily B of \(\tau \) is said to be a base for \(\tau \) if each member of \(\tau \) is a union of the members of B. Equivalently, B is a base for \(\tau \) if for each \((F, A)\not =(0, A)\), there exists \(\{(G_i, A)\in \mathbf{B }: i\in I\}\) such that \((F, A)=\sqcup \{(G_i, A) : i \in I\}\);
(11)
a subset C of \(\tau \) is called a subbase for \(\tau \) if the set is a base for \(\tau \).
$$\begin{aligned}\{\cap _{i=1}^n(F_i,A):(F_i, A)\in \mathbf{C }, n\ge 1\}\end{aligned}$$
(12)
a soft\(T_0\)-space if for each pair of distinct soft points \(a_F\) and \(a_G\), there exists a soft open set \((F_1, A)\) such that either \(a_F\in (F_1, A)\) and \( a_G\not \in (F_1, A)\) or \(a_F\not \in (F_1, A)\) and \(a_G\in (F_1, A)\);
(13)
a soft\(T_1\)-space if for each pair of distinct soft points \(a_F\) and \(a_G\), there exist soft open sets \((F_1, A)\) and \((F_2, A)\) such that \(a_F\in (F_1, A)\), \( a_G\in (F_2, A)\), \(a_F\not \in (F_2, A)\) and \(a_G\not \in (F_1, A)\);
(14)
a soft\(T_2\)-space if for each pair of distinct soft points \(a_F\) and \(a_G\), there exist soft open sets \((F_1, A)\) and \((F_2, A)\) such that \(a_F\in (F_1, A),\)\(a_G\in (F_2, A)\) and \((F_1, A)\sqcap (F_2, A)=(0,A)\);
(15)
a soft regular space if for each soft point \(a_F\) and for each soft closed set (H, A) not containing \(a_F\), there exist soft open sets \((F_1, A)\) and \((F_2, A)\) such that \(a_F\in (F_1, A)\), \( (H, A)\sqsubseteq (F_2, A)\) and \( (F_2, A)\sqcap (F_1, A)=(0,A)\). Also, a soft regular soft \(T_1\)-space is called a soft\(T_3\)-space;
(16)
a soft normal space if for each pair of distinct soft closed sets (F, A) and (H, A), there exist soft open sets \((F_1, A)\) and \((F_2, A)\) such that \( (H, A)\sqsubseteq (F_2, A)\), \((F, A)\sqsubseteq (F_1, A)\) and \( (F_2, A)\sqcap (F_1, A)=(0,A)\). A soft normal soft \(T_1\)-space is called a soft\(T_4\)-space.
Theorem 2.5
Shabir and Ahmad (2015) For a soft topological space \((X, \tau , A)\), the following statements are equivalent:
(i)
\((X, \tau , A)\) is soft regular.
(ii)
For any soft open set (F, A) in \((X, \tau , A)\) and \(a_G\in (F, A),\) there is a soft open set (G, A) such that
$$\begin{aligned} a_G\in cl(G, A)\subseteq (F, A). \end{aligned}$$
Definition 2.6
Georgiou and Megaritis (2014) Let \((X, \tau _X, A)\) and \((Y, \tau _Y ,B)\) be soft topological spaces, \(x \in X\) and \(e : A\longrightarrow B\) be a map. A map \(f : X \longrightarrow Y\) is called softe-continuous at \(x\in X\) if for every \(a\in A \) and every e(a)-soft open neighborhood (G, B) of f(x) in \((Y, \tau _{Y} ,B),\) there exists an a-soft open neighborhood (F, A) of x in \( (X, \tau _X, A)\) such thatIf the map f is soft e-continuous at every point of X, then we say that the map f is softe-continuous.
$$\begin{aligned} \varphi _{f, e}(F, A) \sqsubseteq (G, B). \end{aligned}$$
Proposition 2.7
Georgiou and Megaritis (2014) Let \((X, \tau _{X}, A)\) and \((Y, \tau _{Y} ,B)\) be soft topological spaces, \(B_{Y}\) be a base (subbase) for \((Y, \tau _{Y} ,B)\) and \(e : A \longrightarrow B \) be a map. Then, the following statements are equivalent:
(i)
A map \(f:X\rightarrow Y\) is soft e-continuous.
(ii)
For each \((G, B)\in {B_Y}\), \(\varphi ^{-1}_{f, e}(G, B)\in \tau _X\).
Proposition 2.8
Georgiou and Megaritis (2014) Let \((X, \tau _{X}, A)\) and \((Y, \tau _{Y} ,B)\) be soft topological spaces and e be a bijective map from A to B. Then, the following statements are equivalent:
(i)
A map \(f : X \longrightarrow Y\) is soft e-continuous.
(ii)
\(\varphi _{f, e}(cl(F, A)) \sqsubseteq cl(\varphi _{f, e}((F, A))),\) for every \((F, A) \in SS(X, A).\)
Definition 2.9
Georgiou and Megaritis (2014) Let \((X, \tau _{X}, A)\) and \((Y, \tau _{Y} ,B)\) be soft topological spaces and e be a bijective map from A to B. A bijective map \(f:X\longrightarrow Y\) is called softe-homeomorphism if f and \(f^{-1} \) are soft e-continuous and soft \(e^{-1}\)-continuous, respectively.
3 \(\varepsilon \)-Soft topological semigroups
In this section, we introduce e-right, e-left and \(\varepsilon \)-soft topological semigroups and examine the way topological semigroups and \(\varepsilon \)-soft topological semigroups are related to each other.
Notation. In this paper, we denote a semigroup by \((S,\mu )\), where \(\mu :S\times S\rightarrow S\) is defined by \(\mu (x,y)=xy\). Also, we will use \(\rho _x\) and \(\lambda _x\) as mappings from S to S defined by
$$\begin{aligned} \rho _{x}(y)=yx,\ \ \lambda _{x}(y)=xy. \end{aligned}$$
Definition 3.1
Let \((S,\tau , A)\) be a soft topological space and \((S,\mu )\) be a semigroup, and assume that \(e:A\rightarrow A\) is the identity map. Then, \((S,\tau , A, \mu )\) is said to be an e-right soft topological semigroup if for every \(x\in S\), the map \(\rho _{x}\) is soft e-continuous. If the map \(\lambda _x\) is soft e-continuous for every \(x\in S\), then \((S,\tau , A, \mu )\) is said to be an e-left soft topological semigroup.
The following example reveals the reason \(e:A\rightarrow A\) must be the identity mapping.
Example 3.2
Let \(\mu :{\mathbb {N}}\times {\mathbb {N}}\rightarrow {\mathbb {N}}\) be defined by \(\mu (m,n)=mn.\) Define the soft topology \(\tau _{{\mathbb {N}}}\) on \({\mathbb {N}}\) bywhere \(A=\{0, 1\}\) and \(F_n:A\longrightarrow P({\mathbb {N}})\), for each \(n\in {\mathbb {N}}\), is given by \(F_n(0)=\{n, n+1, n+2,\ldots \}\) and \(F_n(1)=\phi \). Define the map \(e:A\rightarrow A\) by \(e(0)=1\) and \(e(1)=0\). We show that for every \(g\in {\mathbb {N}}\), \(\rho _g\) is not soft e-continuous. To see this, consider \((F _n, A)\in \tau _{{\mathbb {N}}},\)\(g\in {\mathbb {N}}\) and suppose \(\varphi ^{-1}_{\rho _{g}, e}(F_n, A)=(G, A)\). Then,and \(G(0)=\varphi ^{-1}_{\rho _{g}, e}(F_n, A)(0)=\phi .\) Hence, \((G, A)\not \in \tau _{{\mathbb {N}}}\).
$$\begin{aligned} \tau _{{\mathbb {N}}}=\{ (0, A), (1, A)\}\bigcup \{(F_n, A): n\in {\mathbb {N}}\}, \end{aligned}$$
$$\begin{aligned} G(1)= & {} \varphi ^{-1}_{\rho _{g}, e}(F_n, A)(1)=\rho ^{-1}_g(F_n(e(1))\\= & {} \rho ^{-1}_g(F_n(0))=\rho ^{-1}_g(\{n, n+1,\ldots \})\ne \phi , \end{aligned}$$
Now, if we assume that \(e:A\longrightarrow A\) is the identity map, thenfor every \((F_n, A)\in \tau _{{\mathbb {N}}}\), andHence, \((G, A)=\varphi ^{-1}_{\rho _{g}, e}(F_n, A)\in \tau _{{\mathbb {N}}}\). Therefore, \(\rho _g\) is soft e-continuous for every \(g\in {\mathbb {N}}\).
$$\begin{aligned} G(1)=\varphi ^{-1}_{\rho _{g}, e}(F_n, A)(1)=\phi \end{aligned}$$
$$\begin{aligned} G(0)=\varphi ^{-1}_{\rho _{g}, e}(F_n, A)(0)= & {} \rho ^{-1}_g(\{n, n+1,\ldots \}) \ \\= & {} \{x\in {\mathbb {N}}: x.g\in \{n, n+1,\ldots \}\} \\ {}= & {} \{k, k+1,\ldots \}~~~~\hbox { for some}\ k\in {\mathbb {N}} \\ {}= & {} F_k(0). \end{aligned}$$
Obviously, every semigroup is made into an e-right (e-left) soft topological semigroup if we equip it with the discrete soft topology.
