**2. Preliminaries**

#### *2.1. Some Basic Concepts of IFS, FMS*

**Definition 1** ([2])**.** *Let X be a nonempty set. An IFS M in X is given by*

$$M = \{ \langle \mathbf{x}, \mu\_M(\mathbf{x}), \nu\_M(\mathbf{x}) \rangle | \mathbf{x} \in X \}\tag{1}$$

*where μM* : *X* → [0, 1] *and νM* : *X* → [0, 1] *with the condition* 0 ≤ *μM* + *νM* ≤ 1 *for all x* ∈ *X*.

Here *μM*(*x*), *<sup>ν</sup>M*(*x*) ∈ [0, 1] denote the membership and the non-membership functions of the fuzzy set *M*.

**Definition 2** ([10])**.** *A fuzzy multiset M is a generation set of multisets over the universe X, which is denoted by pairs, where the first part of each pair is the element of X, and the second part is the membership of the element relative to M. Note that an element of X may occur more than once in the same or different membership values. For each x* ∈ *X, a membership sequence is defined to be the decreasing ordered sequence of the elements, that is,*

$$\left(\mu^1\_M(\mathfrak{x}), \mu^2\_M(\mathfrak{x}), \dots, \mu^q\_M(\mathfrak{x})\right),$$

*where <sup>μ</sup>*<sup>1</sup>*M*(*x*) ≥ *<sup>μ</sup>*<sup>2</sup>*M*(*x*) ≥···≥ *<sup>μ</sup>qM*(*x*)*. Hence, the FMS M is given by*

$$M = \left\{ \left( \mu\_M^1(\mathbf{x}), \mu\_M^2(\mathbf{x}), \dots, \mu\_M^q(\mathbf{x}) \right) \Big| \mathbf{x} \right\}, \text{ for all } \mathbf{x} \in \mathbf{X}. \tag{2}$$

#### *2.2. Some Concepts of SVNMS*

**Definition 3** ([11])**.** *Let X be a nonempty set with a generic element in X denoted by x. A SVNMS M in X is characterized by three functions: count truth-membership of CTM, count indeterminacy-membership of C IM, and count falsity-membership of CFM, such that CTM*(*x*) : *X* → *R, C IM*(*x*) : *X* → *R, CFM*(*x*) : *X* → *R, for every x* ∈ *X, where R is the set of all real number multisets in the real unit interval* [0, 1]*. Then, a SVNMS M is given by*

$$\mathcal{M} = \left\{ \left< \mathbf{x}, \left( T\_M^1(\mathbf{x}), T\_M^2(\mathbf{x}), \dots, T\_M^k(\mathbf{x}) \right), \left( I\_M^1(\mathbf{x}), I\_M^2(\mathbf{x}), \dots, I\_M^k(\mathbf{x}) \right), \left( F\_M^1(\mathbf{x}), F\_M^2(\mathbf{x}), \dots, F\_M^k(\mathbf{x}) \right) \right> \middle| \mathbf{x} \in X \right\}, \quad \forall X$$

*where the truth-membership sequence T*1*M*(*x*), *<sup>T</sup>*2*M*(*x*), ··· , *<sup>T</sup>kM*(*x*)*, the indeterminacy-membership sequence I*1*M*(*x*), *<sup>I</sup>*2*M*(*x*), ··· , *<sup>I</sup>kM*(*x*)*, and the falsity-membership sequence F*1*M*(*x*), *<sup>F</sup>*2*M*(*x*), ··· , *<sup>F</sup>kM*(*x*) *may be in decreasing order or not. Additionally, the TjM*(*x*), *<sup>I</sup>jM*(*x*), *<sup>F</sup>jM*(*x*) *also satisfies the following condition*

$$0 \le T\_M^{\dot{j}}(\mathbf{x}) + I\_M^{\dot{j}}(\mathbf{x}) + F\_M^{\dot{j}}(\mathbf{x}) \le 3,\text{ for all } \mathbf{x} \in \mathbf{X},\text{ } j = 1, 2, \dots, k.$$

In order to express more concisely, a SVNMS *M* over *X* can be given by

$$M = \left\{ \left< \mathbf{x}, T\_M^j(\mathbf{x}), I\_M^j(\mathbf{x}), F\_M^j(\mathbf{x}) \right> \middle| \mathbf{x} \in X, j = 1, 2, \cdots, k \right\} \tag{3}$$

Furthermore, we represent the set of all SVNMSs on *X* as SVNMS(X).

