On retractions and extension of quasi-overlap and quasi-grouping functions defined on bounded lattices

Abstract

Saminger-Platz, Klement, and Mesiar (2008) extended t-norms from a complete sublattice to its respective lattice using the conventional definition of sublattice. In contrast, Palmeira and Bedregal (2012) introduced a more inclusive sublattice definition, via retractions. They expanded various important mathematical operators, including t-norms, t-conorms, fuzzy negations, and automorphisms. They also introduced De Morgan triples (semi-triples) for these operators and provided their extensions in their groundbreaking work. In this paper, we propose a method of extending quasi-overlap functions and quasi-grouping functions defined on bounded sublattices (in a broad sense) to a bounded superlattice. To achieve that, we use the technique proposed by Palmeira and Bedregal. We also define: quasi-overlap (resp. quasi-grouping) functions generated from quasi-grouping (resp. quasi-overlap) functions and frontier fuzzy negations, De Morgan (semi)triples for the classes of quasi-overlap functions, quasi-grouping functions and fuzzy negations, as well as its respective extensions. Finally we study properties of all extensions defined.

Keywords

Retractions extensions quasi-overlap quasi-grouping bounded lattices

1 Introduction

Quasi-overlap functions were introduced by Paiva et al. in 2021 [1] as a generalization of overlap functions –an important aggregation operator introduced by Bustince et al. in 2010 [2]. Since then, quasi-overlaps have attracted the attention of the academic community, being the center of much research (see [3 –9]). Other important class of aggregation functions that has been investigated in the last years (see [10 –12]) and that is dual to the concept of quasi-overlap is the called quasi-grouping, which was introduced by Qiao and Zhao in [13].

In mathematics, to extend a function defined over a certain domain means to define it over a domain that contains the first one (see [14 –16]). Given a function with certain properties, it is reasonable to ask under what conditions its extension preserves those properties.

In modern fuzzy logic, lattices are used as range of membership degrees, conjunctions are interpreted by triangular norms (t-norms) and disjunctions by triangular conorms (t-conorms).

In [17], Saminger-Platz, Klement and Mesiar provided a way to extend a t-norm from a complete sublattice to the respective lattice. They have considered the ordinary definition of a sublattice, i.e., the sublattice is necessarily a subset of the lattice. In order to relax this situation, Palmeira and Bedregal [18] provided a more general way of defining sublattices, via retractions; namely: given two bounded lattices, M and L, their idea was to provide a way of identifying a copy (retract), K, of M into L, such that K is isomorphic to M. Thus, the lattice structure of M, as well as its properties, are inherited by K via isomorphism. In that work, using their new sublattice definition, they provided extensions for t-norms, t-conorms, t-subnorms, fuzzy negations and automorphisms. Further, they defined De Morgan triples (semi-triples) for the classes of t-norms, t-conorms and fuzzy negations and provided its extensions.

In this article, we introduce a technique for extending quasi-overlap functions and quasi-grouping functions defined on sublattices of bounded lattices. Our approach builds upon the method put forth by Palmeira and Bedregal in [18]. We also: (a) investigate the properties inherited by the extended quasi-overlap functions; (b) define quasi-overlap (resp. quasi-grouping) functions generated from quasi-grouping (resp. quasi-overlap) functions and frontier fuzzy negations as well as investigate properties of its extensions; (c) define De Morgan triples (semi-triples) for the classes of quasi-overlap functions, quasi-grouping functions and fuzzy negations and explore properties of its extensions; (d) investigate the properties of extensions of conjugates of quasi-overlap functions, quasi-grouping functions, fuzzy negations and De Morgan triples (semi-triples).

The paper is organized as follows. In section 2, we recall some basic concepts and results that will be useful for the remainder of this paper. In section 3, we introduce extensions of n-ary functions via retractions, provide an extension for quasi-overlap functions defined on bonded lattices, and demonstrate some properties. In section 4, we supply an extension for quasi-grouping functions defined on bonded lattices, define quasi-overlap (resp. quasi-grouping) functions generated from quasi-grouping (resp. quasi-overlap) functions and frontier fuzzy negations and investigate the properties of its extensions. In section 5, we define De Morgan triples and De Morgan semitriples for quasi-overlap, quasi-grouping functions and fuzzy negations, and investigate the properties of its extensions. In section 6, we explore the relations between automorphism and the provided extensions. Finally, in section 7 you find the final considerations.