Definition 3.3
Let A be a set of parameters and \(e:A\rightarrow A\) be the identity map. We say that \((S, \tau , A, \mu )\) is e-semisoft topological semigroup if \((S, \tau , A, \mu )\) is e-left and e-right soft topological semigroup.
To define an \(\varepsilon \)-soft topological semigroup S, we need a topology on \(S\times S\). This topology is introduced in Proposition 3.5 and is generalized in Proposition 3.6.
Lemma 3.4
Let \(\textit{B}\subseteq SS(X,A)\). Then, B determines a soft topology on (X, A) if
$$\begin{aligned} \sqcup \{(F, A): (F, A)\in \textit{B}\}=(1, A). \end{aligned}$$
Proof
Let \(\tau \) be the family of arbitrary unions of finite intersections from B. Then by (11), B is a subbase for a soft topology on X. \(\square \)
Proposition 3.5
For \(i=1, 2\), let \((X_i, \tau _i, A)\) be a soft topological space and let \(X=X_1\times X_2\). Define \(\tau _\bigtriangleup \) bywhere \(\bigtriangleup =\{(a, a): a\in A\},\) and the map \(F\times G: \bigtriangleup \longrightarrow P(X)\) is defined by \(F\times G(a, a)=F(a)\times G(a)\), for every \((a, a)\in \bigtriangleup \). Then,
$$\begin{aligned} \tau _\bigtriangleup =\{(F\times G, \bigtriangleup ): (F, A )\in \tau _{1}, (G, A )\in \tau _{2}\}, \end{aligned}$$
(i)
\(\tau _\bigtriangleup \) is a soft topology on \((X, \bigtriangleup )\);
(ii)
if \(\varepsilon \) is a map from \(\bigtriangleup \) to A defined by \(\varepsilon (a,a)=a,\) then the projection maps \(P_i:X\rightarrow X_i\), defined by \(P_i(x_1,x_2)=x_i\), are soft \(\varepsilon \)-continuous for \(i=1,2\).
Proof
(i) Assume \(0_i=(0,A)\) and \(1_i=(1, A)\) for \(i=1,2,\) where \(0_i(a)=\phi \) and \(1_i(a)=X_i,\) for each \(a\in A\). Clearly, \((0, \Delta )=(0_1\times 0_2, \bigtriangleup )\in \tau _\bigtriangleup \) and \((1, \Delta )=(1_1\times 1_2,\bigtriangleup )\in \tau _\bigtriangleup \).
Let \((F_1\times G_1, \bigtriangleup )\) and \((F_2\times G_2, \bigtriangleup )\) be in \(\tau _\bigtriangleup .\) Then, \((F_1, A), (F_2, A)\in \tau _{1}\) and \((G_1, A), (G_2,A)\in \tau _{2}.\) For each \((a, a)\in \bigtriangleup \),Since \((F_1\sqcap F_2, A)\in \tau _{1}\) and \( (G_1\sqcap G_2, A)\in \tau _{2}\),Hence, \(\tau _\Delta \) is closed under finite intersections. Let \(\{(F_\alpha \times G_\alpha , \bigtriangleup ):\alpha \in I\}\) be a family of members of \(\tau _{\bigtriangleup }\). Put \((H, \bigtriangleup )=\sqcup _{\alpha \in I}(F_\alpha \times G_\alpha , \bigtriangleup )\), where \((F_\alpha , A)\in \tau _{1}\) and \((G_\alpha , A)\in \tau _{2}\). For every \((a, a)\in \bigtriangleup \), we haveBy property (iii) of soft topology, \(\sqcup _{\alpha \in I}(F_\alpha , A)\in \tau _{1}\) and \(\sqcup _{\alpha \in I}(G_\alpha , A)\in \tau _{2}\). Hence, \((H, \bigtriangleup )\in \tau _\bigtriangleup \). Therefore, \(\tau _\bigtriangleup \) is a soft topology on \((X, \bigtriangleup )\).
$$\begin{aligned}&(F_1\times G_1)\cap ( F_2\times G_2)(a, a)\\&\quad =(F_1(a)\times G_1(a))\cap (F_2(a)\times G_2(a))\\&\quad =(F_1\cap F_2)(a)\times (G_1\cap G_2)(a)\\&\quad =((F_1\cap F_2)\times (G_1\cap G_2))(a,a). \end{aligned}$$
$$\begin{aligned} (F_1\times G_1, \bigtriangleup )\sqcap (F_2\times G_2, \bigtriangleup )\in \tau _\bigtriangleup . \end{aligned}$$
$$\begin{aligned} H(a, a)&=\cup _{\alpha \in I}(F_\alpha (a)\times G_\alpha (a))=(\cup _{\alpha \in I}F_\alpha (a))\\&\quad \times (\cup _{\alpha \in I}G_\alpha (a)). \end{aligned}$$
(ii) Let \((F,A)\in \tau _1\) and \(\varphi _{P_1,e}^{-1}(F,A)=(G,\bigtriangleup ).\) Then for every \((a,a)\in \bigtriangleup \),Hence, \(\varphi _{P_1,\varepsilon }^{-1}(F,A)=(F\times 1_2,\bigtriangleup )\in \tau _\bigtriangleup .\) Consequently, \(P_1\) is soft \(\varepsilon \)-continuous. Similarly, \(P_2\) is also soft \(\varepsilon \)-continuous. \(\square \)
$$\begin{aligned} G(a,a)=\varphi _{P_1,\varepsilon }^{-1}(F,A)(a,a)=P_1^{-1}(F(a))=F(a)\times X_2. \end{aligned}$$
In the following proposition, assume that \(\bigtriangleup =\{\{a\}_\alpha :\alpha \in I, a\in A\}\) and that \(\varepsilon :\bigtriangleup \rightarrow A\) is a map given by \(\varepsilon (\{a\}_{\alpha \in I})=a.\)
Proposition 3.6
Let \(\{(X_\alpha , \tau _\alpha , A): \alpha \in I \}\) be a family of soft topological spaces and \(X=\prod _{\alpha \in I}X_\alpha .\) If \(P_\alpha :X\longrightarrow X_\alpha \) is the \(\alpha th\) projection map, i.e., \(P_\alpha \{x_\alpha \}_{\alpha \in I}=x_\alpha ,\) then the following hold:
(i)
For each \((F_\beta ,A)\in \tau _\beta \), and every \(a\in A\) and \(\alpha \in I\), if then \(P_\beta ^{-1}(F_\beta (a))=\prod V_\alpha (a)\) and \(\varphi _{P_\beta ,\varepsilon }^{-1}(F_\beta ,A)=(\prod V_\alpha ,\bigtriangleup ).\)
$$\begin{aligned} V_\alpha (a)= \left\{ \begin{array}{l} F_\beta (a) \quad \alpha =\beta \\ X_\alpha \quad otherwise, \\ \end{array} \right. \end{aligned}$$
(ii)
The set \(B=\{\varphi _{P_\alpha ,\varepsilon }^{-1}(F_\alpha ,A):(F_\alpha ,A)\in \tau _\alpha \}\) is a subbase for a soft topology on X. We call it the \(\bigtriangleup \)-soft topology on X and denote it by \(\tau _\bigtriangleup .\)
(iii)
For each \(\alpha \in I,\)\(P_\alpha :X\rightarrow X_\alpha \) is soft \(\varepsilon \)-continuous.
Proof
(i) The proof is straightforward.
(ii) Let \(1_\alpha =(1, A),\) where \(1_\alpha (a)=X_\alpha \) for every \(a\in A\). If \(\varphi _{P_\alpha ,\varepsilon }^{-1}(1_\alpha ,A)=(G,\bigtriangleup ),\) then it follows from (i) that for each \(\{a\}_{\alpha \in I}\in \bigtriangleup \), \(G(\{a\}_{\alpha \in I})=P_\alpha ^{-1}(1_{\alpha }(a))=\prod X_{\alpha }=X.\) Hence, \(\varphi _{P_\alpha ,\varepsilon }^{-1}(1_\alpha ,A)=(1,\bigtriangleup ).\) By Lemma 3.4, B is a subbase for a soft topology.
