On the migrativity of uni-nullnorms 1

Abstract

Uni-nullnorms generalize both uninorms and nullnorms. In this paper, we investigate the migrativity property for uni-nullnorms. We characterize uni-nullnorms that are α-migrative over a fixed uni-nullnorm, where the 2-neutral elements of uni-nullnorms can be the same or different. Specifically, the (α, V₁)-migrativity of V when e = e₁, a = a₁, or e ≠ e₁, a = a₁ or e = e₁, a ≠ a₁ are characterized, where V and V₁ are uni-nullnorms with 2-neutral elements {e, 1} _a and {e₁, 1} _{a
₁}, respectively.

Keywords

Uninorms nullnorms 2-uninorms uni-nullnorms migrativity

1. Introduction

The term α-migrative was introduced by Durante and Sarkoci in [6] where they use it to construct t-norms by means of convex combinations. Actually, this property has been presented already in Problem 1.8 (b) of [19]. For a given α ∈]0, 1 [ and a mapping T : [0, 1] ² → [0, 1], T is said to be α-migrative if $T (α x, y) = T (x, α y), \forall x, y \in [0, 1] .$ (1)

Though this definition seems to be rather general, Durante and Sarkoci [6] deals with t-norms and t-subnorms. Replacing the the product αx in Eq. (1) eq-amig by a t-norm T₁, Fodor and Rudas [8] extended the α-migrativity to (α, T₁)-migrativity for t-norms that can be written as $T (T_{1} (α, x), y) = T (x, T_{1} (α, y)), \forall x, y \in [0, 1] .$ (2)

As has been pointed out in [4 , 18], the α-migrativity refers to a certain ratio interchangeability between coordinates which makes it possible to play important roles in image processing and decision making. Therefore, researchers have paid much attention to the migrativity of various binary operations. The migrativity property (and its generalizations) has been studied for t-norms (see [6–9 , 20]), for t-conorms (see [15, 17]), for t-subnorms (see [31]), for uninorms (see [16 , 25–28]), for nullnorms (see [17 , 33]), for copulas (see [18]), for overlap functions (see [22, 23]) and for aggregation functions (see [3, 4]).

Uninorms (see [32]) and nullnorms (see [5], coincide with t-operators in [14]) are important generalizations of t-norms and t-conorms, and have proved to be useful in many fields such as fuzzy sets and fuzzy logic, image processing, decision making, information fusion, etc (see [1 , 21]). Letting a uninorm and a nullnorm share the same underlying t-conorm (resp. t-norm), Sun et al. [29, 30] introduced the concept of a uni-nullnorm (resp. null-uninorm) which is a special case of 2-uninorms introduced in [2]. Uni-nullnorms and null-uninorms are a pair of dual operations and generalize both uninorms and nullnorms. As combinations of uninorms and nullnorms, uni-nullnorms may play important and interesting roles not only in theoretical investigations but also in practical applications. Notice that the migrativity property for uninorms and nullnorms has attracted considerable attention (see [16 , 33]). Mas et al. [16] characterized uninorms with neutral element e that are α-migrative over a fixed uninorm with the same neutral element e. Migrative uninorms and nullnorms over a fixed t-norm and over a fixed t-conorm were introduced and characterized in [17]. Qin and Ruiz-Aguilera [24] characterized uninorms with neutral element e that are α-migrative over idempotent uninorms with neutral element e₁ different from e. Su et al. [25 –28] investigated and characterized uninorms with neutral element e that are α-migrative over uninorms with neutral element e₁ different from e. Zong et al. [33] characterized nullnorms with absorbing element a that are α-migrative over nullnorms with absorbing element a₁ when a = a₁ and a ≠ a₁. Since uni-nullnorms generalize uninorms and nullnorms, conducting researches on the migrativity of uni-nullnorms is meaningful which can generalize and unify corresponding results for t-norms, t-conorms, uninorms and nullnorms.

Up to now, the migrativity of uni-nullnorms have not been touched. In this paper we shall investigate the migrativity of uni-nullnorms. We shall study α-migrative uni-nullnorms over a fixed uni-nullnorm, where the 2-neutral elements of these uni-nullnorms may be the same or different. The structure of the rest of this paper is organized as follows. In Section 2, we recall some fundamentals about uni-nullnorms. In Section 3, we characterize uni-nullnorms with 2-neutral element {e, 1} _a that are α-migrative over uni-nullnorms with the same 2-neutral element {e, 1} _a. In Section 4, we characterize uni-nullnorms with 2-neutral element {e, 1} _a that are α-migrative over uni-nullnorms with 2-neutral element {e₁, 1} _{a
₁} different from {e, 1} _a. Discussion and conclusions are drawn in Section 5.

2. Preliminaries

Throughout this paper, a [0, 1] → [0, 1] mapping is called a unary operator, a [0, 1] ² → [0, 1] mapping is called a binary operator.

Definition 2.1. (Akella [2]). For a commutative binary operator V, {e₁, e₂} _a with 0 ≤ e₁ ≤ a ≤ e₂ ≤ 1 is called the 2-neutral element of V if V (e₁, x) = x for all x ≤ a and V (e₂, x) = x for all x ≥ a.

Definition 2.2. (Akella [2]). A commutative, associative, increasing binary operator V having a 2-neutral element {e₁, e₂} _a is a 2-uninorm.

Definition 2.3. (Sun et al. [29]). A uni-nullnorm V is a 2-uninorm having a 2-neutral element {e, 1} _a with a being an annihilator over [e, 1], i.e., V (a, x) = a for all x ∈ [e, 1].

Note that a uni-nullnorm is a special case of a 2-uninorm, where the underlying upper uninorm of the 2-uninorm is a t-norm. Uni-nullnorms generalize both uninorms and nullnorms. For a uni-nullnorm V with a 2-neutral element {e, 1} _a, V is a uninorm if a = 1, a nullnorm when e = 0, a t-conorm if e = 0 and a = 1, a t-norm whenever e = a = 0 or e = a = 1. A uni-nullnorm with a 2-neutral element {e, 1} _a is proper if 0 < e < a < 1.

Proposition 2.4. (Sun et al. [29]). For any proper uni-nullnorm V with a 2-neutral element {e, 1} _a, V (0, a) = V (0, 1) ∈ {0, a}.

We call a uni-nullnorm V disjunctive if V (0, a) = a, since it acts like the well-known fuzzy disjunction named the maximum t-conorm on the point (0, a), i.e., V (0, a) = a = max(0, a). If V (0, a) =0 then V is called conjunctive, by a similar reason.