**Definition 4** ([11])**.** *Let M* ∈ *SVNMS*(*X*)*, for every element x included in M, the length of x is defined as the cardinal number of CTM*(*x*) *or C IM*(*x*)*, or CFM*(*x*)*, and is expressed as l*(*x* : *<sup>M</sup>*)*. That is, l*(*x* : *M*) = |*CTM*(*x*)| = |*C IM*(*x*)| = |*CFM*(*x*)|*. Suppose M*, *N* ∈ *SVNMS*(*X*)*, then, l*(*x* : *M*, *N*) = max{*l*(*x* : *<sup>M</sup>*), *l*(*x* : *<sup>N</sup>*)}.

**Definition 5** ([11])**.** *An absolute SVNMS M is a SVNMS, whose TjM* (*x*) = 1*, IjM* (*x*) = 0 *and FjM* (*x*) = 0*, for all x* ∈ *X and j* = 1, 2, ··· , *l*(*x* : *M* ).

**Definition 6** ([11])**.***A null SVNMS* Φ *is a SVNMS, whose Tj*Φ (*x*) = 0*, Ij*Φ (*x*) = 1 *and Fj*Φ (*x*) = 1*, for all x* ∈ *X and j* = 1, 2, ··· , *l*(*x* : Φ ).

Let *M*, *N* ∈ *SVNMS*(*X*). In order to further study the operations between *M* and *N*, we must verify that *l*(*x* : *M*) = *l*(*x* : *N*) is true for every *x* ∈ *X*, if not, we use a sufficient number of zeroes to fill the truth-membership values and a sufficient number of ones to fill the indeterminacy-membership values and falsity-membership values of the smaller-length sequences, respectively, so that the lengths of sequences are equal to facilitate computing.

**Definition 7** ([11])**.** *Let M* = *x*, *TjM*(*x*), *<sup>I</sup>jM*(*x*), *<sup>F</sup>jM*(*x*)HHH*x* ∈ *X*, *j* = 1, 2, ··· , *l*(*x* : *M*)*and N* = *<sup>x</sup>*, *TjN*(*x*), *<sup>I</sup>jN*(*x*), *<sup>F</sup>jN*(*x*)HHH*x* ∈ *X*, *j* = 1, 2, ··· , *l*(*x* : *N*) *be two SVNMSs in X. Then, we have*

*(1) Inclusion: M* ⊆ *N if and only if TjM*(*x*) ≤ *TjN*(*x*)*, <sup>I</sup>jM*(*x*) ≥ *<sup>I</sup>jN*(*x*)*, <sup>F</sup>jM*(*x*) ≥ *<sup>F</sup>jN*(*x*) *for j* = 1, 2, ··· ,*l*(*<sup>x</sup>* : *M*, *<sup>N</sup>*)*; (2) Equality: M* = *N if and only if M* ⊆ *N and N* ⊆ *M;*