(iii) By (ii), it is obvious that \(P_\alpha \) is soft \(\varepsilon \)-continuous. \(\square \)
Definition 3.7
Let \((S, \mu )\) be a semigroup and \(\tau \) be a soft topology on (S, A). Let \(\bigtriangleup =\{(a,a):a\in A\}\) and \(\varepsilon :\bigtriangleup \rightarrow A\) be given by \(\varepsilon (a,a)=a.\) We say that \((S, \tau , A,\mu ) \) is an \( \varepsilon \)-soft topological semigroup if the map \(\mu :(S\times S,\tau _\bigtriangleup )\longrightarrow (S,\tau )\) is soft \(\varepsilon \)-continuous.
It is easy to prove that every \(\varepsilon \)-soft topological semigroup is an e-semisoft topological semigroup.
Theorem 3.8
Let \((S, \tau ,\mu )\) be a topological semigroup. Then, \((S, \tau (S), A,\mu )\), whereis an \(\varepsilon \)-soft topological semigroup.
$$\begin{aligned}&\tau (S)=\{(F_U, A): U\in \tau , F_U:A\\&\quad \longrightarrow P(S)\ \mathrm {is}\ \mathrm {given}\ \mathrm {by}\ F_U(a)=U, \forall a\in A\}, \end{aligned}$$
Proof
It is easy to see that \(\tau (S)\) is a soft topology on S. Consider \(\mu :S\times S\longrightarrow S\) given by \(\mu (x, y)=xy\). Take \((H, \bigtriangleup )=\varphi ^{-1}_{\mu , \varepsilon }(F_U, A)\), where \((F_U, A)\in \tau (S)\). For every \((a, a)\in \bigtriangleup ,\)Since \(\mu \) is continuous, there exist open subsets \(U_1\) and \(U_2\) of S such that \(\mu ^{-1}(U)=U_1\times U_2\). Thus, \(H(a, a)=F_{U_1}(a)\times F_{U_2}(a).\) This implies that \((H, \bigtriangleup )=(F_{U_1}\times F_{U_2}, \bigtriangleup )\) is in the \(\bigtriangleup \)-soft topology on S. \(\square \)
$$\begin{aligned} H(a, a)=\mu ^{-1}(F_U(\varepsilon (a, a))=\mu ^{-1}(F_U(a))=\mu ^{-1}(U). \end{aligned}$$
Example 3.9
Let \(S=M_n({\mathbb {R}})\) be the semigroup of all n by n matrices with real entries. We consider \(\tau \) as the subspace topology of \({\mathbb {R}}^{n^2}\) on S. Then, \((S,\tau ,\mu )\) is a topological semigroup. By Theorem 3.8, \((S, \tau (S), A, \mu )\) is an \(\varepsilon \)-soft topological semigroup.
Proposition 3.10
Let \((S, \tau , A, \mu )\) be an \(\varepsilon \)-soft topological semigroup. Then for every \(a\in A, \tau _a=\{F(a): (F, A)\in \tau \}\) is a topology on S such that \((S,\tau _a,\mu )\) is a topological semigroup.
Proof
Let a be an element of A. It is easy to check that \(\tau _a=\{F(a): (F, A)\in \tau \}\) is a topology on S. Take any \((x, y)\in S\times S\) and assume that (F, A) is an a-soft open neighborhood of xy, so that \(\mu (x, y)\in F(a)\). Since \(\mu \) is soft \(\varepsilon \)-continuous, there is an (a, a)-soft open neighborhood of (x, y), say \((G\times H, \bigtriangleup )\), such that \(\varphi _{\mu , \varepsilon }(G\times H,\bigtriangleup )\sqsubseteq (F, A)\). Hence, \( \mu (G(a)\times H( a))\subseteq F(a)\) which implies that \(\mu \) is continuous. \(\square \)
Corollary 3.11
Let \((S, \tau , A,\mu )\) be an \(\varepsilon \)-soft topological semigroup that is also a soft \(T_i\)-space, for \(i=0, 1, 2 ,3 , 4.\) If \(a\in A,\) then \((S,\tau _a,\mu )\) is a \(T_i\)-topological semigroup.
Proof
Let a be an element of A. By Proposition 3.10, \((S,\tau _a,\mu )\) is a topological semigroup. We restrict ourselves to the case of a soft \(T_1\)-space; the other cases can be proved similarly. Let \((S, \tau , A,\mu )\) be a soft \(T_1\)-space and x, y be distinct points in S. Let \(a_F, a_H\) be soft points of (S, A) such that \(F(a)=\{x\},\)\(H(a)=\{y\}\) and \(F(b)=H(b)=\phi ,\) for \(b\not =a\in A.\) Since \((S, \tau , A)\) is a soft \(T_1\)-space, there exist \((G_1, A), (G_2, A) \in \tau \) such that \(a_F\in (G_1, A), a_H\in (G_2, A), a_F\not \in (G_2, A)\) and \( a_H\not \in (G_1, A)\). Thus, \(G_1(a)\) and \(G_2(a)\) are elements of \(\tau _a\) such that\(\square \)
$$\begin{aligned} x\in G_1(a), y\in G_2(a),\ \mathrm {and}\ x\not \in G_2(a), y\not \in G_1(a). \end{aligned}$$
Theorem 3.12
Let \(\{(S_i, \tau _i, A,\mu _i): i\in I \}\) be a family of \(\varepsilon \)-soft topological semigroups and \(S=\prod _{i\in I}S_i\) be the Cartesian product of the semigroups \(S_i.\) Then, \((S, \tau _{2\bigtriangleup }, A,\mu )\) is also an \(\varepsilon \)-soft topological semigroup, where \(2\bigtriangleup =\{(\{a\}_{i\in I},\{a\}_{i\in I}):\{a\}_{i\in I}\in \bigtriangleup \}\) and
$$\begin{aligned} \tau _{2\bigtriangleup }=\{(F\times G, 2\bigtriangleup ): (F, \bigtriangleup ), (G, \bigtriangleup )\in \tau _\bigtriangleup \}. \end{aligned}$$
Proof
Let \(\mu :S\times S\longrightarrow S\) be given by \(\mu (x, y)=xy\). Let \(x=\{x_i\}_{i\in I}\) and \(y=\{y_i\}_{i\in I}\) be elements of S and \((G, \bigtriangleup )=(\prod _{i\in I}G_i, \bigtriangleup )\) be a base element of the \(\bigtriangleup \)-soft topology on (S, A) containing \(\mu (x,y).\) Then for some finite subset \( J=\{i_1, \cdots , i_n\}\) of I, \( (G_i, A)=1_i\) if \(i\in I\setminus J.\) If \(p=\{a\}_{i\in I}\in \bigtriangleup ,\) then for each \(i\in J, (G_i, A)\) is an a-soft open neighborhood of \(x_iy_i\) . Since \(S_i\) is an \(\varepsilon \)-soft topological semigroup, there exist a-soft open neighborhoods \((F_i', A)\) of \(x_i\) and \((H_i', A)\) of \(y_i\) such that \(\varphi _{\mu _i, \varepsilon }(F_i'\times H_i', \bigtriangleup )\sqsubseteq (G_i, A).\)
Take \((F, \bigtriangleup )=(\prod _{i\in I}F_i, \bigtriangleup )\) and \((H, \bigtriangleup )=(\prod _{i\in I}H_i, \bigtriangleup )\), whereThen, these soft sets are p-soft open neighborhoods of x and y, respectively, in the product semigroup S. It follows immediately that \((F\times H, 2\bigtriangleup )\) is in \(\tau _{2\bigtriangleup },\) contains (x, y) andHence, \(\mu \) is soft \(\varepsilon \)-continuous at (x, y). \(\square \)
$$\begin{aligned} F_i = \left\{ \begin{array}{l} F_i' \quad i\in J \\ 1_i \quad \mathrm{otherwise}, \\ \end{array} \right. H_i = \left\{ \begin{array}{l} H_i' \quad i\in J \\ 1_i \quad \mathrm{otherwise}. \end{array} \right. \end{aligned}$$
$$\begin{aligned} \varphi _{\mu , \varepsilon }(F\times H, 2\bigtriangleup )\sqsubseteq (G, \bigtriangleup ). \end{aligned}$$
Notice that, in Theorem 3.12, we restricted ourselves to the case of \(\varepsilon \)-soft topological semigroups. A similar result holds for e-semisoft topological semigroups, e-right soft topological semigroups and e-left soft topological semigroups.