For any uni-nullnorm V with a 2-neutral element {e, 1} _a and for all x, y ∈ [0, 1], define binary operators $T_{V}^{l}$ , S_V, $T_{V}^{r}$ , U_V and N_V by

$T_{V}^{l} (x, y) = \frac{V (ex, ey)}{e},$ (3) $S_{V} (x, y) = \frac{V (e + (a - e) x, e + (a - e) y) - e}{a - e},$ (4) $T_{V}^{r} (x, y) = \frac{V (a + (1 - a) x, a + (1 - a) y) - a}{1 - a},$ (5) $U_{V} (x, y) = \frac{V (ax, ay)}{a},$ (6) $N_{V} (x, y) = \frac{V (e + (1 - e) x, e + (1 - e) y) - e}{1 - e} .$ (7)

Proposition 2.5. (Akella [2] and Sun et al. [29]] For any uni-nullnorm V with a 2-neutral element {e, 1} _a, we have:

(i) $T_{V}^{l}$ and $T_{V}^{r}$ are t-norms, S_V is a t-conorm.

(ii) U_V is a uninorm with a neutral element $\frac{e}{a}$ .

(iii) N_V is a nullnorm with an annihilator $\frac{a - e}{1 - e}$ .

(iv) min(x, y) ≤ V (x, y) ≤ max(x, y) for all (x, y) ∈ [0, e] × [e, a] ∪ [e, a] × [0, e].

(v) V (x, y) = a for all (x, y) ∈ [e, a] × [a, 1] ∪ [a, 1] × [e, a].

(vi) min(x, y) ≤ V (x, y) ≤ a for all (x, y) ∈ [0, e] × [a, 1] ∪ [a, 1] × [0, e].

Proposition 2.5 shows that the structure of a uni-nullnorm over the squares [0, e] ², [e, a] ², [a, 1] ², [0, a] ² and [e, 1] ² are closely related to a t-norm, a t-conorm, a t-norm, a uninorm and a nullnorm, respectively. We call $T_{V}^{l}$ , S_V, $T_{V}^{r}$ , U_V and N_V the underlying lower left t-norm, t-conorm, upper right t-norm, uninorm and nullnorm of V, respectively. The underlying uninorm and nullnorm of a proper uni-nullnorm share the same underlying t-conorm.

Proposition 2.6. Let V be a uni-nullnorm with a 2-neutral element {e, 1} _a. Then V (x, y) = V (a, y) for all (x, y) ∈ [a, 1] × [0, a] and V (x, y) = V (x, a) for all (x, y) ∈ [0, a] × [a, 1].

Proof. It suffices to show that V (x, y) = V (a, y) for all (x, y) ∈ [a, 1] × [0, a] because V is commutative. Form Proposition 2.5 it follows that V (x, e) = a for all x ≥ a. Then for all (x, y) ∈ [a, 1] × [0, e], we obtain V (x, y) = V (x, V (e, y)) = V (V (x, e) , y) = V (a, y), this, together with the fact V (x, y) = a for all (x, y) ∈ [a, 1] × [e, a], yields that V (x, y) = V (a, y) for all (x, y) ∈ [a, 1] × [0, a]. □

The migrativity property has been studied for t-norms, for t-subnorms, for uninorms and for nullnorms. Similar to the migrativity property of uninorms and nullnorms, we can give the definition of migrativity for two uni-nullnorms.

Definition 2.7. Let V and V₁ be uni-nullnorms and α ∈ [0, 1]. We call V is (α, V₁)-migrative or α-migrative over V₁ if $V (V_{1} (α, x), y) = V (x, V_{1} (α, y)), \forall x, y \in [0, 1] .$ (8)

Note that the migrative equation for t-norms or uninorms has been further generalized involving two fixed t-norms or uninorms (see [8, 16]). We state it for uni-nullnorms as follow.

Definition 2.8. Let V, V₁ and V₂ be uni-nullnorms and α ∈ [0, 1]. We call V is (α, V₁, V₂)-migrative if $V (V_{1} (α, x), y) = V (x, V_{2} (α, y)), \forall x, y \in [0, 1] .$ (9)

3. Migrative uni-nullnorms with the same 2-neutral element

In this section, we consider the migrativity of uni-nullnorms with the same 2-neutral element. The uni-nullnorms V, V₁ and V₂ considered in this section are always assumed to have the same 2-neutral element {e, 1} _a.

Theorem 3.1. Let V, V₁ and V₂ be uni-nullnorms with the same 2-neutral element {e, 1} _a and α ∈ [0, 1]. Then V is (α, V₁, V₂)-migrative if and only if V (α, x) = V₁ (α, x) = V₂ (α, x) for all x ∈ [0, 1].

Proof. If V (α, x) = V₁ (α, x) = V₂ (α, x) for all x ∈ [0, 1], then, by the commutativity and associativity of V, the assertion V is (α, V₁, V₂)-migrative is obvious.

Conversely, suppose V is (α, V₁, V₂)-migrative. To prove V (α, x) = V₁ (α, x) = V₂ (α, x) for all x ∈ [0, 1], we need to distinguish two cases: α ≤ a and α > a.

Case 1: α ≤ a. Since α ≤ a, we have V₂ (α, x) ≤ V₂ (a, 1) = a and V₁ (α, x) ≤ V₁ (a, 1) = a for all x ∈ [0, 1]. Then, for all x ∈ [0, 1], from the (α, V₁, V₂)-migrativity of V it follows that V (α, x) = V (V₁ (α, e) , x) = V (e, V₂ (α, x)) = V₂ (α, x) and V₁ (α, x) = V (V₁ (α, x) , e) = V (x, V₂ (α, e)) = V (x, α) . Therefore, when α ≤ a, V (α, x) = V₁ (α, x) = V₂ (α, x) for all x ∈ [0, 1].

Case 2: α > a. We shall consider two subcases: x ≥ e and x < e.

Case 2.1: x ≥ e. Because α > a, for all x ≥ e, we have V₁ (α, x) ≥ V₁ (α, e) = a and V₂ (α, x) ≥ V₂ (α, e) = a. Then the (α, V₁, V₂)-migrativity of V implies that V₁ (α, x) = V (V₁ (α, x) , 1) = V (x, V₂ (α, 1)) = V (α, x) and V (α, x) = V (V₁ (α, 1) , x) = V (1, V₂ (α, x)) = V₂ (α, x). Therefore V (α, x) = V₁ (α, x) = V₂ (α, x) for all x ∈ [e, 1].

Case 2.2: x < e. Because α > a, for all x < e, we get V₁ (α, x) ≤ V₁ (α, e) = a and V₂ (α, x) ≤ V₂ (α, e) = a, then the (α, V₁, V₂)-migrativity of V and Proposition 2.6 implies that V₁ (α, x) = V (V₁ (α, x) , e) = V (x, V₂ (α, e)) = V (α, x) and V (α, x) = V (a, x) = V (V₁ (α, e) , x) = V (e, V₂ (α, x)) = V₂ (α, x) . Thus V (α, x) = V₁ (α, x) = V₂ (α, x) for all x ∈ [0, e].