2 Prelim Preliminaries

This section presents an overview of some key concepts for the paper. We assume that the reader has knowledge about partial order or partially ordered set –poset (see [19], Definition 1.2), linear order or linearly ordered set (see [19], Definition 1.3), bottom and top elements (see [19], Definition 1.21), bounded posets, (complete) lattice (see [19], Definition 2.4) and bounded lattice (see [19], Definition 2.12).

Given a partial order 〈X, ≤ _X〉 we use the following notation: if X has bottom and top elements they will be denote by 0_X and 1_X, respectively; the symbols ∨_X and ∧_X will be used for the join and the meet operations, respectively. Moreover, a bounded lattice 〈L, ≤ _L, 0_L, 1_L〉 and its algebraic counterpart, 〈L, ≥ _L, ∧ _L, ∨ _L, 0_L, 1_L〉, are used interchangeably. In both cases we use just L to denote the structures.

2.1 Homomorphisms and retractions

In what follows we recall the concepts of homomorphism. For more details, see [18].

Definition 2.1. [Increasingness and decreasingness] Given two partial orders 〈X, ≤ _X〉 and 〈Y, ≤ _Y〉, a function f : X → Y is said to be: (a) increasing (resp. strictly increasing) if x₁ ≤ _Xx₂ implies f (x₁) ≤ _Yf (x₂) (resp. x₁ < _Xx₂ implies f (x₁) < _Yf (x₂)); (b) decreasing (resp. strictly decreasing) if x₁ ≤ _Xx₂ implies f (x₁) ≥ _Yf (x₂) (resp. if x₁ < _Xx₂ implies f (x₁) > _Yf (x₂)).

Definition 2.2. [Lattice ord-homomorphism] Let L and M be bounded lattices. An increasing function f : L → M is said to be a lattice ord-homomorphism whenever f (0_L) =0_M and f (1_L) =1_M.

Definition 2.3. [Lattice alg-homomorphism] Let L and M be bounded lattices. A function f : L → M is said to be a lattice alg-homomorphism if it satisfies the following conditions.

f (x ∧ _Ly) = f (x) ∧ _Mf (y), for all x, y ∈ L;

f (x ∨ _Ly) = f (x) ∨ _Mf (y), for all x, y ∈ L;

f (0_L) =0_M and f (1_L) =1_M.

Proposition 2.4. [[18]] Every alg-homomorphism is an ord-homomorphism.

Since every alg-homomorphism is an ord-homomorphism, from now on we will use the term homomorphism to mean ord-homomorphism.

Definition 2.5. [LM-retraction and retract] Given two functions r : L → M and s : M → L, the pair (r, s) is called a LM-retraction whenever r ∘ s = Id_M. In this case M is said to be a retract of L and s is said to be a pseudo-inverse of r.

Definition 2.6. [Sublattice and superlattice] Given L and M bounded lattices, M is said to be a sublattice of L in a broad sense if M is a retract of L 1 . In this context, we also say that L is a superlattice with respect to M.

Example 2.7. Let L, M be the bounded lattices shown in Fig. 1 and r : L → M, s : M → L given by setting $\begin{matrix} r (1_{L}) & = 1_{M}; & s (1_{M}) & = 1_{L}; \\ r (d) & = 1_{M}; & s (f) & = b; \\ r (b) & = f; & s (g) & = c; \\ r (c) & = g; & s (e) & = a; \\ r (a) & = e; & s (0_{M}) & = 0_{L} . \\ r (0_{L}) & = 0_{M} . \end{matrix}$ It is easy to see that r ∘ s = Id_M. Therefore, (r, s) is a LM-retraction and M is a sublattice of L (or, equivalently, L is a superlattice with respect to M).

Fig. 1

Bounded lattices L and M.

In what follows, we will use (r, s) to denote a LM-retraction, L to refer to a bounded lattice and M to indicate a sublattice (as in Definition 2.8) of it.

Definition 2.8. [Lower/upper LM-retractions] A LM-retraction (r, s) is said to be

lower (resp. strict lower) if s ∘ r ≤ _LId_L (resp. s ∘ r < _LId_L);

upper (resp. strict upper) if s ∘ r ≥ _LId_L (resp. s ∘ r > _LId_L).

Observe that for any LM-retraction (r, s), s is injective. Here, we also use the term LM-retraction if r and s are lattice homomorphisms.

Proposition 2.9. r (x) =1_M iff x = 1_L whenever (r, s) is a lower LM-retraction. Dually, r (x) =0_M iff x = 0_L whenever (r, s) is an upper LM-retraction.