Lemma 3.13
Let \((S,\mu )\) be a semigroup and \(\tau \) be a soft topology on S. If \(e:A\rightarrow A\) is the identity map, then for arbitrary elements x and y of S and every soft set (F, A),
(i)
\(\varphi _{\rho _{xy},e}(F,A)=\varphi _{\rho _{y},e}(\varphi _{\rho _{x},e}(F,A)),\ \ \varphi ^{-1}_{\rho _{xy},e}(F,A)=\varphi ^{-1}_{\rho _{y},e}(\varphi ^{-1}_{\rho _{x},e}(F,A)),\) and
(ii)
\(\varphi _{\lambda _{xy},e}(F,A)=\varphi _{\lambda _{y},e}\circ \varphi _{\lambda _{x},e}(F,A),\ \ \varphi ^{-1}_{\lambda _{xy},e}(F,A)=\varphi ^{-1}_{\lambda _{y},e}(\varphi ^{-1}_{\lambda _{x},e}(F,A)).\)
Proof
(i) Let \(\varphi _{\rho _{x},e}(F,A)=(H,A),\)\(\varphi _{\rho _{y},e}(H,A)=(G,A)\) and \(\varphi _{\rho _{xy},e}(F,A)=(K,A).\) Then, \(\varphi _{\rho _{y},e}\circ \varphi _{\rho _{x},e}(F,A)=(G,A).\) For each \(b\in A,\) by Proposition 2.3, \(H(b)=\rho _x F(e^{-1}(b))=\rho _xF(b).\) Hence,(ii) The proof is similar to that of (i). \(\square \)
$$\begin{aligned} G(b)= & {} \rho _yHe^{-1}(b)=\rho _y\circ \rho _xF(b)=\rho _y\{zx:z\in F(b)\}\\ {}= & {} \{zxy:z\in F(b)\}=\rho _{xy}F(b)=K(b). \end{aligned}$$
Corollary 3.14
Let \((S,\mu )\) be a semigroup and \(\tau \) be a soft topology on S. Then, the setsare subsemigroups of S.
$$\begin{aligned} \rho (S)= & {} \{x\in S: \rho _x\ \mathrm {is}\ \mathrm {soft}\ e-\mathrm {continuous}\}\ \mathrm {and}\ \ \Lambda (S)\\= & {} \{x\in S:\lambda _x\ \mathrm {is}\ \mathrm {soft}\ e-\mathrm {continuous}\} \end{aligned}$$
Proof
By Lemma 3.13, the proof is straightforward. \(\square \)
Theorem 3.15
Let \((S,\tau ,A, \mu )\) be an e-right soft topological semigroup and (N, A) be a soft set over S such that If \((S, \tau )\) is a soft regular space, then there is a soft topology \(\tau _k\) on S such that \(\tau \subseteq \tau _k\) and \((S,\tau _k, A,\mu )\) is an \(\varepsilon \)-soft topological semigroup.
(i)
N(a) is a subset of the set \(\Lambda (S)\) and \( |N(a)|\le |A|\), for every \(a\in A\), and
(ii)
\(cl(N, A)=(1, A)\).
Proof
Let k be a cardinal number such that \(|A|\le k.\) Ifthen B generates a soft topology \(\tau _k\) on S. Obviously, \(\tau \subseteq \tau _k.\) Let \(x, y\in S,\)\(a\in A\) and \((U, A)\in \tau _k\) be an a-soft open neighborhood of \(\mu (x,y)=xy\). Then for some soft set \(\sqcap _{i\in I}(F_i, A)\) in B, \(xy\in \sqcap _{i\in I}(F_i, A)\sqsubseteq (U, A).\) Suppose \(i\in I,\) then \(xy \in F_i(a)\). Since \((S, \tau )\) is a soft regular space, by Theorem 2.5, there is \((W_i, A)\in \tau \) such that \( xy\in W_i(a)\) and \(cl(W_i, A)\sqsubseteq (F_i, A).\) If \((W,A)=\sqcap _{i\in I}(W_i,A),\) thenSince \(\rho _{y}(x)=xy\in W(a)\) and \(\rho _{y}\) is soft e-continuous, we find \((V, A)\in \tau \) such that \(x\in V(a)\) and \(\varphi _{\rho _y,e}(V,A)\sqsubseteq (W, A).\) For every \(b\in A,\)Let \(b\in A\). By (ii), \(V(b)\cap N(b)\ne \phi .\) If \(z_b\in V(b)\cap N(b),\) then \(\lambda _{z_b}(y)=z_{b}y\in V(b)y\subseteq W(b).\) By (i), there exists \((W_{z_b}, A)\in \tau \) such that \( y\in W_{z_b}(b)\) andPutClearly, \(y\in G(a)\). By (i), \((G_b,A)\in \tau _k\), and since \(|A|\le k\), we conclude that \((G, A)\in \tau _k\). On the other hand, if b is an arbitrary element of A, then by (1),This implies that for each \(g\in G(b)\),Since \(cl(N, A)=(1, A)\), we obtain \(cl((N, A)\cap (V, A))=(V, A).\) That \(\rho _g\) is soft e-continuous, by Proposition 2.8, impliesand soBy (2),If \((H_1, A)=\varphi _{\rho _{g}, e}(V, A),\) then for each \(b\in A,\)Thus, \(V(b)G(b)\subseteq U(b).\) Since \((V, A)\in \tau \subseteq \tau _k\) is an a-soft open neighborhood of x and \((G, A)\in \tau _k\) is an a-soft open neighborhood of y, the soft set \((V\times G, \bigtriangleup )\in \tau _\bigtriangleup =\{(U\times V,\bigtriangleup ): (U, A)\in \tau _k, (V,A)\in \tau _k\}\) is an (a, a)-soft open neighborhood of (x, y) . If \((H, A)=\varphi _{\mu , \varepsilon }(V\times G, \bigtriangleup ),\) then for each \(b\in A,\)Therefore, \(\varphi _{\mu , \varepsilon }(V\times G, \bigtriangleup )\sqsubseteq (U,A).\)\(\square \)
$$\begin{aligned} B=\{\sqcap _{i\in I}(F_i, A):(F_i, A)\in \tau , {|I|}\le k\}, \end{aligned}$$
$$\begin{aligned} xy\in W(a)\ \ \mathrm {and}\ cl(W, A)\sqsubseteq \sqcap _{i\in I}(F_i, A)\sqsubseteq (U,A). \end{aligned}$$
$$\begin{aligned} V(b)y=\rho _{y}(V(e^{-1}(b)))\subseteq W(b). \end{aligned}$$
$$\begin{aligned} \varphi _{\lambda _{z_{b}},e}(W_{z_b}, A)\sqsubseteq (W, A). \end{aligned}$$
(1)
$$\begin{aligned} (G_b,A)= & {} \sqcap \{(W_{z_b}, A):z_b\in V(b)\cap N(b)\}\ \mathrm {and}\ (G,A)\\= & {} \sqcap _{b\in A}(G_b,A). \end{aligned}$$
$$\begin{aligned} (V(b)\cap N(b))G(b)\subseteq W(b). \end{aligned}$$
$$\begin{aligned} \varphi _{\rho _{g}, e}((N, A)\cap (V, A))\sqsubseteq (W, A). \end{aligned}$$
(2)
$$\begin{aligned} \varphi _{\rho _{g}, e}(cl((N, A)\cap (V, A)))\sqsubseteq cl(\varphi _{\rho _{g}, e}((N, A)\cap (V, A))) \end{aligned}$$
$$\begin{aligned} \varphi _{\rho _{g}, e} (V, A)\sqsubseteq cl(\varphi _{\rho _{g}, e}((N, A)\cap (V, A))). \end{aligned}$$
$$\begin{aligned} \varphi _{\rho _{g}, e} ((V, A))\sqsubseteq cl(W, A)\sqsubseteq (U, A))). \end{aligned}$$
$$\begin{aligned} V(b)g=\rho _{g}(V(e^{-1}(b)))=H_1(b)\subseteq U(b). \end{aligned}$$
$$\begin{aligned} H(b)= & {} \mu (V\times G(\varepsilon ^{-1}(b)))\\= & {} \mu (V(b), G(b))=V(b)G(b)\subseteq U(b). \end{aligned}$$
Corollary 3.16
If in Theorem 3.15, A is a finite set, then \((S,\tau , A,\mu )\) is an \(\varepsilon \)-soft topological semigroup.