Combining the above two subcases, we know V (α, x) = V₁ (α, x) = V₂ (α, x) for all x ∈ [0, 1] when α > a.

From the above discussions, we have V (α, x) = V₁ (α, x) = V₂ (α, x) for all x ∈ [0, 1]. □

Theorem 3.1 shows that for uni-nullnorms V, V₁ and V₂ having the same 2-neutral element such that V is (α, V₁, V₂)-migrative, V (α, ·), V₁ (α, ·) and V₂ (α, ·) must coincide. Obviously, this characterization holds when V₁ = V₂. Since uni-nullnorms generalize both uninorms and nullnorms, Theorem 3.1 generalizes Proposition 4 in [16] and Theorem 3 in [33].

At the end of this section, we shall discuss some properties for uni-nullnorms V and V₁ such that V is (α, V₁)-migrative. First, we have the following results when V = V₁ or when α ∈ {0, a, 1}.

Proposition 3.2. Let V and V₁ be uni-nullnorms with the same 2-neutral element {e, 1} _a.

(i) V is (α, V)-migrative for all α ∈ [0, 1].

(ii) V is (e, V₁)-migrative.

(iii) If V and V₁ are disjunctive, then V is both (a, V₁)-migrative and (1, V₁)-migrative.

(iv) If V₁ is disjunctive, then V is (1, V₁)-migrative if and only if V is disjunctive.

(v) If V₁ is disjunctive, then V is (a, V₁)-migrative if and only if V is disjunctive.

(vi) If V₁ is conjunctive, then V is (0, V₁)-migrative if and only if V is conjunctive.

Proof. (i) The assertion is evident since V is commutative and associative.

(ii) Note that V (e, x) = V₁ (e, x) = x for all x ∈ [0, a] and V (e, x) = V₁ (e, x) = a for all x ∈ [a, 1]. Then, by Theorem 3.1, V is (e, V₁)-migrative.

(iii) If V and V₁ is disjunctive, then V (a, x) = V₁ (a, x) = a for all x ∈ [0, 1] and V (1, x) = V₁ (1, x) = a for all x ∈ [0, a]. From the fact V (1, x) = V₁ (1, x) = x for all x ∈ [a, 1] and Theorem 3.1 we deduce V is both (a, V₁)-migrative and (1, V₁)-migrative.

(iv) Suppose V₁ is disjunctive and V is (1, V₁)-migrative. Then we have V (a, 0) = V (V₁ (1, a) , 0) = V (a, V₁ (1, 0)) = V (a, a) = a . i.e., V is disjunctive.

Conversely, since V and V₁ are disjunctive, we have V (1, x) = V₁ (1, x) = a for all x ∈ [0, a]. Moreover, V (1, x) = V₁ (1, x) = x for all x ∈ [a, 1]. Then, by Theorem 3.1, V is (1, V₁)-migrative.

(v) If V₁ is disjunctive and V is (a, V₁)-migrative, then V (a, 0) = V (V₁ (a, a) , 0) = V (a, V₁ (a, 0)) = V (a, a) = a, i.e., V is disjunctive.

The converse is evident since V (a, x) = V₁ (a, x) = a holds for all x ∈ [0, 1].

(vi) If V₁ is conjunctive and V is (0, V₁)-migrative, then V (0, a) = V (V₁ (0, e) , a) = V (e, V₁ (0, a)) = V (e, 0) =0, i.e., V is conjunctive.

Conversely, since V and V₁ are conjunctive, it is clear that V (0, x) = V₁ (0, x) =0 for all x ∈ [0, 1]. Then, by Theorem 3.1, V is (0, V₁)-migrative. □

The following result relates the migrativity of uni-nullnorms with the migrativity of the underlying uninorms, nullnorms, t-norms and t-conorms.

Proposition 3.3. Let α ∈ [0, 1], V and V₁ be uni-nullnorms with the same 2-neutral element {e, 1} _a, T^l, S, T^r, U and N (resp. $T_{1}^{l}$ , S₁, $T_{1}^{r}$ , U₁ and N₁) be the underlying lower left t-norm, t-conorm, upper right t-norm, uninorm and nullnorm of V (resp. V₁), respectively. Suppose V is (α, V₁)-migrative, we have:

(i) U is $(\frac{α}{a}, U_{1})$ -migrative when α ≤ a.

(ii) N is $(\frac{α - e}{1 - e}, N_{1})$ -migrative when α ≥ e.

(iii) T^l is $(\frac{α}{e}, T_{1}^{l})$ -migrative when α ≤ e.

(iv) S is $(\frac{α - e}{a - e}, S_{1})$ -migrative when e ≤ α ≤ a.

(v) T^r is $(\frac{α - a}{1 - a}, T_{1}^{r})$ -migrative when α ≥ a.

Proof. We prove (i) only, the others can be proved similarly. If α ≤ a, then $V (x, y) = aU (\frac{x}{a}, \frac{y}{a})$ and $V_{1} (x, y) = {aU}_{1} (\frac{x}{a}, \frac{y}{a})$ for all x, y ≤ a. Therefore the (α, V₁)-migrativity of V yields $aU (U_{1} (\frac{α}{a}, \frac{x}{a}), \frac{y}{a}) = aU (\frac{x}{a}, U_{1} (\frac{α}{a}, \frac{y}{a}))$ for all x, y ∈ [0, a]. Taking $\frac{x}{a} = x^{'}$ and $\frac{y}{a} = y^{'}$ , we obtain $U (U_{1} (\frac{α}{a}, x^{'}), y^{'})$ $= U (x^{'}, U_{1} (\frac{α}{a}, y^{'}))$ for all x′, y′ ∈ [0, 1], i.e., U is $(\frac{α}{a}, U_{1})$ -migrative. □

4. Migrative uni-nullnorms with different 2-neutral elements

In this section, V and V₁ are assumed to be uni-nullnorms with 2-neutral elements {e, 1} _a and {e₁, 1} _{a
₁}, respectively. We consider the migrativity for uni-nullnorms with different 2-neutral elements. There are there subcases, that is, e ≠ e₁, a = a₁ or e = e₁, a ≠ a₁ or e ≠ e₁, a ≠ a₁. In this paper, we mainly concern the two cases: e ≠ e₁, a = a₁ or e = e₁, a ≠ a₁.

4.1 Characterizations on the (α, V₁)-migrativity of V when e ≠ e₁ and a = a₁

In this subsection, we consider the (α, V₁)-migrativity of V when e ≠ e₁ and a = a₁. Unless otherwise specified, we assume e ≠ e₁ and a = a₁ in this subsection. When α ≤ a, we have the following characterization for the (α, V₁)-migrativity of V.

Theorem 4.1. If α ≤ a, then V is (α, V₁)-migrative if and only if V₁ (α, x) = V (λ, x) for all x ∈ [0, 1] with λ = V₁ (α, e).