Proof. Let (r, s) be a lower LM-retraction. Since r and s are homomorphisms, it follows that x = 1_L ⇒ r (x) = r (1_L) =1_M

and $\begin{matrix} r (x) = 1_{M} & \Rightarrow x \geq_{L} (s \circ r) (x) = s (1_{M}) = 1_{L} \\ \Rightarrow x = 1_{L} . \end{matrix}$ The proof that if (r, s) is an upper LM-retraction then r (x) =0_M iff x = 0_L is dually analogous. □

2.2 Aggregation functions

The content of this part can also be found in [20, 21].

Definition 2.10. If 〈X, ≤_X, 0_X, 0_X〉 is a bounded poset, an increasing n-ary function A : Xⁿ → X is called an aggregation function whenever A (0_X, …, 0_X) =0_X and A (1_X, …, 1_X) =1_X.

Next we recall some important properties of aggregation functions. Let us begin by revisiting the concept of bivariate function, essential for characterizing Archimedean and nilpotent aggregation functions.

Definition 2.11. [Power of a Bivariate Function] The power of a bivariate function f : X² → X is defined as: $f^{(1)} (x) = x;$ (1) $f^{(n + 1)} (x) = f (x, f^{(n)} (x)), for n \in ℤ_{+}^{*} .$ (2)

Definition 2.12. Let A : X² → X be an aggregation function.

If $〈 X, \leq_{X} 〉$ is a bounded poset, A is said to be Archimedean if, for all (x, y) ∈ (X \ {0_X, 1_X}) ², there exists $n \in ℤ_{+}^{*}$ such that A⁽ⁿ⁾ (x) < y 2 . An element x ∈ X \ {0_X, 1_X} is said to be a nilpotent element of A if there exists $n \in Z_{+}^{*}$ such that A⁽ⁿ⁾ (x) =0_X. If each element x ∈ X \ {0_X, 1_X} is a nilpotent element of A, then A is said to be a nilpotent aggregation. An element a ∈ X \ {0_X, 1_X} is a zero divisor of A if for all i ∈ {1, …, n} there exists some (x₁, …, x_i-1, a, x_i+1, …, x_n) ∈ (X \ {0_X}) ⁿ such that A (x₁, …, x_i-1, a, x_i+1, …, x_n) =0_X 3 .

If $〈 X, \leq_{X} 〉$ is a poset that has a top element, A satisfies the cancellation law if, for x, y, z ∈ X, it holds: A (x, y) = A (x, z) ⇒ x = 0_X or y = z.

An element x ∈ X is said to be an idempotent element of A if A (x, …, x) = x. If each element x ∈ X is an idempotent element of A, then A is said to be an idempotent aggregation.

Example 2.13.

A : ^unit2 → unit, given by A (x, y) = min {x, y} · max {x, y}, is an Archimedean aggregation function: A⁽ⁿ⁾ (x) = xⁿ and $lim_{n \to + \infty} x^{n} = 0$ for all (x, y) ∈ ([0, 1]\{0, 1}) ² and $2 < n \in ℤ_{+}^{*}$ . So, there exists $0 < r \in ℝ$ such that n > r implies A⁽ⁿ⁾ (x) < y.

$A (x, y) = {\begin{matrix} \frac{2 xy}{x + y}, & if x + y \neq 0; \\ 0, & if x + y = 0 . \end{matrix}$ is an aggregation function on ^unit2 satisfying the cancellation law.

If $〈 L, \leq_{L} 〉$ is a bounded lattice, the meet operation, x ∧ y, and the join operation, x ∨ y, both defined on L², are idempotent aggregations.

A (x, y) = max(x + y - 1, 0) is a nilpotent aggregation on [0, 1] ² 4 . Indeed, if 0 < x ≤ 0.5, then A⁽²⁾ (x) = A (x, x) = max(2x - 1, 0) =0. On the other hand, if 0.5 < x < 1, then $A^{(n)} (x) = {\begin{matrix} nx - (n - 1), & if x \geq \frac{n - 1}{n}, \\ 0, & if x < \frac{n - 1}{n} . \end{matrix}$

Once $lim_{n \to + \infty} \frac{n - 1}{n} = 1$ , for all ɛ > 0 there exists $0 < r \in ℝ$ such that n > r implies $x < \frac{n - 1}{n}$ . So, A⁽ⁿ⁾ (x) =0.

0.999 is a zero divisor element of the aggregation A (x, y) = max {x + y - 1, 0}: for all x < 0.001 it follows that A (0.999, x) = max {x - 0.001, 0} =0. However, a = 1 is not a zero divisor.

3 Extensions of quasi-overlap functions

Subsequently, we revisit the concept of quasi-overlap function (cf. [1]).