Proof
Let \(k=|A|.\) Clearly, \(B=\{\sqcap _{i\in I}(F_i, A):(F_i, A)\in \tau , {|I|}\le k\}\subseteq \tau ,\) and hence, \(\tau =\tau _k.\)\(\square \)
Theorem 3.17
Let \((S,\tau ,A, \mu )\) be an e-left soft topological semigroup and (N, A) be a soft set over S such that If \((S, \tau _S)\) is a soft regular space, then there is a soft topology \(\tau _k\) on S such that \(\tau \subseteq \tau _k\) and \((S,\tau _k, A,\mu )\) is an \(\varepsilon \)-soft topological semigroup. Moreover, if A is a finite set, then \((S,\tau , A,\mu )\) is an \(\varepsilon \)-soft topological semigroup.
(i)
N(a) is a subset of the set \(\rho (S)\) and \( |N(a)|\le |A|\), for every \(a\in A\), and
(ii)
\(cl(N, A)=(1, A)\).
Proof
The proof is similar to that of Theorem 3.15. \(\square \)
4 Point open soft topology
A useful series of e-right, e-left and e-semisoft topological semigroups appears when considering semigroups of the formTo see this, we first introduce the point open soft topological space \((Sf(X,Y,e'),A)\), and then, we study some properties of this soft space. In Theorem 4.4, we prove that the soft topological space \(Sf(X,X,e')\) along with the operation \((f,g)\rightarrow f\circ g\) is an e-right soft topological semigroup, and in Proposition 4.6, we show that this space is not necessarily an e-left topological semigroup. Theorem 4.10 indicates the difference between \(\varepsilon \)-soft topological semigroups and e-semisoft topological semigroups.
$$\begin{aligned} Sf(X,X)=\{f: f\ \mathrm {is}\ \mathrm {a}\ \mathrm {map}\ \mathrm {from}\ X\ \mathrm {to}\ X\}. \end{aligned}$$
Let \((X, \tau _X, A)\) and \((Y, \tau _Y, B)\) be soft topological spaces and \(Sf(X, Y, e^\prime )\) be the collection of all maps from X to Y, where \(e^\prime :A\longrightarrow B\) is a map of parameters.
Assume that \(x\in X\) and \((G, B)\in \tau _Y\). Define the map \((G, B)^x:A\longrightarrow P(S\textit{f}(X, Y, e^\prime ))\) byfor every \(a\in A\). Then, \(((G, B)^x, A)\in SS(Sf(X, Y, e^\prime ), A)\). If \(B=\{(G, B)^x, A): x\in X, (G, B)\in \tau _Y\}\), then by Lemma 3.4, the set B determines a soft topology on \(Sf(X, Y, e^\prime )\), because \(\sqcup B=(1, A)\). We call this the point open soft topology and denote it by \(\tau _p\). The soft topological space \(((Sf(X, Y, e^\prime ), \tau _p, A)\) will be denoted by \(((Sf_p(X, Y, e^\prime ), A).\)
$$\begin{aligned} (G, B)^x(a)=\{f\in Sf(X, Y, e^\prime ): f(x)\in G(e^\prime (a))\}, \end{aligned}$$
A point open soft topological space \(((Sf_p(X, X, e^\prime ), A)\) has a standard base \(\beta \) which consists of the soft sets \((O(x_1,\ldots , x_n, G_1,\ldots , G_n),A),\) where \(x_1,\ldots , x_n\) are pairwise distinct points of X, \((G_i, A)\not =(0, A)\) is in \(\tau _X\) for \(i=1,\ldots , n\) andis a map which is defined byfor each \(a\in A\).
$$\begin{aligned} O(x_1,\ldots , x_n, G_1,\ldots , G_n):A\longrightarrow P(Sf_p(X, X, e^\prime )) \end{aligned}$$
$$\begin{aligned}&O(x_1,\ldots , x_n, G_1,\ldots , G_n)(a)\\&\quad =\{f\in Sf_p(X, X, e^\prime ):f(x_i)\in G_i(e^\prime (a))\}, \end{aligned}$$
Note that \(\prod _{x\in X}X_x=Sf(X,X),\) where \(X_x=X\) for every \(x\in X.\) Hence, we obtain the following result.
Proposition 4.1
If \((X,\tau _X,A)\) is a soft topological space and \(e':A\rightarrow A\) is the identity map, thenMoreover, the set \(\{((F,A)^x\circ \varepsilon ,\bigtriangleup ):(F,A)\in \tau _X,x\in X\}\) is a base for \(\tau _\bigtriangleup .\)
$$\begin{aligned} \varphi ^{-1}_{\rho _x, \varepsilon }(F, A)=(F,A)^x\circ \varepsilon . \end{aligned}$$
Proof
Let (F, A) be a soft set over X and \((H,\bigtriangleup )=\varphi ^{-1}_{\rho _x, \varepsilon }(F, A).\) Then for every \(\{a\}\in \bigtriangleup ,\)\(\square \)
$$\begin{aligned} H(\{a\})= & {} \rho _{x}^{-1}(F(\varepsilon \{a\}))=\rho _x^{-1}(F(a))\\= & {} \{f\in Sf(X,X):f(x)\in F(e'(a))\}\\= & {} (F,A)^x(a)=(F,A)^x\circ \varepsilon \{a\}. \end{aligned}$$
Proposition 4.2
Let \((X, \tau _X, A)\) be a soft topological space and \((Y,\tau _Y,B)\) be a soft topological \(T_i\)-space for \(i=0,1,2.\) Then, \((Sf_p(X, Y, e^\prime ), A)\) is also a soft \(T_i\)-space.
Proof
Let \((Y, \tau _Y, B)\) be a soft \(T_2\)-space and \(a_G, a_H\) be soft points in \((Sf_p(X, Y, e^\prime ), A).\) Then for every \(a^\prime \in A-\{a\}\), \(G(a^\prime )=H(a^\prime )=\phi \) and \(G(a)\not =H(a)\). Define two maps \(H_1\) and \(H_2\) from B to p(Y) byIf \(b=e^\prime (a),\) then \(H_1(b)\not =H_2(b)\). Hence, \(b_{H_1}\not =b_{H_2}\) are soft points in \((Y,\tau _Y, B).\) Since \((Y, \tau _Y, B)\) is a soft \(T_2\)-space, there exist \((F_1, B)\) and \((F_2, B)\) in \(\tau _Y\) such that \(b_{H_1}\in (F_1, B) \), \(b_{H_2}\in (F_2, B)\) and \((F_1, B)\sqcap (F_2, B)=(0, B)\). Hence \(F_1(b^{'})\cap F_2(b^{'})=\phi \) for any \(b^{'}\in B.\) On the other hand, since \(H_1(b)\not =H_2(b),\) there exist \(f\in S\textit{f}_p(X, Y, e^\prime )\) and \(x\in X\) such thatWe assume that \(f(x)\in H_2(b)\) and \( f(x)\not \in H_1(b)\). It is easy to prove that \((F_1, B)^x(a)\cap (F_2, B)^x(a)=\phi .\) Hence,If \(g\in H(a),\) then \(g(x)\in H_1(e^{'}(a))=H_1(b)\subseteq F_1(b).\) So \(H(a)\subseteq (F_1, B)^x(a).\) Similarly, \(G(a)\subseteq (F_2, B)^x(a)\). The relations \(H(a)\subseteq (F_1, B)^x(a)\) and \(G(a)\subseteq (F_2, B)^x(a)\) imply that \(a_{H}\in ((F_1, B)^x, A)\) and \(a_G\in ((F_2, B)^x, A).\) Therefore, \(S\textit{f}_p(X, Y, e^\prime )\) is a soft \(T_2\)-space. The proof of other cases is similar. \(\square \)
$$\begin{aligned} H_1(b)=\left\{ \begin{array}{lr} \{f(x): x\in X, f\in H(a)\}&{} b=e^\prime (a)\\ \phi &{} \mathrm{otherwise} \end{array} \right. \\ H_2(b)=\left\{ \begin{array}{lr} \{f(x): x\in X, f\in G(a)\}&{} b=e^\prime (a)\\ \phi &{} \mathrm{otherwise}. \end{array}\right. \end{aligned}$$
$$\begin{aligned} f(x)\in H_2(b)- H_1(b)\ or\ f(x)\in H_1(b)- H_2(b). \end{aligned}$$
$$\begin{aligned} ((F_1, B)^x, A)\sqcap ((F_2, B)^x, A)=(0, A). \end{aligned}$$
Proposition 4.3
Let \((X, \tau _X, A)\) be a soft topological space. If a soft topological space \((Y, \tau _Y, B)\) is a soft \(T_3\)-space or a soft \(T_4\)-space, then \((S\textit{f}_p(X, Y, e^\prime ), A)\) has the same property.