*(3) Complement: Mc* = *x*, *<sup>F</sup>jM*(*x*), 1 − *<sup>I</sup>jM*(*x*), *TjM*(*x*)HHH*x* ∈ *X*, *j* = 1, 2, ··· , *l*(*x* : *<sup>M</sup>*)*; (4) Union: M* ∪ *N* = *x*, *TjM*(*x*) ∨ *TjN*(*x*), *<sup>I</sup>jM*(*x*) ∧ *<sup>I</sup>jN*(*x*), *<sup>F</sup>jM*(*x*) ∧ *<sup>F</sup>jN*(*x*)HHH*x* ∈ *X*, *j* = 1, 2, ··· , *l*(*x* : *M*, *<sup>N</sup>*)*; (5) Intersection: M* ∩ *N* = *x*, *TjM*(*x*) ∧ *TjN*(*x*), *<sup>I</sup>jM*(*x*) ∨ *<sup>I</sup>jN*(*x*), *<sup>F</sup>jM*(*x*) ∨ *<sup>F</sup>jN*(*x*)HHH*x* ∈ *X*, *j* = 1, 2, ··· , *l*(*x* : *M*, *<sup>N</sup>*)*; (6) Addition: M* ⊕ *N* = *x*, *TjM*(*x*) + *TjN*(*x*) − *TjM*(*x*)*TjN*(*x*), *<sup>I</sup>jM*(*x*)*IjN*(*x*), *<sup>F</sup>jM*(*x*)*FjN*(*x*)HHH*x* ∈ *X*, *j* = 1, 2, ··· , *l*(*x* : *M*, *<sup>N</sup>*)*; (7) Multiplication: M* ⊗ *N* = *x*, *TjM*(*x*)*TjN*(*x*), *<sup>I</sup>jM*(*x*) + *<sup>I</sup>jN*(*x*) − *<sup>I</sup>jM*(*x*)*IjN*(*x*), *<sup>F</sup>jM*(*x*) + *<sup>F</sup>jN*(*x*) − *<sup>F</sup>jM*(*x*)*FjN*(*x*)HHH *x* ∈ *X*, *j* = 1, 2, ··· , *l*(*<sup>M</sup>*, *<sup>N</sup>*)}.

**Definition 8** ([14])**.** *Let M* = *xi*, *TjM*(*xi*), *<sup>I</sup>jM*(*xi*), *<sup>F</sup>jM*(*xi*)HHH*xi* ∈ *X*; *i* = 1, 2, ··· , *n*; *j* = 1, 2, ··· , *l*(*x* : *M*) *and N* = *xi*, *TjN*(*xi*), *<sup>I</sup>jN*(*xi*), *<sup>F</sup>jN*(*xi*)HHH*xi* ∈ *X*; *i* = 1, 2, ··· , *n*; *j* = 1, 2, ··· , *l*(*x* : *N*) *be two SVNMSs in X* = {*<sup>x</sup>*1, *x*2, ··· , *xn*}*. Now, we propose the generalized distance measure between M and N as follows:*

$$D\_{P}(M,N) = \left[\frac{1}{n}\sum\_{i=1}^{n}\frac{1}{N\_{l}}\sum\_{j=1}^{l\_{l}}\left(\left|T\_{M}^{l}(\mathbf{x}\_{l}) - T\_{N}^{l}(\mathbf{x}\_{l})\right|^{P} + \left|I\_{M}^{l}(\mathbf{x}\_{l}) - I\_{N}^{l}(\mathbf{x}\_{l})\right|^{P} + \left|F\_{M}^{l}(\mathbf{x}\_{l}) - F\_{N}^{l}(\mathbf{x}\_{l})\right|^{P}\right)\right]^{\frac{1}{P}},\tag{4}$$

*where li* = *l*(*xi* : *M*, *N*) = max{*l*(*xi* : *<sup>M</sup>*), *l*(*xi* : *N*)} *for i* = 1, 2, ··· , *n*.

If *P* = 1, 2, it reduces to the Hamming distance and the Euclidean distance, which are usually applied to real science and engineering areas.

Based on the relationship between the distance measure and the similarity measure, we can introduce two distance-based similarity measures between *M* and *N*:

$$S\_1(M, N) = 1 - D\_P(M, N),\tag{5}$$

$$S\_2(M, N) = \frac{1 - D\_P(M, N)}{1 + D\_P(M, N)}.\tag{6}$$

(*M* ∩ *Q*)*;*

#### **3. Some New Properties of SVNMS**

The operation of SVNMS is discussed in depth and certain theoretical results are obtained. On this basis, this section generalizes the union and intersection operations of two SVNMSs to the general case, that is, for any indicator set. In addition, this section presents the arithmetic properties of SVNMSs.

