Almost uniformly convergence on MV-algebra of intuitionistic fuzzy sets

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Introduction
The year 2023 is the 40-th anniversary of the invention of the concept and theory of intuitionistic fuzzy sets by K. T. Atanassov in the paper [1].As an intuitionistic fuzzy set A on Ω he understands a pair (µ A , ν A ) of mappings µ A , ν A : Ω → [0, 1] such that µ A +ν A ≤ 1 Ω .The concept of the intuitionistic fuzzy sets is the generalization of the concept of the fuzzy sets introduced by L. Zadeh (see [13,14]).Namely if µ A : Ω −→ [0, 1] is a fuzzy set, then A = (µ A , 1 − µ A ) is the corresponding intuitionistic fuzzy set.Sometimes we need to work with intuitionistic fuzzy events.An intuitionistic fuzzy event is an intuitionistic fuzzy set A = (µ A , ν A ) such that µ A , ν A : Ω → [0, 1] are S-measurable (see [2,3,8]).The family of all IF-events on (Ω, S) will be denoted by F.
In papers [7,9] Riečan constructed the suitable M V -algebra (M, ⊕, ⊙, ¬, (0 Ω , 1 Ω ), (1 Ω , 0 Ω )) to the intuitionistic fuzzy space (F, m).In this paper we study an almost uniformly convergence for a sequence of observables on mentioned M V -algebra and we formulate some definitions of this convergence.As an example of the use of almost uniformly convergence we prove a variation of the Egorov's theorem for M V -algebra of intuitionistic fuzzy sets.This theorem says about a connection between almost everywhere convergence and almost uniformly convergence.We define a partial binary operation ⊖ called difference on M V -algebra of intuitionistic fuzzy sets.We are inspired by the results of B. Riečan in paper [6].There he studied an almost uniformly convergence in D-posets.
Remark that in a whole text we use a notation "IF" in short as the phrase "intuitionistic fuzzy".

MV-algebra of intuitionistic fuzzy sets
In this section we study the properties of the M V -algebra of IF-sets.In papers [7,9] B. Riečan showed that any IF-space F can be embedded to a convenient M V -algebra.Now we recall the basic notions about M V -algebras.By the Mundici theorem any M V -algebra can be defined by the help of an l-group (see [11]).
Definition 2.1 ( [11]).By an ℓ-group we shall mean the structure (G, +, ≤) such that the following properties are satisfied: For each ℓ-group G, an element u ∈ G is said to be a strong unit of G, if for all a ∈ G there is an integer n ≥ 1 such that nu ≥ a (nu is the sum u + . . .+ u with n).
Example 2.1.Let (Ω, S) be a measurable space, S be a σ-algebra.Consider Then (G, +, ≤) is an ℓ-group with the neutral element 0 = (0 Ω , 1 Ω ), and the lattice operations 11]).An M V -algebra is an algebraic system (M, ⊕, ⊙, ¬, 0, u), where ⊕, ⊙ are binary operations, ¬ is a unary operation, 0, u are fixed elements, which can be obtained by the following way: there exists a lattice group (G, +, ≤) such that M = {x ∈ G; 0 ≤ x ≤ u}, where 0 is the neutral element of G, u is a strong unit of G, and Here ∨, ∧ are the lattice operations with respect to the order and ¬a is the opposite element of the element a with respect to the operation of the group.

Definition 2.3 ([11]
).An M V -algebra M is said to be σ-complete if its underlying lattice is σ-complete, i.e., every non-empty countable subset of M has a supremum in M .
Every finite M V -algebra M is σ-complete -indeed, M is complete, in the sense that every non-empty subset of M has a supremum in M .
A finitely additive state is a state, if moreover We say that m is faithful (also called, strictly positive By a state on an M V -algebra M we understand each monotone mapping m : M → [0, 1] i.e.A ≤ B ⇒ m(A) ≤ m(B) satisfying the following conditions: Following proposition says about the properties of a state m on the M V -algebra M .Proposition 2.1 ( [11]).Let m be a finitely additive state on an MV-algebra M .Then we have: (iv) m is also a valuation with respect to the underlying lattice order of M ; stated otherwise, for all a, b ∈ M , we have Each state on M V -algebra is sub-σ-additive.

Lemma 2.1 ([5]
).Let m be a state on MV-algebra M .Then Now we recall the definition of n-dimensional observable in MV-algebras.

Definition 2.5 ([11]
).Let M be an MV-algebra.An n-dimensional observable of M is a map x : B(R n ) → M satisfying the following conditions: When n = 1 we say that x is an observable.
The condition (ii) above states that, whenever A ∩ B = ∅, then x(A ∪ B) = x(A) + x(B) in the ℓ-group with strong unit corresponding to M .

Almost uniformly convergence in MV-algebra of IF-sets
In this section we study an almost uniformly convergence of observables in MV-algebra of IF-sets constructed in Example 2.2.We show some definitions of this convergence.Definition 3.1.Let (M, ⊕, ⊙, ¬, (0 Ω ,1 Ω ),(1 Ω ,0 Ω )) be the M V-algebra constructed in Example 2.2 and m be a state.We say that the sequence (x n ) ∞ 1 of the observables converges m-almost uniformly to 0, if The Definition 3.1 can be rewritten in the following form.Definition 3.2.Let (M, ⊕, ⊙, ¬, (0 Ω ,1 Ω ),(1 Ω ,0 Ω )) be the M V-algebra constructed in Example 2.2 and m be a state.We say that the sequence (x n ) ∞ 1 of the observables converges m-almost uniformly to 0, if Now we define a partial binary operation ⊖ called difference on the M V -algebra of IF-sets and we formulate a definition of almost uniformly convergence using this partial binary operation.We are inspired by paper [6].There B. Riečan studied an almost uniformly convergence in D-posets.
Let (M, ⊕, ⊙, ¬, (0 It is easy to see, that and m be a state.We say that the sequence (x n ) ∞ 1 of the observables converges m-almost uniformly to 0, if In [12] F. Chovanec proved that every M V -algebra M is a D-poset, where b ⊖ a = b ⊙ ¬a.Recall that D-poset is partially ordered set D with the greatest element 1 D and with a partial binary operation ⊖ such that b ⊖ a is defined if and only if a ≤ b and satisfying the following conditions (see [12]): In paper [5] we formulated an almost uniformly convergence for a family of IF-events F. We proved a variation of the Egorov's theorem, too.The results were the generalization of the results in [4], because if Proof.Let a sequence of the observables (x n ) ∞ 1 converges m-almost everywhere to 0. By Definition 2.13 in [11] we have for every p, i.e. lim p→∞ m A p k = 1.By (1) we have that for every α > 0 and every p there exists

Conclusion
The paper is concerned in a probability theory on the M V -algebra (M, ⊕, ⊙, ¬, (0 Ω ,1 Ω ),(1 Ω ,0 Ω )) constructed in Example 2.2.We formulated three definitions of m-almost uniformly convergence for a sequence of observables in the M V -algebra M. We defined a partial binary operation ⊖ called difference on mentioned M V -algebra M. Therefore the M V -algebra M is a D-poset (M, ≤, ⊖, (1 Ω , 0 Ω )).We proved the Egorov's theorem and we showed the connection between an almost everywhere convergence and an almost uniformly convergence of observables in M Valgebra M.