Definition 3.1. [Quasi-overlap function] An increasing and commutative operation O : L² → L is called a quasi-overlap function whenever it satisfies the following boundary conditions:

O (x, y) =0_L iff x = 0_L or y = 0_L;

O (x, y) =1_L iff x = 1_L and y = 1_L.

Example 3.2.

If there exists at least an element ξ ∈ L \ {0_L, 1_L}, the function O_ξ : L² → L, defined as $O_{ξ} (x, y) = {\begin{matrix} 0_{L}, & if x = 0_{L} or y = 0_{L}, \\ 1_{L}, & if x = 1_{L} and y = 1_{L}, \\ ξ, & otherwise, \end{matrix}$ is a quasi-overlap function on L [22].

If 0_L is a meet-irreducible element of L 5 , the binary meet function, given by x ∧ y for all x, y ∈ L, is a quasi-overlap function on L [1].

The functions A (x, y) = min {x, y} · max {x, y} and $A (x, y) = {\begin{matrix} \frac{2 xy}{x + y}, & if x + y \neq 0; \\ 0, & if x + y = 0 . \end{matrix}$ are quasi-overlap on the unit.

In what follows we propose a method of extending functions defined on sublattices (in a broad sense) of a bounded lattice via retractions.

Definition 3.3. [Extension] Given a LM-retraction (r, s) and a function f : Mⁿ → M, we will say that a function f^E : Lⁿ → L extends f from M to L if it is such that $f^{E} \circ ({\overset{︷}{s \times \dots \times s}}^{n times}) = s \circ f$ (3) and $r \circ f^{E} = f \circ ({\overset{︷}{r \times \dots \times r}}^{n times}) .$ (4)

Theorem 3.4. Given a quasi-overlap O : M² → M and a LM-retraction (r, s) that satisfies

$r (x) = 0_{M} \Leftrightarrow x = 0_{L}$ (5) and

$r (x) = 1_{M} \Leftrightarrow x = 1_{L},$ (6) the function O^E : L² → L, given by $O^{E} (x, y) = s (O (r (x), r (y))),$ (7) is a quasi-overlap function that extends O from M to L.

Proof. From Eq. (7) and once r ∘ s = Id_M, it follows that:

$\begin{matrix} [O^{E} \circ (s \times s)] (x, y) & = s (O (r (s (x)), r (s (y)))) \\ = s (O (x, y)) \\ = (s \circ O) (x, y) \end{matrix}$ for all (x, y) ∈ M × M. Similarly, [r ∘ O^E] (x, y) = [O ∘ (r × r)] (x, y) for all (x, y) ∈ L × L. Therefore, O^E extends O from M to L. Once O, r and s are increasing and O is commutative, then O^E is also increasing and commutative. Further, for all (x, y) ∈ L × L, we have

O^E (x, y) =0_L

⇔s (O (r (x) , r (y))) = 0_L [FromEq . eqrefeq : extensofquas3 .]

⇔O (r (x) , r (y)) = 0_M [FromEq . eqrefeq : extensofquas1 .]

⇔r (x) =0_M or r (y) =0_M [FromDefinition3.1 .]

⇔x = 0_L or y = 0_L . [FromEq . eqrefeq : extensofquas1 .]

Similarly, O^E (x, y) =1_L iff x = 1_L and y = 1_L□.

As a direct consequence of Proposition 2.9 and Theorem 3.4, we have the following:

Corollary 3.5. Given a quasi-overlap O : M² → M and a LM-retraction (r, s), the function O^E : L² → L defined in Eq. (7) is a quasi-overlap function that extends O from M to L if one of the following conditions holds.

(r, s) is a lower LM-retraction satisfying Eq. (5);

(r, s) is an upper LM-retraction satisfying Eq. (6).

It is worth noting that r and s of Theorem 3.4 are not unique. For each pair (r, s), O^E will represent a distinct extension of O from M to L. This implies that there are multiple ways to extend O. See the example below.

Example 3.6. Let M = {0, 2, 4, . . . . , n} where n is an even number and L = {0, 1, 2, . . . . , n + 1}, both equipped with the natural order on non-negative integers numbers, and O : M² → M the quasi-overlap given by O (x, y) = min(x, y).

For each i = 1, . . . , n, consider r_i : L → M defined by $r_{i} (x) = {\begin{matrix} x, & if { \end{matrix} x is even, x + 1, & if {x is odd and x < i, x - 1, & if {x is odd and x \geq i,$ and s_i : M → L defined by $s_{i} (x) = {\begin{matrix} x - 1, & if { \end{matrix} x < i, x + 1, & if {x \geq i .$ Then, r_i ∘ s_i = Id_M and O^E (x, y) = s_i (O (r_i (x) , r_i (y))) is a quasi-overlap that extends O from M to L for all i = 1, . . . . , n.