Proof. Suppose α ≤ a. For all x ∈ [0, 1], we have V₁ (α, x) ≤ V₁ (a, 1) = a, then from the (α, V₁)-migrativity of V it follows that V₁ (α, x) = V (V₁ (α, x) , e) = V (x, V₁ (α, e)) = V (λ, x) .

Conversely, if V₁ (α, x) = V (λ, x) for all x ∈ [0, 1], then we know V (V₁ (α, x) , y) = V (V (λ, x) , y) = V (x, V (λ, y)) = V (x, V₁ (α, y)) holds for all x, y ∈ [0, 1], i.e., V is (α, V₁)-migrative. □

It is worth mentioning that when a = a₁ = 1, the uni-nullnorms V and V₁ become uninorms. Taking a = a₁ = 1 in Theorem 4.1, we obtain a characterization for migrative uninorms which has been presented already as Lemma 1 in [24]. We point out that some results characterizing migrative uninorms in [26, 28] can be obtained from Theorem 4.1. For example, as consequences of Theorem 4.1, we have the following two results.

Corollary 4.2. If α ∈]0, 1 [\ {e₁}, a = a₁ = 1, V (x, y) = max(x, y) when min(x, y) ≤ e ≤ max(x, y) and V₁ (x, y) = min(x, y) when min(x, y) ≤ e₁ ≤ max(x, y), then V is not (α, V₁)-migrative.

Proof. Suppose α ∈]0, 1 [\ {e₁}, a = a₁ = 1, V (x, y) = max(x, y) when min(x, y) ≤ e ≤ max(x, y) and V₁ (x, y) = min(x, y) when min(x, y) ≤ e₁ ≤ max(x, y). Assume to the contrary that V is (α, V₁)-migrative, that is V₁ (α, x) = V (V₁ (α, e) , x) for all x ∈ [0, 1] by Theorem 4.1. On the one hand, when α ≤ e₁, we have V₁ (α, e) ≤ V₁ (e₁, e) = e, then V₁ (α, 1) = min(α, 1) = α and V (V₁ (α, e) , 1) = max(V₁ (α, e) , 1) =1, a contradiction. On the other hand, when α > e₁, we have V₁ (α, e) ≥ V₁ (e₁, e) = e, then V₁ (α, 0) = min(α, 0) =0 and V (V₁ (α, e) , 0) = max(V₁ (α, e) , 0) = V₁ (α, e), a contradiction. Thus V is not (α, V₁)-migrative. □

Corollary 4.3. Suppose α ∈]0, 1 [\ {e₁}, a = a₁ = 1, V (x, y) = min(x, y) when min(x, y) ≤ e ≤ max(x, y) and V₁ (x, y) = min(x, y) when min(x, y) ≤ e₁ ≤ max(x, y). If V is (α, V₁)-migrative, then α < min(e, e₁). Moreover, V is (α, V₁)-migrative if and only if V₁ (α, x) = V (α, x) for all x ∈ [0, 1]

Proof. Suppose α ∈]0, 1 [\ {e₁}, a = a₁ = 1, V (x, y) = min(x, y) when min(x, y) ≤ e ≤ max(x, y), V₁ (x, y) = min(x, y) when min(x, y) ≤ e₁ ≤ max(x, y) and V is (α, V₁)-migrative. Assume to the contrary that α ≥ min(e, e₁). If min(e, e₁) = e₁ and α > e₁, then V₁ (α, e) ≥ V₁ (e₁, e) = e, therefore V₁ (α, e₁) = α and V (V₁ (α, e) , e₁) = min(V₁ (α, e) , e₁) = e₁, a contradiction. If min(e, e₁) = e and α ≥ e, then α < e₁, otherwise α > e₁ which implies that V₁ (α, e) = min(α, e) = e, and hence α = V₁ (α, e₁) = V (V₁ (α, e) , e₁) = V (e, e₁) = e₁, a contradiction. Thus α < e₁. Consequently, V₁ (α, 1) = min(α, 1) = α and V (V₁ (α, e) , 1) ∈ {V₁ (α, e) , 1}, then V₁ (α, e) = α, and hence V (V₁ (α, e) , 1) = V (α, 1) =1, i.e., V (V₁ (α, e) , 1)≠ V₁ (α, 1), this is impossible. Thus α < min(e, e₁).

Moreover, suppose V is (α, V₁)-migrative, we have α < min(e, e₁), then V₁ (α, e) ≤ V₁ (e₁, e) = e and V₁ (α, 1) = min(α, 1) = α, and hence α = V₁ (α, 1) = V (V₁ (α, e) , 1) = min(1, V₁ (α, e)) = V₁ (α, e). Using Theorem 4.1, we know V is (α, V₁)-migrative if and only if V₁ (α, x) = V (V₁ (α, e) , x) = V (α, x) for all x ∈ [0, 1]. □

Note that Corollary 4.2 is just Theorem 3.2.1 in [28] and Corollary 4.3 is just Theorem 3.1 in [26]. Note also that when a = a₁ = 1 and e₁ = 1, if α = 1, then V₁ (α, x) = V₁ (1, x) = x and V (λ, x) = V (V₁ (α, e) , x) = V (V₁ (1, e) , x) = V (e, x) = x for all x ∈ [0, 1], by Theorem 4.1 we know V is (1, V₁)-migrative; if α = 0, then V₁ (α, x) = V₁ (0, x) =0 and V (λ, x) = V (V₁ (α, e) , x) = V (V₁ (0, e) , x) = V (0, x) for all x ∈ [0, 1], using Theorem 4.1 we know V is (0, V₁)-migrative if and only if V (0, x) for all x ∈ [0, 1], i.e., V (0, 1) =0. We mention that the two consequences of Theorem 4.1 are exactly Lemma 1 in [17]. Propositions 4 and 5 in [17] indicated that a conjunctive uninorm V with neutral element e ∈]0, 1 [ locally internal on the boundary is α-migrative over a t-norm V₁ when α ∈]0, e [ if and only if V₁ (α, x) = V (V₁ (α, e) , x) for all x ∈ [0, 1]. Obviously, this result is just a special case of Theorem 4.1 when a = a₁ = 1 and e₁ = 1.

In what follows, we shall characterize the (α, V₁)-migrativity of V when α > a. First, we have the following necessary condition for the (α, V₁)-migrativity of V when α > a.

Proposition 4.4. Let α > a and V be (α, V₁)-migrative. Then $\begin{matrix} V_{1} (α, x) = {\begin{matrix} V (α, x), x \geq e_{1}, \\ V (λ, x), x \leq a, \end{matrix} \end{matrix}$ where λ = V₁ (α, e). Furthermore, V₁ (α, x) = V (α, x) = V (λ, x) = a for all x ∈ [e₁, a].