Proof
Let \((Y, \tau _Y, B)\) be a soft \(T_4\)-space. By (16), it is a soft normal and a soft \(T_1\)-space. By Theorem 4.2, \((S\textit{f}_p(X, Y, e^\prime ), A)\) is a soft \(T_1\)-space . We claim that \((S\textit{f}_p(X, Y, e^\prime ), A)\) is a soft normal space. To see this, let (H, A) and (G, A) be soft closed sets in \((S\textit{f}_p(X, Y, e^\prime ), A)\) such thatConsider \(a\in A\) and define two maps \(H_1\) and \(G_1\) from B to P(Y) bySince \((Y, \tau _Y, B)\) is a soft \(T_4\)-space and \((H_1, B)\sqcap (G_1, B)=(0, B),\) the soft points \(b_{H_1}, b_{G_1}\) are soft closed. That \((Y, \tau _Y, B)\) is a soft normal space gives us \((H_2, B)\) and \((G_2, B)\) in \(\tau _Y\) such thatSince \( H_1(e^\prime (a))\not =G_1(e^\prime (a),\) we may assume without loss of generality that there exist \(f\in H(a)\) and \(x\in X\) such that \(f(x)\not \in G(a)\). Now, \(((H_2, B)^x, A) \) and \(((G_2, B)^x, A)\) are elements of \(\tau _p\) such that (H, A) is in \(((H_2, B)^x, A) \), (G, A) is in \(((G_2, B)^x, A)\) and \(((H_2, B)^x, A) \sqcap ((G_2, B)^x, A) =(0, A).\) The proof of the other case is similar. \(\square \)
$$\begin{aligned}&(H, A)\not =(0, A), (G, A)\not =(0, A)\\&\quad \hbox { and }(H, A) \sqcap (G, A)=(0, A). \end{aligned}$$
$$\begin{aligned} H_1(b)=\left\{ \begin{array}{lr} \{f(x): x\in X \quad f\in H(a)\}&{} b=e^\prime (a)\\ \phi &{} b\not =e^\prime (a) \end{array}\right. \\G_1(b)=\left\{ \begin{array}{lr} \{f(x): x\in X, f\in G(a)\}&{} b=e^\prime (a)\\ \phi &{} b\not =e^\prime (a). \end{array}\right. \end{aligned}$$
$$\begin{aligned}&(H_1, B)\sqsubseteq (H_2, B), (G_1, B)\sqsubseteq (G_2, B),\\&(H_2, B)\sqcap (G_2, B)=(0, B). \end{aligned}$$
Theorem 4.4
Let \((X, \tau _X, A)\) be a soft topological space and \(e':A\rightarrow A\) be a map of parameters. Then, \((Sf_p(X, X, e^\prime ), A)\) with the operation \((f, g)\longrightarrow f\circ g\) is an e-right soft topological semigroup, where \(e:A\rightarrow A\) is the identity map.
Proof
Clearly, \((Sf_p(X, X, e^\prime ), \circ )\) is a semigroup. We claim that for every \(f\in Sf_p(X, X, e^\prime )\), the map \(\rho _f(g)=g\circ f\) is a soft e-continuous map from \(Sf_p(X, X, e^\prime )\) to \(Sf_p(X, X, e^\prime )\). Let \(g\in Sf_p(X, X, e^\prime ), a\in A\) and \(((G, A)^x, A)\) be an e(a)-soft open neighborhood of \(\rho _f(g)\). Then,implies that \(g\circ f(x)\in (G, A)(e^\prime (a)).\) So \( ((G, A)^{f(x)}, A)\) is an a-soft open neighborhood of g in \(Sf_p(X, X, e^\prime ).\) Let \(\varphi _{\rho _{f}, e}((G, A)^{f(x)}, A)=(H,A).\) Then for each \(a\in A,\)Hence, \(\varphi _{\rho _{f}, e}((G, A)^{f(x)}, A)\sqsubseteq ((G, A)^x, A)\). By Proposition 2.6, \(\rho _f\) is soft e-continuous in g. \(\square \)
$$\begin{aligned} \rho _f(g)=g\circ f\in (G, A)^x(e(a))=(G, A)^x(a) \end{aligned}$$
$$\begin{aligned} H(a)= & {} \rho _f(G,A)^{f(x)}(a)\\= & {} \rho _f\{g\in Sf_p(X, X, e^\prime ) : g\circ f(x)\in G(e^{'}(a))\}\\= & {} \{g\circ f\in Sf_p(X, X, e^\prime ):g\circ f(x)\in G(e^{'}(a))\}\\\subseteq & {} (G,A)^x(a). \end{aligned}$$
Lemma 4.5
Let \((X,\tau _X, A)\) be a soft topological space and \(e:A\rightarrow A\) be a map. Then,
(i)
the set is a soft topology on (X, A),
$$\begin{aligned} \tau _e=\{(U\circ e,A): (U,A)\in \tau _X\} \end{aligned}$$
(ii)
if e is a bijective map and \((F\circ e^{-1},A)\sqsubseteq (U,A),\) then \((F,A)\sqsubseteq (U\circ e,A),\) and
(iii)
if the identity map \(I:(X,\tau _X)\rightarrow (X,\tau _X)\) is soft e-continuous and the map e is a bijection, then \(\tau _e\subseteq \tau _X.\)
Proof
(i) It is easy to show that (0, A) and (1, A) are in \(\tau _e.\) Let \((U\circ e,A)\) and \((V\circ e,A)\) be elements of \(\tau _e\), and assume that \((U\circ e,A)\sqcap (V\circ e,A)=(H,A)\) and \((U,A)\sqcap (V,A)=(K,A)\in \tau _X.\) Then for every \(a\in A,\)Hence, \((H,A)=(K\circ e,A)\in \tau _e.\) Similarly, we can show that \(\tau _e\) is closed under arbitrary unions.
$$\begin{aligned} H(a)&=U\circ e(a)\cap V\circ e(a)\\&=U(e(a))\cap V(e(a))=K(e(a)). \end{aligned}$$
(ii) If \(e:A\rightarrow A\) is a bijection and \(a\in A,\) then there is only one member b in A such that \(a=e^{-1}(b).\) We havewhich implies that \((F,A)\sqsubseteq (U\circ e,A).\)
$$\begin{aligned} F(a)=F(e^{-1}(b)\subseteq U(b)=U(e(a))=U\circ e(a), \end{aligned}$$
(iii) Let \((G,A)\in \tau _e.\) Then for some \((U,A)\in \tau _X,\)\((G,A)=(U\circ e,A).\) Suppose \(a\in A\) and x is a member of G(a). Since the identity map \(I:(X,\tau _X)\rightarrow (X,\tau _X)\) is soft e-continuous and \(I(x)=x\in U(e(a)),\) there exists \((F,A)\in \tau _X\) such thatOn the other hand, by Proposition 2.3, \(\varphi ^{-1}_{I,e^{-1}}(F,A)\sqsubseteq (U,A).\) If \((H,A)=\varphi ^{-1}_{I,e^{-1}}(F,A),\) then for each \(b\in A,\)Hence, \((F\circ e^{-1},A)\sqsubseteq (U,A).\) By (ii), \((F,A)\subseteq (U\circ e,A).\) Therefore, \((G,A)\in \tau _X.\)\(\square \)
$$\begin{aligned} x\in F(a)\ \ \mathrm {and}\ \ \varphi _{I,e}(F,A)\sqsubseteq (U,A). \end{aligned}$$
$$\begin{aligned} F\circ e^{-1}(b)=I\circ F\circ e^{-1}(b)=H(b)\subseteq U(b). \end{aligned}$$
Proposition 4.6
Let \((X, \tau _X, A)\) be a soft topological space, \(e:A\rightarrow A\) be the identity map and \(e':A\rightarrow A\) be a bijective map. Then, \(f\in Sf_p(X, X, e^\prime )\) is soft \(e^\prime \)-continuous if and only if the map \(\lambda _f(g)=f\circ g\) is a soft e-continuous function from \( (Sf_p(X, X, e^\prime ), A)\) to \( (Sf_p(X, X, e^\prime ), A)\).