Let r′, r″ : L → M defined by $r^{'} (x) = {\begin{matrix} x, & if { \end{matrix} x is even, 0, & if {x is odd,$ $r^{″} (x) = {\begin{matrix} x, & if { \end{matrix} x is even, x - 1, & if {x is odd,$ and s : M → L be defined by s (x) = x. Then, r′ ∘ s = r″ ∘ s = Id_M, and O^E (x, y) = s (O (r′ (x) , r′ (y))) and O^E (x, y) = s (O (r″ (x) , r″ (y))) are quasi-overlap functions that extend O from M to L.

Given r : L → M defined by $r (x) = {\begin{matrix} x, & if { \end{matrix} x is even, x - 1, & if {x is odd,$ and s′, s″ : M → L defined by s′ (x) = x and s″ (x) = x + 1, we have r ∘ s′ = r ∘ s″ = Id_M. Thus, O^E (x, y) = s′ (O (r (x) , r (y))) and O^E (x, y) = s″ (O (r (x) , r (y))) are quasi-overlap functions that extend O from M to L.

Next, we demonstrate that the extension O^E, defined in Eq. (7) of Theorem 3.4, preserves some properties of O.

Proposition 3.7. If O is a quasi-overlap function on M and (r, s) is a lower LM-retraction satisfying Eq. (5), then the following assertions hold.

O^E is Archimedean whenever O is Archimedean and r is strict;

s (a) is a nilpotent element of O^E whenever a is a nilpotent element of O;

s (a) is also an idempotent element of O^E whenever a is an idempotent element of O;

s (a) is also a zero divisor of O^E whenever a is a zero divisor of O.

Proof. Initially, it is worth observing that, from Proposition 2.9, Eq. (6) is satisfied.

Let O be Archimedean and consider (a, b) ∈ (L \ {0_L, 1_L}) ². Using Eq. (5) and Eq. (6), we can deduce that (r (a) , r (b)) ∈ (M \ {0_M, 1_M}) ². Once O is Archimedean, there exists $n \in ℤ_{+}^{*}$ such that O⁽ⁿ⁾ (r (a)) < _Mr (b). Considering that s is an injective homomorphism and (r, s) is a strict lower LM-retraction, it follows that: O^E⁽ⁿ⁾ (a) = s (O⁽ⁿ⁾ (r (a))) < _Ls (r (b)) < _Lb.

If a ∈ M \ {0_M, 1_M} is a nilpotent element of O, then there exists $n \in ℤ_{+}^{*}$ such that O⁽ⁿ⁾ (a) =0_M. As s is an injective homomorphism, we have s (a) ∈ L \ {0_L, 1_L} and it is a nilpotent element of O^E. To see why, suppose that s (a) =0_L. Then we would have a = r (s (a)) = r (0_L) =0_M, which is a contradiction. Therefore, s (a) ≠0_L. Similarly we can show that s (a) ≠1_L. Further, $\begin{matrix} {O^{E}}^{(n)} (s (a)) & = s (O^{(n)} (r (s (a)))) \\ = s (O^{(n)} (a)) \\ = s (0_{M}) = 0_{L} . \end{matrix}$

If a ∈ M is an idempotent element of O, it follows that:

$\begin{matrix} O^{E} (s (a), s (a)) & = s (O (r (s (a)), r (s (a)))) \\ = s (O (a, a)) \\ = s (a) . \end{matrix}$

Suppose a ∈ M \ {0_M, 1_M} is a zero divisor element of O and consider (a, x) ∈ (M \ {0_M}) ² such that O (a, x) =0_M. From proof of (ii), we have s (a) ∈ L \ {0_L, 1_L} and s (x) ∈ L \ {0_L}. Therefore, (s (a) , s (x)) ∈ (L \ {0_L}) ² and $\begin{matrix} O^{E} (s (a), s (x)) & = s (O (r (s (a)), r (s (x)))) \\ = s (O (a, x)) \\ = s (0_{M}) = 0_{L} . \end{matrix}$

□

Proposition 3.8. Let (r, s) be a lower LM-retraction satisfying Eq. (5) and O be a quasi-overlap on M that satisfies the cancellation law. O^E (Eq. (7)) satisfies the cancellation law iff r is injective.