Proof. On the one hand, for all x ≥ e₁, it holds that V₁ (α, x) ≥ V₁ (α, e₁) = a, then V₁ (α, x) = V (V₁ (α, x) , 1) = V (x, V₁ (α, 1)) = V (α, x) . On the other hand, when x ≤ a, we have V₁ (α, x) ≤ V₁ (α, a) = a. Then from the (α, V₁)-migrativity of V it follows that V₁ (α, x) = V (V₁ (α, x) , e) = V (x, V₁ (α, e)) = V (λ, x) . Therefore, for all x ∈ [e₁, a], we obtain V₁ (α, x) = V (α, x) = V (λ, x) = a. □

Based on Proposition 4.4, by distinguishing the two cases e > e₁ and e < e₁, we can characterize the (α, V₁)-migrativity of V when α > a as follows.

Theorem 4.5. If α > a and e > e₁, then V is (α, V₁)-migrative if and only if V₁ (α, x) = V (α, x) for all x ∈ [0, 1].

Proof. The sufficiency is obvious whenever V (α, x) = V₁ (α, x) holds for all x ∈ [0, 1] since V is commutative and associative.

As for the necessity, by Proposition 4.4 we only need to show V₁ (α, x) = V (α, x) when x < e₁. Since e₁ < e, we have V (α, e) = a. For all x < e₁ < e, using Propositions 2.6 and 4.4, we deduce V₁ (α, x) = V (λ, x) = V (V₁ (α, e) , x) = V (a, x) = V (α, x). □

Theorem 4.6. If α > a and e < e₁, then V is (α, V₁)-migrative if and only if $\begin{matrix} V_{1} (α, x) = {\begin{matrix} V (λ, x), x \leq a, \\ V (α, x), x > a, \end{matrix} \end{matrix}$ where λ = V₁ (α, e).

Proof. With the help of Proposition 4.4, it suffices to show that V is (α, V₁)-migrative when V₁ (α, x) = V (λ, x) for all x ≤ a and V₁ (α, x) = V (α, x) for all x > a. We need to consider the following cases:

When x, y ≤ a, it is obvious that V (V₁ (α, x) , y) = V (V (λ, x) , y) = V (x, V (λ, y)) = V (x, V₁ (α, y)) .

When x, y > a, we have V (V₁ (α, x) , y) = V (V (α, x) , y) = V (x, V (α, y)) = V (x, V₁ (α, y)) .

When x ≤ a < y, from the fact e ≤ λ = V₁ (α, e) ≤ a and Proposition 2.6, we obtain V (V₁ (α, x) , y) = V (V (λ, x) , y) = V (x, V (λ, y)) = V (x, a) and V (x, V₁ (α, y)) = V (x, V (α, y)) = V (V (α, x) , y) = V (V (a, x) , y) = V (x, V (a, y)) = V (x, a) .

When y ≤ a < x, it can be proved similarly.

Combining the above four cases, we claim V is (α, V₁)-migrative. □

Theorems 4.1, 4.1 and 4.1 characterize the (α, V₁)-migrativity of V when e ≠ e₁ and a = a₁. When a = a₁ = 1, the uni-nullnorms V and V₁ become uninorms, thus Theorem 4.1 generalizes corresponding characterizations for migrative uninorms over t-norms, t-conorms or uninorms in [17 , 28].

4.2 Characterizations on the (α, V₁)-migrativity of V when e = e₁ and a ≠ a₁

In this subsection, we consider the (α, V₁)-migrativity of V when e = e₁ and a ≠ a₁. Unless otherwise specified, we assume e = e₁ and a ≠ a₁ in this subsection. We shall distinguish two cases: a₁ < a and a₁ > a. First, we consider the case a₁ < a which can be divided into three subcases: α ≤ a₁ < a, a₁ < α < a and a₁ < a ≤ α. When α ≤ a₁ < a, we have the following characterization for the (α, V₁)-migrativity of V.

Theorem 4.7. If α ≤ a₁ < a, then V is (α, V₁)-migrative if and only if V₁ (α, x) = V (α, x) for all x ∈ [0, 1].

Proof. The sufficiency is obvious, we prove the necessity only. Suppose α ≤ a₁ < a and V is (α, V₁)-migrative. For all x ∈ [0, 1], it follows that V₁ (α, x) ≤ V₁ (a₁, 1) = a₁ < a, and then V₁ (α, x) = V (V₁ (α, x) , e) = V (x, V₁ (α, e)) = V (x, α). □

As consequences of Theorem 4.7, we have the following two results.

Corollary 4.8. If e ≤ α ≤ a₁ < a, then V is not (α, V₁)-migrative.

Proof. Assume to the contrary that V is (α, V₁)-migrative, then Theorem 4.7 yields that a₁ = V₁ (α, 1) = V (α, 1) = a, a contradiction. □

Note that when e = e₁ = 0, V and V₁ become nullnorms. Therefore, Corollary 4.8 generalizes Lemma 6 in [33].

Corollary 4.9. If α ≤ a₁ < a and V, V₁ are disjunctive, then V is not (α, V₁)-migrative.

Proof. Assume to the contrary that V is (α, V₁)-migrative, then from Theorem 4.7 it follows that a₁ = V₁ (α, 1) = V (α, 1) = a, a contradiction. □

When a₁ < α < a, V cannot be (α, V₁)-migrative as is shown below.

Theorem 4.10. If a₁ < α < a, then V is not (α, V₁)-migrative.

Proof. Suppose V is (α, V₁)-migrative. Then we have α = V (α, e) = V (V₁ (α, 1) , e) = V (1, V₁ (α, e)) = V (1, a₁) = a, a contradiction. Therefore V is not (α, V₁)-migrative. □

Note that when e = e₁ = 0, V and V₁ become nullnorms. Therefore, Theorem 4.10 generalizes Lemma 7 in [33].

When a₁ < a ≤ α, we have the following characterization for the (α, V₁)-migrativity of V.

Theorem 4.11. If a₁ < a ≤ α, then V is (α, V₁)-migrative if and only if $\begin{matrix} V_{1} (α, x) = {\begin{matrix} V (a_{1}, x), x \leq a, \\ V (α, x), x \geq a . \end{matrix} \end{matrix}$

Proof. Suppose V is (α, V₁)-migrative. First, we prove V₁ (α, x) = V (a₁, x) for all x ≤ a which shall be divided into two subcases x ≤ a₁ and x ≥ a₁. For all x ≤ a₁, we have V₁ (α, x) ≤ V₁ (α, a₁) = a₁ < a. Then form the (α, V₁)-migrativity of V it follows that V₁ (α, x) = V (V₁ (α, x) , e) = V (x, V₁ (α, e)) = V (x, a₁). For all x ∈ [a₁, a], we have $V_{1} (α, x) \leq V_{1} (α, a) = a_{1} + (1 - a_{1}) T_{V_{1}}^{r} (\frac{α - a_{1}}{1 - a_{1}}, \frac{a - a_{1}}{1 - a_{1}}) \leq min (α, a) = a$ . By the (α, V₁)-migrativity of V, we obtain V₁ (α, x) = V (V₁ (α, x) , e) = V (x, V₁ (α, e)) = V (x, a₁). Thus V₁ (α, x) = V (a₁, x) for all x ≤ a, especially, we have V₁ (α, a) = V (a₁, a). Now, we prove V₁ (α, x) = V (x, α) for all x ≥ a. Since V₁ (α, a) = V (a₁, a) = a, for all x ≥ a, we have V₁ (α, x) ≥ V₁ (α, a) = a, and then V₁ (α, x) = V (V₁ (α, x) , 1) = V (x, V₁ (α, 1)) = V (x, α).