Proof
Let \(f: X\longrightarrow X\) be soft \(e^\prime \)-continuous. Assume that \(x\in X, (G, A)\in \tau _X\) and \(((G, A)^x, A)\) is an element of the subbase of the point open soft topology on \( (S\textit{f}_p(X, X, e^\prime ), A).\) If \(\varphi ^{-1}_{f,e^{'}}(G,A)=(L,A),\) then for each \(a\in A,\)We show thatLet \(\varphi ^{-1}_{\lambda _{f}, e}((G, A)^x, A)=(H,A)\) and \(a\in A\). Then,Hence,By Proposition 2.7, \((L,A)=\varphi ^{-1}_{f, e^{\prime }}(G, A)\in \tau _X\), and therefore, \( \varphi ^{-1}_{\lambda _{f},e}((G, A)^x,A)\in \tau _p\). By Proposition 2.7, \(\lambda _f\) is soft e-continuous.
$$\begin{aligned} (L,A)^x(a)= & {} \{g\in Sf(X,X, e^\prime ):g(x)\in L( e^\prime (a))\}\\= & {} \{g\in Sf(X,X,e^\prime ):g(x)\in f^{-1}G(e^\prime (a))\}. \end{aligned}$$
$$\begin{aligned} \varphi ^{-1}_{\lambda _{f}, e}((G, A)^x, A)=((L,A)^x,A). \end{aligned}$$
$$\begin{aligned} \begin{array}{ll} H(a)&{}=\{g\in Sf_p(X, X, e^\prime ):g\in \lambda ^{-1}_f((G, A)^x(e(a)))\} \\ &{}=\{g\in Sf_p(X, X, e^\prime ):f\circ g\in ((G, A)^x(a))\} \\ {} &{}=\{ g\in (Sf_p(X, X, e^\prime ): f\circ g(x)\in G(e^\prime (a)\} \\ &{}=\{ g\in (Sf_p(X, X, e^\prime ): g(x)\in f^{-1}G(e^\prime (a)\}\\ &{}= (L,A)^x(a). \end{array} \end{aligned}$$
$$\begin{aligned} \varphi ^{-1}_{\lambda _{f}, e}((G, A)^x, A)=((L,A)^x,A). \end{aligned}$$
Conversely, suppose that \(\lambda _f\) is soft e-continuous. If f is not soft \(e^\prime \)-continuous at \(x_0\in X\), then for some \(a\in A\), an \(e^\prime (a)\)-soft open neighborhood of f(x), say (G, A), and each a-soft open neighborhood (F, A) of \(x_0\) in \(\tau _X\),Since \(\lambda _f\) is soft e-continuous at the identity map \(I:X\rightarrow X,\) for some a-soft open neighborhood \(((G, A)^{x_0}, A)\) of f, there exists \((O(x_1,\ldots , x_n, U_1,\ldots , U_n), A)\) in \(\tau _p\) containing I such thatWe show that the last relation cannot be true. To see this, we must prove thatCase 1. Let \(x_0\in \{ x_1,\ldots , x_n\}\), and assume, for example, that \(x_0=x_1.\) Put \((W,A)=(U_1\circ e',A).\) By Lemma 4.5, (W, A) is an a-soft open neighborhood of \(x_0\) in \(\tau _X.\) By (3), we conclude that \(\varphi _{f,e'}(W,A)\not \sqsubseteq (G,A).\) If \(\varphi _{f,e'}(W,A)=(K,A),\) then for some \(b\in A,\)\(K(b)\not \subseteq G(b).\) Since \(e'\) is a bijection, there exists \(c\in A\) such that \(e'(c)=b\) and \(K(b)=f\circ W(c).\) Consider \(y_0\in W(c)\) such that \(f(y_0)\not \in G(e'(c))\) and define the map \(g:X\rightarrow X\) byThat the identity map I is in \((O(x_1,\ldots ,x_n, U_1,\ldots ,U_n),A)\) givesfor every \(2\le i\le n\). Also, \(g(x_0)=y_0\in W(c)=U_1(e'(c)).\) Therefore, \(g\in (O(x_1,\ldots ,x_n, U_1,\ldots ,U_n),A).\) Thus, (4) is true. This implies thatwhich is a contradiction.
$$\begin{aligned}&\varphi _{f,e^{\prime }}(F, A)\not \sqsubseteq (G, A),\ i.e.,\ \ \varphi _{f, e^{\prime }}(F, A)\nonumber \\&\quad - (G, A)\not =(0, A). \end{aligned}$$
(3)
$$\begin{aligned} \varphi _{\lambda _{f}, e}(O(x_1,\ldots , x_n, U_1,\ldots , U_n), A)\sqsubseteq ((G, A)^{x_0}, A). \end{aligned}$$
$$\begin{aligned}&\exists c\in A\ \exists g\in O(x_1,\ldots ,x_n, U_1,\ldots ,U_n)(c)\ \exists y_0\nonumber \\&\quad \in X\ s.t.\ g(x_0)=y_0,\ f(y_0)\not \in G(e{'}(c)). \end{aligned}$$
(4)
$$\begin{aligned} g(x)=\left\{ \begin{array}{lr} x &{} x\not =x_0\\ y_0 &{} x=x_0. \end{array}\right. \end{aligned}$$
$$\begin{aligned} g(x_i)=x_i=I(x_i)\in U_i(e'(c)), \end{aligned}$$
$$\begin{aligned} \varphi _{\lambda _{f}, e}(O(x_1,\ldots , x_n, U_1,\ldots , U_n), A)\not \sqsubseteq ((G, A)^{x_0}, A), \end{aligned}$$
Case 2. Let \(x_0\not \in \{ x_1,\ldots , x_n\}\) and \((F, A)\in \tau _X\) be an a-soft open neighborhood of \(x_0.\) Then by (3),If \((K,A)=\varphi _{f,e^{\prime }}(F, A),\) then there exist \(b,c\in A\) such that \(e'(c)=b\) and \(f\circ F(c)=K(b)\not \subseteq G(b)=G(e'(c)).\) Consequently, for some \(y_0\in F(c),\)\(f(y_0)\) is not in \(G(e'(c)).\) Define the map \(g:X\rightarrow X\) by \(g(x_0)=y_0\) and \(g(x)=x,\) for all \(x_0\not =x\in X.\) Since \( g(x_i)=x_i=I(x_i)\in U_i\circ e'(c),\) for every \(1\le i\le n,\) we conclude that \((O(x_1,\ldots ,x_n,U_1,\ldots ,U_n),A)\in \tau _p\) is a c-soft open neighborhood of g. But from \(\lambda _f(g)(x_0)=f(y_0)\not \in G(e'(c)),\) we getwhich is a contradiction.
$$\begin{aligned} \varphi _{f,e^{\prime }}(F, A)\not \sqsubseteq (G, A),\ i.e.,\ \ \varphi _{f, e^{\prime }}(F, A)- (G, A)\not =(0, A). \end{aligned}$$
$$\begin{aligned} \varphi _{\lambda _{f}, e}(O(x_1,\ldots , x_n, U_1,\ldots , U_n), A)\not \sqsubseteq ((G, A)^{x_0}, A), \end{aligned}$$
Therefore, by cases (1) and (2), f is soft e -continuous. \(\square \)
Theorem 4.7
Let \((X, \tau _X, A)\) be a soft topological space, \(e:A\rightarrow A\) be the identity map and \(e':A\rightarrow A\) be a bijection. Then, \( (S\textit{f}_p(X, X, e^\prime ), A)\) is not an e-left soft topological semigroup if and only if there exists \(f\in S\textit{f}_p(X, X, e^\prime )\) which is not soft \(e^\prime \)-continuous.
Proof
By Proposition 4.6, the proof is straightforward. \(\square \)
Definition 4.8
Let \((X,\tau _X,A)\) be a soft topological space.
(i)
If \(a_F\) is a soft point, then \((U,A)\in \tau _X\) is called a \(\textit{soft T-neighborhood}\) of \(a_F\) if \(a_F\in (U,A)\) and for every \(a\not =b\in A,\)\( F(a)\cap U(b)=\phi \).
(ii)
The soft topological space \((X,\tau _X,A)\) is called a \(\textit{soft T-space}\) if for every two soft points \(a_F\not = a_H,\) there exist soft T-neighborhoods (U, A) and (V, A) of \(a_F\) and \(a_H,\) respectively, such that
$$\begin{aligned}&\forall b\in A-\{a\}:\ \ F(a)\cap V(b)=\phi \ \ \mathrm {and}\\&\quad H(a)\cap U(b)=\phi . \end{aligned}$$
Clearly, every soft T-space is a soft \(T_2\)-space. The following example presents a soft \(T_2\)-space which is not a soft T-space.