Proof. If O^E satisfies the cancellation law but r is not injective, then there exist y, z ∈ L such that r (y) = r (z) with y ¬ = z. Given any x ∈ L, we have: O^E (x, y) = s (O (r (x) , r (y))) = s (O (r (x) , r (z))) = O^E (x, z). Taking x ¬ =0_L, once O^E satisfies the cancellation law, it follows that y = z, which is a contradiction. Thus, r is injective. Now, suppose that r is injective. Let a, b, c ∈ L be such that O^E (a, b) = O^E (a, c). Then, s (O (r (a) , r (b))) = s (O (r (a) , r (c))), and since s is injective, it follows that O (r (a) , r (b)) = O (r (a) , r (c)). By the cancellation law of O on M, we have r (a) =0_M or r (b) = r (c), and thus x = 0_L or b = c. Therefore, O^E satisfies the cancellation law.□

4 Extensions of quasi-grouping functions and fuzzy negations

In the following we review the concept of a quasi-grouping function (cf. [10, 13]).

Definition 4.1. [Quasi-grouping function] An increasing and commutative operation G : L² → L is called a quasi-grouping function whenever it satisfies the following boundary conditions:

G (x, y) =0_L iff x = 0_L and y = 0_L;

G (x, y) =1_L iff x = 1_L or y = 1_L.

Example 4.2.

Fixed ξ ∈ L \ {0_L, 1_L}, the binary operator G_ξ : L² → L, given by

$G_{ξ} (x, y) = {\begin{matrix} 0_{L}, & if x = 0_{L} and y = 0_{L}, \\ 1_{L}, & if x = 1_{L} or y = 1_{L}, \\ ξ, & otherwise, \end{matrix}$

is a quasi-grouping function.

If 1_L is co-prime element of L 6 , the binary join function, given by x ∨ y for all x, y ∈ L, is a quasi-grouping function [13].

Theorem 4.3. Given a quasi-grouping function G : M² → M and a LM-retraction (r, s) that satisfies Eq. (5) and Eq. (6), the function G^E : L² → L, given by $G^{E} (x, y) = s (G (r (x), r (y))),$ (8) is a quasi-grouping that extends G from M to L.

Proof. By definition of G^E it follows that $\begin{matrix} [G^{E} \circ (s \times s)] (x, y) & = G^{E} (s (x), s (y)) \\ = s (G (r (s (x)), r (s (y)))) \\ = (s \circ G) (x, y) \end{matrix}$ for all (x, y) ∈ M × M. Analogously, [r ∘ G^E] (x, y) = [G ∘ (r × r)] (x, y) for all (x, y) ∈ L × L. Therefore, G^E extends G from M to L. Once G, r and s are increasing and G is commutative, then G^E is also increasing and commutative. Moreover, for all (x, y) ∈ L × L, it holds:

G^E (x, y) =0_L

⇔s (G (r (x) , r (y))) = 0_L [FromEq . eqrefeq : extensofqgroup .]

⇔G (r (x) , r (y)) = 0_M [FromEq . eqrefeq : extensofquas1 .]

⇔r (x) =0_M and r (y) =0_M [FromDefinition4.1 .]

⇔x = 0_L and y = 0_L [FromEq . eqrefeq : extensofquas1 .]

Similarly, G^E (x, y) =1_L iff x = 1_L or y = 1_L.□

Next, we revisit the concept of fuzzy negation (cf. [23]).

Definition 4.4. [Fuzzy negation] A function N : L → L is called a fuzzy negation if it satisfies the following properties:

N is decreasing;

N (0_L) =1_L and N (1_L) =0_L.

Further, we call N:

strong if it has the involutive property: $N (N (x)) = x, \forall x \in L;$ (9)

frontier if it is such that: $N (x) \in {0_{M}, 1_{M}} iff x \in {0_{L}, 1_{M}} .$ (10)

Regarding extension of fuzzy negations, in [18] Palmeira and Bedregal demonstrated the following:

Proposition 4.5. If N : M → M is a fuzzy negation and (r, s) is a LM-retraction, then N^E : L × L → L, given by $N^{E} (x) = s (N (r (x))),$ (11) is a fuzzy negation that extends N from M to L.

Proposition 4.6. Let N₁, N₂ : L → L be frontier fuzzy negations, O : L² → L a quasi-overlap function and G : L² → L a quasi-grouping function. Then,

O_G,N₁,N₂ : L² → L, defined by $O_{G, N_{1}, N_{2}} (x, y) = N_{1} (G (N_{2} (x), N_{2} (y))),$ (12) is a quasi-overlap function;

G_O,N₁,N₂ : L² → L, defined by $G_{O, N_{1}, N_{2}} (x, y) = N_{1} (O (N_{2} (x), N_{2} (y))),$ (13) is a quasi-grouping function.