Conversely, let V₁ (α, x) = V (a₁, x) for all x ≤ a and V₁ (α, x) = V (α, x) for all x ≥ a. We need to distinguish the following three cases.

When x, y ≤ a or x, y ≥ a, V (V₁ (α, x) , y) = V (x, V₁ (α, y)) holds trivially.

When x ≤ a ≤ y, we consider two subcases.

x ≥ e. When x ≥ e, we have a₁ = V (a₁, e) ≤ V (a₁, x) ≤ V (a₁, a) ≤ a. Then V (V₁ (α, x) , y) = V (V (a₁, x) , y) = a = V (x, V (α, y)) = V (x, V₁ (α, y)).

x < e. When x < e, we get V (V₁ (α, x) , y) = V (V (a₁, x) , y) = V (V (a₁, y) , x) = V (a, x) and V (x, V₁ (α, y)) = V (x, V (α, y)). Because V (α, y) ≥ V (a, a) = a, from Proposition 2.6 it follows V (a, x) = V (V (α, y) , x) for all x < e. Therefore V (V₁ (α, x) , y) = V (x, V₁ (α, y)).

When y ≤ a ≤ x, it can be proved similarly.

Combining the above cases, we know V is (α, V₁)-migrative. □

Note that when a₁ < a ≤ α and e = e₁ = 0, we have V₁ (α, x) = a₁ for all x ≤ a₁. Therefore, Theorem 4.11 generalizes Lemma 10 in [33] which characterized the α-migrativity of a nullnorm V over a nullnorm V₁ when a₁ < a < α.

As a consequence of Theorem 4.11, we have the following property on V₁ when V is (α, V₁)-migrative and a₁ < a ≤ α.

Proposition 4.12 If a₁ < a ≤ α and V is (α, V₁)-migrative, then we have (i) V₁ (α, e) = a₁ = V (a₁, a₁); (ii) V₁ (a, x) = a for all x ≥ α.

Proof. (i) From Theorem 4.11 it follows that V₁ (α, e) = V (a₁, e) = a₁ and V (a₁, a₁) = V₁ (α, a₁) = a₁. Thus V₁ (α, e) = a₁ = V (a₁, a₁).

(ii) From Theorem 4.11 we know V₁ (α, a) = V (α, a) = a. Then for all x ≥ α, V₁ (a, 1) ≥ V₁ (a, x) ≥ V₁ (a, α) = a, i.e., V₁ (a, x) = a for all x ≥ α. □

In what follows, we shall characterize the (α, V₁)-migrativity of V when a < a₁. We divide the case a < a₁ into four subcases: a < a₁ ≤ α, a < α < a₁, e ≤ α ≤ a < a₁ and α < e ≤ a < a₁. Actually, when a < a₁ ≤ α or a < α < a₁, V cannot be (α, V₁)-migrative, see the following two results.

Theorem 4.13. If a < α < a₁, then V is not (α, V₁)-migrative.

Proof. Suppose to the contrary that V is (α, V₁)-migrative. Then α = V (α, 1) = V (V₁ (α, e) , 1) = V (e, V₁ (α, 1)) = V (e, a₁) = a, a contradiction. Thus V is not (α, V₁)-migrative. □

Note that when e = e₁ = 0, V and V₁ become nullnorms. Therefore, Theorem 4.13 generalizes Lemma 9 in [33].

Theorem 4.14. If a < a₁ ≤ α, then V is not (α, V₁)-migrative.

Proof. Assume to the contrary that V is (α, V₁)-migrative, then a = V (α, a) = V (V₁ (α, 1) , a) = V (1, V₁ (α, a)) = V (1, a₁) = a₁, which is impossible. Therefore V is not (α, V₁)-migrative. □

Note that when e = e₁ = 0, V and V₁ become nullnorms. Therefore, Theorem 4.14 generalizes Lemma 8 in [33].

Now, we shall characterize the (α, V₁)-migrativity of V when e ≤ α ≤ a < a₁. First, we give some properties on V and V₁ when V is (α, V₁)-migrative and e ≤ α ≤ a < a₁.

Proposition 4.15. If e ≤ α ≤ a < a₁ and V is (α, V₁)-migrative, then V₁ (α, a) = a.

Proof. It suffices to show that V₁ (α, a) ≤ a because V₁ (α, a) ≥ V₁ (e, a) = a. Assume to the contrary, V₁ (α, a) > a. Since V is (α, V₁)-migrative, we obtain a = V (a₁, a) = V (V₁ (α, 1) , a) = V (1, V₁ (α, a)) = V₁ (α, a) > a, a contradiction. Therefore V₁ (α, a) ≤ a and hence V₁ (α, a) = a. □

Proposition 4.16. If e ≤ α ≤ a < a₁ and V is (α, V₁)-migrative, then V (a₁, x) = V (a₁, y) for all x, y ≥ a₁. Furthermore, V (a₁, a₁) = a₁.

Proof. Since V is (α, V₁)-migrative, for all x, y ≥ a₁ it follows that V (a₁, y) = V (V₁ (α, x) , y) = V (x, V₁ (α, y)) = V (x, a₁) . Furthermore, V (a₁, a₁) = V (a₁, 1) = a₁. □

Based on Propositions 2.6 and 4.15, we have the following characterization for the (α, V₁)-migrativity of V when e ≤ α ≤ a < a₁.

Theorem 4.17. Suppose e ≤ α ≤ a < a₁. Then V is (α, V₁)-migrative if and only if $\begin{matrix} V_{1} (α, x) = {\begin{matrix} V (α, x), x \leq a, \\ V (a_{1}, x), x > a . \end{matrix} \end{matrix}$

Proof. Suppose V is (α, V₁)-migrative. From Proposition 4.15 we have V₁ (α, x) ≤ V₁ (α, a) = a for all x ≤ a. Then for all x ≤ a, V (α, x) = V (V₁ (α, e) , x) = V (e, V₁ (α, x)) = V₁ (α, x) . For all x > a, it holds that V₁ (α, x) ≥ V₁ (α, a) = a, then V (a₁, x) = V (V₁ (α, 1) , x) = V (1, V₁ (α, x)) = V₁ (α, x) .