**Remark 1.** *The union and intersection operations of the two SVNMSs can be extended to general case, that is, for any index set T, if Mt* ∈ *SVNMS*(*X*)*,* ∀*t* ∈ *T, we can define*

$$\cup\_{t \in T} M\_t = \left\{ \left< \mathbf{x}, \vee\_{t \in T} T\_{M\_l}^j(\mathbf{x}), \wedge\_{t \in T} I\_{M\_l}^j(\mathbf{x}), \wedge\_{t \in T} F\_{M\_l}^j(\mathbf{x}) \right> \middle| \mathbf{x} \in X, j = 1, 2, \dots, l\_x \right\},$$

*and*

$$\cap\_{t \in T} M\_t = \left\{ \left< \mathbf{x}, \wedge\_{t \in T} T\_{M\_t}^j(\mathbf{x}), \vee\_{t \in T} I\_{M\_t}^j(\mathbf{x}), \vee\_{t \in T} F\_{M\_t}^j(\mathbf{x}) \right> \, \middle| \, \mathbf{x} \in X, j = 1, 2, \dots, l\_x \right\},$$

*where lx* = max{ *l*(*x* : *Mt*)|*t* ∈ *<sup>T</sup>*}.

**Proposition 1.** *Let M, N and Q be three SVNMSs in X. We have the following operational properties:*

*(1) Commutation: M* ∪ *N* = *N* ∪ *M, M* ∩ *N* = *N* ∩ *M; (2) Association: M* ∪ (*N* ∪ *Q*) = (*M* ∪ *N*) ∪ *Q, M* ∩ (*N* ∩ *Q*) = (*M* ∩ *N*) ∩ *Q; (3) Idempotent: M* ∪ *M* = *M, M* ∩ *M* = *M; (4) Absorption: M* ∪ (*M* ∩ *N*) = *M, M* ∩ (*M* ∪ *N*) = *M; (5) Identity: M* ∪ *M* = *M; M* ∩ *M* = *M, M* ∪ Φ = *M, M* ∩ Φ = Φ *; (6) Distribution: M* ∪ (*N* ∩ *Q*) = (*M* ∪ *N*) ∩ (*M* ∪ *Q*)*, M* ∩ (*N* ∪ *Q*) = (*M* ∩ *N*) ∪ *(7) Involution: Mc<sup>c</sup>* = *M, M<sup>c</sup>* = Φ *,* Φ*<sup>c</sup>* = *M; (8) De Morgan:* (*M* ∩ *N*)*<sup>c</sup>* = *Mc* ∪ *Nc ,* (*M* ∪ *N*)*<sup>c</sup>* = *Mc* ∩ *Nc* 

**Remark 2.** *As we know, the complementation can be established in classical set, however, it is not true in SVNMS. For example, let X* = {*<sup>x</sup>*1, *x*2, *<sup>x</sup>*3}*, M* ∈ *SVNMS*(*X*) *as follows:*

.

$$M = \{ \langle \mathbf{x}\_1, (0.5, 0.3), (0.1, 0.1), (0.7, 0.8) \rangle, \langle \mathbf{x}\_2, (0.7, 0.68, 0.62), (0.3, 0.45, 0.5), (0.34, 0.28, 0.49) \rangle, \langle \mathbf{x}\_3, (0.6, 0.28, 0.49) \rangle, \langle \mathbf{x}\_4, (0.6, 0.28, 0.49) \rangle, \langle \mathbf{x}\_5, (0.6, 0.28, 0.49) \rangle, \langle \mathbf{x}\_6, (0.6, 0.28, 0.28) \rangle, \langle \mathbf{x}\_7, (0.6, 0.28, 0.28) \rangle, \langle \mathbf{x}\_8, (0.6, 0.28, 0.28) \rangle, \langle \mathbf{x}\_9, (0.6, 0.28, 0.28) \rangle, \langle \mathbf{x}\_{10}, (0.6, 0.28, 0.28) \rangle, \langle \mathbf{x}\_{11}, (0.6, 0.28, 0.28) \rangle, \langle \mathbf{x}\_{12}, (0.6, 0.28, 0.28) \rangle, \langle \mathbf{x}\_{13}, (0.6, 0.28, 0.28) \rangle, \langle \mathbf{x}\_{14}, (0.6, 0.28, 0.28) \rangle, \langle \mathbf{x}\_{15}, (0.6, 0.28, 0.28) \rangle, \langle \mathbf{x}\_{18}, (0.6, 0.28, 0.28) \rangle, \langle \mathbf{x}\_{19}, (0.6, 0.28, 0.28) \rangle, \langle \mathbf{x}\_{18}, (0.6, 0.28, 0.28) \rangle$$