Proof.

From commutativity of G follows that O_G,N₁,N₂ is commutative. Since N₁ and N₂ are decreasing and G is increasing, given x, y, z ∈ L, we have:

y ≤ _Lz

⇒N₂ (y) ≥ _LN₂ (z)

⇒G (N₂ (x) , N₂ (y)) ≥ _LG (N₂ (x) , N₂ (z))

⇒N₁ (G (N₂ (x) , N₂ (y)))

≤ _LN₁ (G (N₂ (x) , N₂ (z)))

⇒O_G,N₁,N₂ (x, y) ≤ _LO_G,N₁,N₂ (x, z).

Further, for all (x, y) ∈ L², we have:

O_G,N₁,N₂ (x, y) =0_L

⇔N₁ (G (N₂ (x) , N₂ (y))) = 0_L

⇔G (N₂ (x) , N₂ (y)) =1_L [ByEq . eqrefeq : frontierneg .]

⇔N₂ (x) = 1_L or N₂ (y) = 1_L [ByDefinition4.1]

⇔x = 0_L or y = 0_L . [ByEq . eqrefeq : frontierneg .]

Analogously, O_G,N₁,N₂ (x, y) =1_L iff x = 1_L and y = 1_L.

It follows similarly to the proof of (i). □

In [10] (Theorem 5.2), Sun, Pang and Zhang proved that given a strong fuzzy negation N : L → L, a binary operator G : L × L → L is a quasi-grouping function iff there exists a quasi-overlap function O : L × L → L such that G (x, y) = N (O (N (x) , N (y))) , ∀ x, y ∈ L.

In what follows, we generalize this result.

Definition 4.7. A function f : X → X is said to be:

left-invertible, if there exists a function f^Left : X → X, called inverse of f to left, such that f^Left ∘ f = Id_X;

right-invertible, if there exists a function f^Right : X → X, called inverse of f to right, such that f ∘ f^Right = Id_X;

invertible, if it is left-invertible and right-invertible.

Remark 4.8. Regarding Definition 4.7, if f : X → X is invertible, f^Left : X → X is an inverse of f to left and f^Right : X → X is an inverse of f to right, then: $\begin{matrix} f^{Left} & = f^{Left} \circ {Id}_{X} \\ = f^{Left} \circ (f \circ f^{Right}) \\ = (f^{Left} \circ f) \circ f^{Right} \\ = {Id}_{X} \circ f^{Right} \\ = f^{Right} . \end{matrix}$

Theorem 4.9. If L admits a left-invertible frontier fuzzy negation whose left-inverse is a frontier fuzzy negation, a binary operator G : L × L → L is a quasi-grouping function iff there exists a quasi-overlap function O : L × L → L and frontier fuzzy negations N₁, N₂ : L → L such that G (x, y) = G_O,N₁,N₂ (x, y) , ∀ x, y ∈ L.

Proof. Let N : L → L be a frontier fuzzy negation with left-inverse N′ : L → L that is also a frontier fuzzy negation. Suppose G : L × L → L is a quasi-grouping function. Then, for all x, y ∈ L, we have: $\begin{matrix} G (x, y) & = N^{'} (N (G (N^{'} (N (x)), N^{'} (N (y))))) \\ = N^{'} (O_{G, N, N^{'}} (N (x), N (y))) \\ = G_{O_{G, N, N^{'}}, N^{'}, N} (x, y) \end{matrix}$ The reciprocal follows from Proposition 4.6(ii). □

Analogously, we have the following result for quasi-overlap functions:

Proposition 4.10. If L admits a left-invertible frontier fuzzy negation whose left-inverse is a frontier fuzzy negation, a binary operator G : L × L → L is a quasi-grouping function iff there exists a quasi-overlap function O : L × L → L and frontier fuzzy negations N₁, N₂ : L → L such that O (x, y) = O_G,N₁,N₂ (x, y) , ∀ x, y ∈ L.