Conversely, suppose V₁ (α, x) = V (α, x) for all x ≤ a and V₁ (α, x) = V (a₁, x) for all x > a. We need to distinguish four cases.

When x, y ≤ a, V (V₁ (α, x) , y) = V (V (α, x) , y) = V (x, V (α, y)) = V (x, V₁ (α, y)).

When x, y > a, V (V₁ (α, x) , y) = V (V (a₁, x) , y) = V (x, V (a₁, y)) = V (x, V₁ (α, y)).

When x ≤ a < y, from Proposition 2.6 it holds that V (a₁, x) = V (a, x), then we deduce that V (V₁ (α, x) , y) = V (V (α, x) , y) = V (x, V (α, y)) = V (x, a) and V (x, V₁ (α, y)) = V (x, V (a₁, y)) = V (V (a₁, x) , y) = V (V (a, x) , y) = V (x, V (a, y)) = V (x, a) .

When y ≤ a < a, analogous to the proof of the case x ≤ a < y, we get V (V₁ (α, x) , y) = V (x, V₁ (α, y)).

Combining the four cases, we know V is (α, V₁)-migrative. □

Note that when α ≤ a < a₁ and e = e₁ = 0, we have V₁ (α, x) = a₁ for all x ≥ a₁. Therefore, Theorem 4.17 generalizes Lemma 11 in [33] which characterized the α-migrativity of a nullnorm V over a nullnorm V₁ when α < a < a₁.

At last, we shall characterize the (α, V₁)-migrativity of V when α < e ≤ a < a₁. If α < e ≤ a < a₁, form Proposition 2.6 we know V₁ (α, 1) = V₁ (α, a₁), then α ≤ V₁ (α, 1) ≤ a₁. We will distinguish two cases: V₁ (α, 1) ≤ a and a < V₁ (α, 1) ≤ a₁. When V₁ (α, 1) ≤ a and α < e ≤ a < a₁, we have the following characterization for the (α, V₁)-migrativity of V.

Theorem 4.18. Suppose α < e ≤ a < a₁ and V₁ (α, 1) ≤ a. Then V is (α, V₁)-migrative if and only if V₁ (α, x) = V (α, x) for all x ∈ [0, 1].

Proof. Suppose α < e ≤ a < a₁ and V₁ (α, 1) ≤ a. If V₁ (α, x) = V (α, x) for all x ∈ [0, 1], then the assertion V is (α, V₁)-migrative is quite evident.

Conversely, if V is (α, V₁)-migrative, for all x ∈ [0, 1], we get V (α, x) = V (V₁ (α, e) , x) = V (e, V₁ (α, x)) = V₁ (α, x) since V₁ (α, x) ≤ V₁ (α, 1) ≤ a. □

As for characterizing the (α, V₁)-migrativity of V when α < e ≤ a < a₁ and a < V₁ (α, 1) ≤ a₁, we first give the following property regarding on the value of V and V₁ at the point (α, a) when V is (α, V₁)-migrative.

Proposition 4.19. Suppose α < e ≤ a < a₁, V₁ (α, 1) > a and V is (α, V₁)-migrative. Then V₁ (α, a) = V (α, a) = a.

Proof. Suppose V₁ (α, 1) > a, then from the (α, V₁)-migrativity of V, it follows that a = V (V₁ (α, 1) , e) = V (1, V₁ (α, e)) = V (1, α) . Using Proposition 2.6, we have V (a, α) = V (1, α) = a. Moreover, since V₁ (α, a) ≤ V₁ (e, a) = a, it holds that V (α, a) = V (V₁ (α, e) , a) = V (e, V₁ (α, a)) = V₁ (α, a) . Thus V₁ (α, a) = V (α, a) = a. □

Based on Proposition 4.19, we have the following characterization for the (α, V₁)-migrativity of V when α < e ≤ a < a₁ and a < V₁ (α, 1).

Theorem 4.20. Suppose α < e ≤ a < a₁ and V₁ (α, 1) > a. Then V is (α, V₁)-migrative if and only if

$\begin{matrix} V_{1} (α, x) = {\begin{matrix} V (α, x), x \leq a, \\ V (μ, x), x \geq a, \end{matrix} \end{matrix}$ where μ = V₁ (α, 1).

Proof. Assume V is (α, V₁)-migrative, for all x ≤ a, it holds that V₁ (α, x) ≤ V₁ (e, a) = a, then V (α, x) = V (V₁ (α, e) , x) = V (e, V₁ (α, x)) = V₁ (α, x) . For all x ≥ a, from Proposition 4.19 it follows that V₁ (α, x) ≥ V₁ (α, a) = a, then V (μ, x) = V (V₁ (α, 1) , x) = V (1, V₁ (α, x)) = V₁ (α, x) .

Conversely, suppose V₁ (α, x) = V (α, x) for all x ≤ a and V₁ (α, x) = V (μ, x) for all x ≥ a. Then V₁ (α, a) = V (α, a) = V (μ, a) = a. To prove V is (α, V₁)-migrative, we shall consider four cases.

When x, y ≤ a, V (V₁ (α, x) , y) = V (V (α, x) , y) = V (x, V (α, y)) = V (x, V₁ (α, y)).

When x, y > a, V (V₁ (α, x) , y) = V (V (μ, x) , y) = V (x, V (μ, y)) = V (x, V₁ (α, y)).

When x ≤ a < y, from Proposition 2.6 it follows that V (α, y) = V (α, a) = a and V (μ, x) = V (a, x). Then V (V₁ (α, x) , y) = V (V (α, x) , y) = V (x, V (α, y)) = V (x, a) and V (x, V₁ (α, y)) = V (x, V (μ, y)) = V (V (μ, x) , y) = V (V (a, x) , y) = V (x, V (a, y)) = V (x, a).

When y ≤ a < x, similar to the proof of the case x ≤ a < y, we get V (V₁ (α, x) , y) = V (x, V₁ (α, y)).

Thus V is (α, V₁)-migrative. □

Theorems 4.7, 4.10, 4.11, 4.13, 4.14, 4.17, 4.18 and 4.20 characterize the (α, V₁)-migrativity of V when e = e₁ and a ≠ a₁. Note that when e = e₁ = 0, the uni-nullnorms V and V₁ become nullnorms, thus Corollary 4.8, Theorems 4.7, 4.10, 4.11, 4.13, 4.14, 4.17 generalize corresponding characterizations for migrative nullnorms over nullnorms in [33].

5. Discussion and conclusions

5.1 Discussion

The migrativity property has been studied for t-norms, for t-conorms, for uninorms and for nullnorms (see [6–9 , 33]). Uni-nullnorms generalize t-norms, t-conorms, uninorms and nullnorms. In this paper, we further generalized the migrativity property for uni-nullnorms. We characterized uni-nullnorms that are α-migrative over a fixed uni-nullnorm with the same or different 2-neutral element.