*Obviously,*

*M* ∪ *Mc* = {*<sup>x</sup>*1,(0.7, 0.8),(0.1, 0.1),(0.5, 0.3),*<sup>x</sup>*2,(0.7, 0.68, 0.62),(0.3, 0.45, 0.5),(0.34, 0.28, 0.49), *<sup>x</sup>*3,(0.67, 0.5, 0.7),(0.2, 0.3, 0.4),(0.4, 0.5, 0.3)} = *M* ;

*M* ∩ *Mc* = {*<sup>x</sup>*1,(0.5, 0.3),(0.9, 0.9),(0.7, 0.8),*<sup>x</sup>*2,(0.34, 0.28, 0.49),(0.7, 0.55, 0.5),(0.7, 0.68, 0.62), *<sup>x</sup>*3,(0.4, 0.5, 0.3),(0.8, 0.7, 0.6),(0.67, 0.5, 0.7)} = Φ .

#### **4. Decomposition Theorem and Representation Theorem of SVNMS**

In this section, the notions of cut sets, strong cut sets of SVNMS are defined. Some properties of cut sets are proposed. We also investigate decomposition theorem and representation theorem of SVNMS based on cut sets.

## *4.1. Decomposition Theorem*

**Definition 9.** *Let X* = {*<sup>x</sup>*1, *x*2, ··· , *xn*}*, A* ∈ *SVNMS*(*X*) *and α*, *β*, *γ* ∈ [0, 1] *with* 0 ≤ *α* + *β* + *γ* ≤ 3*. The α*-*cut set of truth value function generated by A is defined as follows:*

$$A^a = \left\{ \mathbf{x}\_{\bar{i}} \in X \middle| T\_A^j(\mathbf{x}\_{\bar{i}}) \ge \mathbf{a}; \mathbf{i} = 1, 2, \dots, \mathbf{a}; \mathbf{j} = 1, 2, \dots, l(\mathbf{x}\_{\bar{i}} : A) \right\}; \tag{7}$$

*The strong α*-*cut set of truth value function generated by A is defined as follows:*

$$A^{a+} = \left\{ \mathbf{x}\_i \in X \Big| T\_A^j(\mathbf{x}\_i) > a; i = 1, 2, \dots, n; j = 1, 2, \dots, l(\mathbf{x}\_i; A) \right\};\tag{8}$$

*The β*-*cut set of indeterminacy value function generated by A is defined as follows:*

$$A\pounds = \left\{ \mathbf{x}\_{i} \in X \Big| I\_{A}^{j}(\mathbf{x}\_{i}) \le \pounds; i = 1, 2, \dots, n; j = 1, 2, \dots, l(\mathbf{x}\_{i}; A) \right\};\tag{9}$$

*The strong β*-*cut set of indeterminacy value function generated by A is defined as follows:*

$$A\beta + = \left\{ \mathbf{x}\_i \in X \Big| l\_A^j(\mathbf{x}\_i) < \beta; i = 1, 2, \dots, n; j = 1, 2, \dots, l(\mathbf{x}\_i : A) \right\};\tag{10}$$

*The γ*-*cut set of falsity value function generated by A is defined as follows:*

$$A\_{\gamma} = \left\{ \mathbf{x}\_{i} \in X \Big| F\_{A}^{j}(\mathbf{x}\_{i}) \le \gamma; i = 1, 2, \dots, n; j = 1, 2, \dots, l(\mathbf{x}\_{i} : A) \right\};\tag{11}$$

*The strong γ*-*cut set of falsity value function generated by A is defined as follows:*

$$A\_{\gamma+} = \left\{ \mathbf{x}\_{i} \in X \Big| F\_{A}^{j}(\mathbf{x}\_{i}) < \gamma; i = 1, 2, \dots, n; j = 1, 2, \dots, l(\mathbf{x}\_{i}; A) \right\}. \tag{12}$$