Example 4.9
Let \(X = \{x_1, x_2\}, A = \{a, b\}\) and \(\tau = \{ (0, A), (1, A), (F_1, A), (F_2, A)\},\) whereDefine the maps \(H_1,H_2, G_1\) and \(G_2\) from A to P(X) byThen, \(a_{H_1},a_{H_2},b_{G_1}\) and \(b_{G_2}\) are the only disjoint soft points in SS(X, A). Obviously, \(a_{H_1},b_{G_1}\in (F_1,A)\) and \(a_{H_2},b_{G_2}\in (F_2,A).\) Since \((F_1,A)\sqcap (F_2,A)=(0,A),\)\((X,\tau ,A)\) is a soft \(T_2\)-space. On the other hand, \(H_1(a)\cap F_2(b)\not =\phi \) implies that \((X,\tau ,A)\) is not a soft T-space.
$$\begin{aligned} F_1 = \{(a, \{x_1\}), (b,\{x_2\})\}, F_2 = \{(a,\{x_2\}, (b,\{x_1\})\}. \end{aligned}$$
$$\begin{aligned} H_1=\{(a,\{x_1\}),(b,\phi )\},\ \ H_2=\{(a,\{x_2\},(b,\phi )\},\\G_1=\{(a,\phi ),(b,\{x_1\})\},\ \ G_2=\{(a,\phi ),(b,\{x_2\})\}. \end{aligned}$$
Now, suppose that \(\tau {=}\{ (0, A), (1, A), (F_1, A),\ldots , (F_7, A)\}\), whereand \(F_7=\{(a,\{x_1\}),(b,\{x_2\})\}.\) It is easy to see that \((X,\tau ,A)\) is a soft T-space.
$$\begin{aligned} F_1= & {} \{(a,\phi ),(b,\{x_1\})\},\ F_2=\{(a,\phi ),(b,\{x_2\})\},\\ F_3= & {} \{(a,\phi ),(b,X)\}\\ F_4= & {} \{(a,\{x_1\}),(b,\phi )\},\\ F_5= & {} \{(a,\{x_2\}),(b,\phi )\},\ F_6=\{(a,X),(b,\phi )\} \end{aligned}$$
Theorem 4.10
Let \((X, \tau _X, A)\) be a soft T-space, \(e:A\rightarrow A\) be the identity map and \(e':A\rightarrow A\) be a bijective map. Then, the following are equivalent:
(i)
\((X, \tau _X, A)\) is a discrete soft topological space.
(ii)
\((S\textit{f}_p(X, X, e^\prime ), A)\) is an \(\varepsilon \)-soft topological semigroup.
(iii)
\((S\textit{f}_p(X, X, e^\prime ), A)\) is an e-semi soft topological semigroup.
Proof
The proofs of \((i\Rightarrow ii)\) and \((ii\Rightarrow iii)\) are straightforward. We show that (iii) implies (i). Let \((S\textit{f}_p(X, X, e^\prime ), A)\) be an e-semisoft topological semigroup. For every \(f\in (S\textit{f}_p(X, X, e^\prime ), A),\) the left action \(\lambda _f\) from \((S\textit{f}_p(X, X, e^\prime ), A)\) to \( (S\textit{f}_p(X, X, e^\prime ), A)\) is soft e-continuous. By Proposition 4.6, \(f:X\rightarrow X\) is soft \(e^\prime \)-continuous. Consider \(x_0\in X,\) and let \(a_F\) be a soft point such that \(F(a)=\{x_0\}.\) We show that \(a_F\in \tau _X.\) To see this, take \(x_1\in X\) such that \(x_0\not =x_1\) and define \( f:X\rightarrow X\) byLet \(a_H\) be a soft point such that \(H(a)=\{x_1\}.\) Since \((X,\tau _X,A)\) is a soft T-space, there exist soft T-neighborhoods (U, A) and (W, A) such that \(a_F\in (U,A)\) and \(a_H\in (W,A).\) So, \(x_1\not \in U(a)\) and \(x_0\not \in W(a).\) Let \(\varphi ^{-1}_{f,e'}(U,A)=(K,A).\) If \(x\in f^{-1}(U(a)),\) then \(f(x)\in U(a).\) Hence, \(f(x)=x_0.\) Assume \(e'(b)\not =a\) and \(x\in f^{-1}(U\circ e'(b)).\) If \(f(x)=x_1,\) then \(x_1\in H(a)\cap U(e'(b))=\phi ,\) a contradiction. If \(f(x)=x_0,\) then \(x_0\in F(a)\cap U(e'(b))=\phi ,\) a contradiction. Hence, \(f(x)\not =x_0,x_1.\) Thus for each \(b\in A,\)This implies that \(\varphi ^{-1}_{f,e^{'}}(U,A)=(F,A).\) Since f is soft \(e^\prime \)-continuous, \(a_F=(F,A)\in \tau _X\). Now, it is easy to see that \((X,\tau _X,A)\) is a discrete space. \(\square \)
$$\begin{aligned} f(x)=\left\{ \begin{array}{ll} x_0 &{} x=x_0 \\ x_1 &{} x\not =x_0. \end{array}\right. \end{aligned}$$
$$\begin{aligned} K(b)=f^{-1}(U\circ e'(b))=\left\{ \begin{array}{ll} \{x_0\} &{} e'(b)=a \\ \phi &{} e'(b)\not =a. \end{array}\right. \end{aligned}$$
Example 4.11
Let \(( X , \tau _{X}, E)\) be a soft T-space andBy Theorem 4.4, \((Sf_pc(X, X,e^\prime ), A, \circ )\) is an e-semisoft topological semigroup, but Theorem 4.10 shows that it is not necessarily an \(\varepsilon \)-soft topological semigroup.
$$\begin{aligned}&Sfc(X, X,e^\prime )=\{ f \in S\textit{f}_p(X, X, e^\prime ): \\&\quad f\hbox { is soft }e^\prime \hbox {-continuous}\}. \end{aligned}$$
Theorem 4.12
Let \((S, \tau _S, B,\mu )\) be an \(\varepsilon \)-soft topological semigroup and \((X, \tau _X, A)\) be a soft topological space. Define \(\mu ':Sf(X, S, e^\prime )\times Sf(X, S, e^\prime )\longrightarrow Sf(X, S, e^\prime )\) byfor every \(x\in X\). Then, \((Sf_p(X, S, e^\prime ), A ,\mu ')\) is an \(\varepsilon \)-soft topological semigroup.
$$\begin{aligned} \mu (f , g)(x)=f(x)g(x), \end{aligned}$$
Proof
Clearly, \((Sf(X, S, e^\prime ), \mu ')\) is a semigroup. We prove that \(\mu '\) is soft \(\varepsilon \)-continuous. Let \(a\in A\) and \(((F, B)^x, A) \) be an a-soft open neighborhood of \(\mu '(f , g).\) Then, \( f(x)g(x)\in F(e^{'}(a)).\) Since \(\mu :S\times S\longrightarrow S\) is soft \(\varepsilon \)-continuous, there exist \(e^\prime (a)\)-soft open neighborhoods \((H_1, B)\) and \((H_2,B)\) of f(x) and g(x), respectively, such that \(\varphi _{\mu , e}(H_1\times H_2, \Delta )\sqsubseteq (F, B)\). If \((M_1, A)=((H_1, B)^x, A)\) and \((M_2, A)=((H_2, B)^x, A),\) then they are a-soft open neighborhoods of f and g, respectively, such that \(\varphi _{\mu , e}(M_1\times M_2, \Delta )\sqsubseteq ((F,B)^x, A)\). Therefore, \((Sf_p(X, S, e^\prime ), A)\) is an \(\varepsilon \)-soft topological semigroup. \(\square \)
Corollary 4.13
If \((S, \tau _S, A) \) is an \(\varepsilon \)-soft topological semigroup, then \((S\textit{f}_{p}(S, S, e^\prime ), A)\) with the operation defined in the previous theorem is an \(\varepsilon \)-soft topological semigroup. Moreover, if \((S, \tau _{S}, A) \) is a soft T-space, then \((S, \tau _{S},A) \) is a discrete soft topological space.
5 Conclusion and suggestions for future work
Soft sets play an important role in the solution of some problems in economics, engineering and physics which include uncertainty and incomplete information. Soft algebraic structures and soft topological spaces are two important areas of study in the theory of soft sets. Moreover, the important objects of soft sets represent a blend of algebraic structures and soft topological structures. In these objects, the continuity of the involved operations is beneficial. In this paper, the authors introduced e-right, e-left, e-semi and \(\varepsilon \)-soft topological semigroups and studied some of their soft topological properties. The point open soft topological space \((Sf(X,Y,e'),A)\) has given a useful series of e-right, e-left, e-semi and \(\varepsilon \)-soft topological semigroups and has indicated the differences between them. To achieve these goals, the authors needed to define the concepts of \(\Delta \)-soft topology, point open soft topology, compact open soft topology, soft T-Space, and to study their properties carefully.
Suggested studies
We propose the study of
1.
e-right (e-left, e-semi) soft topological groups and \( \varepsilon \)-soft topological groups;
2.
separation axioms on \(\varepsilon \)-soft topological groups;
3.
uniformities on \(\varepsilon \)-soft topological groups.
Acknowledgements
The authors would like to express their sincere gratitude to the referees for their valuable suggestions and comments.
Compliance with ethical standards
Conflict of interest
The authors declare that they have not conflict of interest.
Human participants
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