Theorem 3.1 characterized the (α, V₁)-migrativity of V when V and V₁ are uni-nullnorms with the same 2-neutral element {e, 1} _a. If a = 1, then V and V₁ become uninorms, and Theorem 3.1 generalizes Proposition 4 in [16] which characterized uninorms with neutral element e that are α-migrative over a fixed uninorm with the same neutral element e. If e = 0, then V and V₁ become nullnorms, and Theorem 3.1 generalizes Theorem 3 in [33] which characterized nullnorms with absorbing element a that are α-migrative over nullnorms with the absorbing element a. If e = a = 1, then V and V₁ become t-norms, and Theorem 3.1 generalizes Theorem 3 in [8] which characterized t-norms that are α-migrative over t-norms. If e = a = 0, then V and V₁ become t-conorms, and Theorem 3.1 generalizes Proposition 3 in [15] which characterized t-conorms that are α-migrative over t-conorms.

Theorems 4.1, 4.5 and 4.6 characterized the (α, V₁)-migrativity of V when V and V₁ are uni-nullnorms with 2-neutral elements {e, 1} _a and {e₁, 1} _a (e₁ ≠ e), respectively. If a = 1, then V and V₁ become uninorms, and Theorem 3 generalizes Lemma 1 in [24] which characterized uninorms with neutral element e that are α-migrative over uninorms with neutral element e₁. Moreover, as has been pointed out in Subsection 4.1, Theorem 4.1 generalizes corresponding characterizations for migrative uninorms over t-norms, t-conorms and uninorms in [17 , 28]. When a < 1 and e > e₁ = 0, Theorems 4.1 and 4.5 give characterizations for uni-nullnorms that are α-migrative over nullnorms. When a < 1 and e₁ > e = 0, Theorems 4.1 and 4.6 give characterizations for nullnorms that are α-migrative over uni-nullnorms.

Theorems 4.7, 4.10, 4.11, 4.13, 4.14, 4.17, 4.18 and 4.20 characterized the (α, V₁)-migrativity of V when V and V₁ are uni-nullnorms with 2-neutral elements {e, 1} _a and {e, 1} _{a
₁} (a₁ ≠ a), respectively. If e = 0, then V and V₁ become nullnorms, and these results generalize corresponding characterizations for nullnorms with absorbing element a that are α-migrative over nullnorms with absorbing element a₁ in [33]. When e > 0 and a₁ = 1, these results give characterizations for uni-nullnorms that are α-migrative over uninorms. When e > 0 and a = 1, these results give characterizations for uninorms that are α-migrative over uni-nullnorms.

5.2 Conclusions and future work

In this paper, we investigated the migrativity for uni-nullnorms. Suppose V and V₁ are uni-nullnorms with 2-neutral elements {e, 1} _a and {e₁, 1} _{a
₁}, respectively. We characterized the (α, V₁)-migrativity of V when e = e₁, a = a₁ or e ≠ e₁, a = a₁ or e = e₁, a ≠ a₁. Let λ = V₁ (α, e) and μ = V₁ (α, 1).

Case 1 e = e₁ and a = a₁ : V is (α, V₁)-migrative if and only if V (α, x) = V₁ (α, x) for all x ∈ [0, 1].

Case 2 e ≠ e₁ and a = a₁

Case 2.1 α ≤ a : V is (α, V₁)-migrative if and only if V₁ (α, x) = V (λ, x) for all x ∈ [0, 1].

Case 2.2 α > a

Case 2.2.1 e > e₁ : V is (α, V₁)-migrative if and only if V₁ (α, x) = V (α, x) for all x ∈ [0, 1].

Case 2.2.2 e < e₁ : V is (α, V₁)-migrative if and only if $\begin{matrix} V_{1} (α, x) = {\begin{matrix} V (λ, x), x \leq a, \\ V (α, x), x > a . \end{matrix} \end{matrix}$

Case 3 e = e₁ and a ≠ a₁

Case 3.1 a₁ < a

Case 3.1.1 α ≤ a₁ : V is (α, V₁)-migrative if and only if V₁ (α, x) = V (α, x) for all x ∈ [0, 1].

Case 3.1.2 a₁ < α < a : V is not (α, V₁)-migrative.

Case 3.1.3 α ≥ a : V is (α, V₁)-migrative if and only if $\begin{matrix} V_{1} (α, x) = {\begin{matrix} V (a_{1}, x), x \leq a, \\ V (α, x), x \geq a . \end{matrix} \end{matrix}$

Case 3.2 a₁ > a

Case 3.2.1 α > a : V is not (α, V₁)-migrative.

Case 3.2.2 e ≤ α ≤ a : V is (α, V₁)-migrative if and only if $\begin{matrix} V_{1} (α, x) = {\begin{matrix} V (α, x), x \leq a, \\ V (a_{1}, x), x > a . \end{matrix} \end{matrix}$

Case 3.2.3 α < e and V₁ (α, 1) ≤ a : V is (α, V₁)-migrative if and only if V₁ (α, x) = V (α, x) for all x ∈ [0, 1].

Case 3.2.4 α < e and V₁ (α, 1) > a : V is (α, V₁)-migrative if and only if $\begin{matrix} V_{1} (α, x) = {\begin{matrix} V (α, x), x \leq a, \\ V (μ, x), x \geq a . \end{matrix} \end{matrix}$

Since uni-nullnorms generalize t-norms, t-conorms, uninorms and nullnorms, these characterizations for uni-nullnorms that are α-migrative over uni-nullnorms generalize and unify corresponding characterizations for migrative t-norms, t-conorms, uninorms and nullnorms in [15–17 , 33].

In the future works, we will characterize the (α, V₁)-migrativity of V when e ≠ e₁ and a ≠ a₁, explore properties of migrative uni-nullnorms and investigate properties on α when V is (α, V₁)-migrative. Moreover, we will study migrative left continuous uni-nullnorms and migrative uni-nullnorms with continuous underlying t-norms and t-conorms.

Footnotes

Acknowledgments

The authors are grateful to the anonymous referees for their valuable comments and suggestions which helped to improve the paper.

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On the migrativity of uni-nullnorms 1

Abstract

Keywords

1. Introduction

4. Migrative uni-nullnorms with different 2-neutral elements

4.1 Characterizations on the (α, V1)-migrativity of V when e ≠ e1 and a = a1

4.2 Characterizations on the (α, V1)-migrativity of V when e = e1 and a ≠ a1

5. Discussion and conclusions

5.1 Discussion

5.2 Conclusions and future work

Footnotes

Acknowledgments

References

4.1 Characterizations on the (α, V₁)-migrativity of V when e ≠ e₁ and a = a₁

4.2 Characterizations on the (α, V₁)-migrativity of V when e = e₁ and a ≠ a₁