Next, we can define the (*<sup>α</sup>*, *β*, *<sup>γ</sup>*)-*cut* sets as follows:

$$A^{(\mathbf{a},\boldsymbol{\beta},\cdot,\gamma)} = \left\{ \mathbf{x}\_{i} \in X \Big| T\_{A}^{j}(\mathbf{x}\_{i}) \ge \mathbf{a}, \boldsymbol{I}\_{A}^{j}(\mathbf{x}\_{i}) \le \boldsymbol{\beta}, \boldsymbol{F}\_{A}^{j}(\mathbf{x}\_{i}) \le \gamma; i = 1, 2, \dots, n; j = 1, 2, \dots, l(\mathbf{x}\_{i}:A) \right\};\tag{13}$$

$$A^{(\mathbf{a}+,\mathbf{\bar{s}},\gamma)} = \left\{ \mathbf{x}\_{i} \in X \Big| T\_{A}^{j}(\mathbf{x}\_{i}) > u, I\_{A}^{j}(\mathbf{x}\_{i}) \le \beta, \mathcal{F}\_{A}^{j}(\mathbf{x}\_{i}) \le \gamma; i = 1, 2, \dots, u; j = 1, 2, \dots, l(\mathbf{x}\_{i}:A) \right\}; \tag{14}$$

$$A^{(\mathbf{a}\_r, \mathbf{f} + \dots \mathbf{e}\_r)} = \left\{ \mathbf{x}\_i \in X \Big| \boldsymbol{T}\_A^j(\mathbf{x}\_i) \ge \mathbf{a}\_r \boldsymbol{I}\_A^j(\mathbf{x}\_i) < \boldsymbol{\beta}, \boldsymbol{F}\_A^j(\mathbf{x}\_i) \le \gamma; i = 1, 2, \dots, n; j = 1, 2, \dots, l(\mathbf{x}\_i; A) \right\};\tag{15}$$

$$A^{(\mathbf{a}\_r \mathbf{f}\_r^\perp \gamma +)} = \left\{ \mathbf{x}\_i \in X \Big| T\_A^j(\mathbf{x}\_i) \ge \mathbf{a}\_r I\_A^j(\mathbf{x}\_i) \le \beta\_r F\_A^j(\mathbf{x}\_i) < \gamma; i = 1, 2, \dots, n; j = 1, 2, \dots, l(\mathbf{x}\_i : A) \right\};\tag{16}$$

$$A^{(\mathbf{a}+,\mathbf{\bar{p}}+,\mathbf{\gamma})} = \left\{ \mathbf{x}\_{i} \in \mathcal{X} \middle| T\_{A}^{j}(\mathbf{x}\_{i}) > \mathbf{a}, \mathbf{l}\_{A}^{j}(\mathbf{x}\_{i}) < \mathfrak{f}, \mathbf{\bar{F}}\_{A}^{j}(\mathbf{x}\_{i}) \le \gamma; i = 1, 2, \dots, n; j = 1, 2, \dots, l(\mathbf{x}\_{i}:A) \right\}; \tag{17}$$

$$A^{(\mathbf{a}+,\mathbf{\bar{p}},\gamma+)} = \left\{ \mathbf{x}\_{l} \in X \Big| T\_{A}^{\circ}(\mathbf{x}\_{l}) > a, \mathbf{l}\_{A}^{\circ}(\mathbf{x}\_{l}) \le \beta, \mathbf{f}\_{A}^{\circ}(\mathbf{x}\_{l}) < \gamma; i = 1, 2, \dots, n; j = 1, 2, \dots, l(\mathbf{x}\_{l}:A) \right\}; \tag{18}$$

$$A^{(\mathbf{a},\mathbf{\hat{p}}+,\gamma+\cdot)} = \left\{ \mathbf{x}\_i \in X \Big| T\_A^j(\mathbf{x}\_i) \ge \mathbf{a}, \mathbf{l}\_A^j(\mathbf{x}\_i) < \beta\_r F\_A^j(\mathbf{x}\_i) < \gamma; i = 1, 2, \dots, n; j = 1, 2, \dots, l(\mathbf{x}\_i:A) \right\};\tag{19}$$

$$A^{\{a+,\emptyset+,\emptyset+,\gamma+\}} = \left\{ \mathbf{x}\_{i} \in \mathcal{X} \Big| T\_{A}^{j}(\mathbf{x}\_{i}) > a, I\_{A}^{j}(\mathbf{x}\_{i}) < \emptyset, F\_{A}^{j}(\mathbf{x}\_{i}) < \gamma; i = 1,2,\cdots,n; j = 1,2,\cdots,l(\mathbf{x}\_{i}:A) \right\}.\tag{20}$$

The *α*-*cut* sets, *β*-*cut* sets, *γ*-*cut* sets of SVNMS satisfy the following properties:

**Theorem 1.** *Let A*, *B* ∈ *SVNMS*(*X*)*, α*, *β*, *γ* ∈ [0, 1] *with* 0 ≤ *α* + *β* + *γ* ≤ 3. *Then,*

$$\begin{aligned} (1) \quad & A \subseteq B \Rightarrow A^{\underline{a}} \subseteq B^{\underline{a}}, \; A\beta \subseteq B\beta, \; A\_{\gamma} \subseteq B\_{\gamma};\\ (2) \quad & (A \cap B)^{\underline{a}} = A^{\underline{a}} \cap B^{\underline{a}}, \; (A \cap B)\beta = A\beta \cap B\beta, \; (A \cap B)\_{\gamma} = A\_{\gamma} \cap B\_{\gamma};\\ (3) \quad & (A \cup B)^{\underline{a}} = A^{\underline{a}} \cup B^{\underline{a}}, \; (A \cup B)\beta = A\beta \cup B\beta, \; (A \cup B)\_{\gamma} = A\_{\gamma} \cup B\_{\gamma};\\ (4) \quad & \left(\bigcap\_{t \in T} A\_{t}\right)^{\underline{a}} = \bigcap\_{t \in T} (A\_{t})^{\underline{a}}, \left(\bigcap\_{t \in T} A\_{t}\right)\beta = \bigcap\_{t \in T} (A\_{t})\beta, \left(\bigcap\_{t \in T} A\_{t}\right)\_{\gamma} = \bigcap\_{t \in T} (A\_{t}) \end{aligned}$$

$$\begin{array}{rcl} \text{(4)} & \left( \underset{t \in T}{\cap} A\_t \right) &=& \underset{t \in T}{\cap} \left( A\_t \right)^a \; \Big( \underset{t \in T}{\cap} A\_t \Big) \nexists \emptyset = \underset{t \in T}{\cap} \left( A\_t \right) \emptyset \; \Big( \underset{t \in T}{\cap} A\_t \Big)\_{\gamma} =& \underset{t \in T}{\cap} \left( A\_t \right) \; \gamma \\\\ \text{(5)} & \left( \underset{\text{(5)}}{\dots} \dots \text{)} \; \Big( \underset{\text{(5)}}{\dots} \dots \underset{\text{(5)}}{\dots} \dots \; \underset{\text{(5)}}{\dots} \dots \; \underset{\text{(5)}}{\dots} \dots \; \Big) \; \Big) \; \end{array}$$

$$(\mathfrak{F}) \quad \left(\bigvee\_{t \in T} A\_t\right)^{\mathfrak{a}} = \bigvee\_{t \in T} (A\_t)^{\mathfrak{a}}, \\
\left(\bigvee\_{t \in T} A\_t\right) \mathfrak{F} = \bigsqcup\_{t \in T} (A\_t) \mathfrak{F}, \\
\left(\bigvee\_{t \in T} A\_t\right)\_{\gamma} = \bigsqcup\_{t \in T} (A\_t)\_{\gamma'};$$

$$(6)\quad a\_1 \ge a\_2, \beta\_1 \le \beta\_2, \gamma\_1 \le \gamma\_2 \Rightarrow A^{a\_1} \subseteq A^{a\_2}, \ A\beta\_1 \subseteq A\beta\_2, \ A\_{\gamma\_1} \subseteq A\_{\gamma\_2}.$$