Proof. Let N : L → L be a frontier fuzzy negation with left-inverse N′ : L → L that is also a frontier fuzzy negation. Suppose O : L × L → L is a quasi-overlap function. Then, for all x, y ∈ L, we have: $\begin{matrix} O (x, y) & = N^{'} (N (O (N^{'} (N (x)), N^{'} (N (y))))) \\ = N^{'} (G_{O, N, N^{'}} (N (x), N (y))) \\ = O_{G_{O, N, N^{'}}, N^{'}, N} (x, y) \end{matrix}$ The reciprocal follows from Proposition 4.6(i).□

Remark 4.11. Regarding Theorem 4.9, note that we can have G (x, y) = G_{O_G,N₁,N₂,N₁,N₂} (x, y) , ∀ x, y ∈ L without N₁ and N₂ being left-invertible. Indeed, consider ξ ∈ L as in Example 3.2(i) and in Example 4.2(i), and N₁, N₂ : L → L frontier fuzzy negations such that N₁ (ξ) = ξ 1 . For all x, y ∈ L, we have: $\begin{matrix} G_{O_{ξ}, N_{1}, N_{2}} (x, y) \\ = N_{1} (O_{ξ} (N_{2} (x), N_{2} (y))) \\ = {\begin{matrix} N_{1} (0_{L}), & if N_{2} (x) = 0_{L} or N_{2} (y) = 0_{L}, \\ N_{1} (1_{L}), & if N_{2} (x) = 1_{L} and N_{2} (y) = 1_{L}, \\ N_{1} (ξ), & otherwise, \end{matrix} \\ = G_{ξ} (x, y) . \end{matrix}$ Analogously, O_{G_ξ,N₁,N₂} (x, y) = O_ξ (x, y), for all x, y ∈ L. Thus, for all x, y ∈ L, $\begin{matrix} G_{(O_{G_{ξ}, N_{1}, N_{2}}), N_{1}, N_{2}} (x, y) & = G_{O_{ξ}, N_{1}, N_{2}} (x, y) \\ = G_{ξ} (x, y) . \end{matrix}$ Therefore, taking G = G_ξ we have G (x, y) = G_{(O_G,N₁,N₂),N₁,N₂} (x, y), for all x, y ∈ L.

Proposition 4.12. Let N₁, N₂ : M → M be frontier fuzzy negations, O : M² → M a quasi-overlap function and G : M² → M a quasi-grouping function. If (r, s) is a LM-retraction satisfying Eq. (5) and Eq. (6), then

O_G,N₁,N₂^E : L² → L, defined by O_G,N₁,N₂^E (x, y) = s (O_G,N₁,N₂ (r (x) , r (y))), is a quasi-overlap function that extends O_G,N₁,N₂ from M to L;

G_O,N₁,N₂^E : L² → L, defined by G_O,N₁,N₂^E (x, y) = s (G_O,N₁,N₂ (r (x) , r (y))), is a quasi-grouping function that extends G_O,N₁,N₂ from M to L.

Proof. It follows from Theorem 3.4 and Theorem 3.4 since, by Proposition 4.6, O_G,N₁,N₂ is a quasi-overlap function and G_O,N₁,N₂ is a quasi-grouping function. □

Theorem 4.13. Let N₁, N₂ : M → M be frontier fuzzy negations, O : M² → M a quasi-overlap function and G : M² → M a quasi-grouping function. If (r, s) is a LM-retraction satisfying Eq. (5) and Eq. (6), then

O_G,N₁,N₂^E (x, y) = O_{G^E,N₁^E,N₂^E} (x, y);

G_O,N₁,N₂^E (x, y) = G_{O^E,N₁^E,N₂^E} (x, y).

Proof.

By Equations (7), (8), (11) and (12) it follows:

O_G,N₁,N₂^E (x, y)

= s (O_G,N₁,N₂ (r (x) , r (y)))

= s (N₁ (G (N₂ (r (x)) , N₂ (r (y)))))

= s ( N₁ (G (r (s (N₂ (r (x)))) , r (s (N₂ (r (y)))))))

= s (N₁ (G (r (N₂^E (x)) , r (N₂^E (y)))))

= s (N₁ (r (s (G (r (N₂^E (x)) , r (N₂^E (y)))))))

= s (N₁ (r (G^E (N₂^E (x) , N₂^E (y)))))

= N₁^E (G^E (N₂^E (x) , N₂^E (y)))

= O_{G^E,N₁^E,N₂^E} (x, y)

It follows analogously to the proof of (i). □

5 De Morgan triples

The De Morgan laws are logical rules that relate conjunctions and disjunctions through negations. More precisely, the following rules: $α \land β \equiv \neg (\neg α \lor \neg β) and α \lor β \equiv \neg (\neg α \land \neg β)$ (14) or $\neg (α \land β) \equiv \neg α \lor \neg β and \neg (α \lor β) \equiv \neg α \land \neg β .$ (15)

The logical equivalences in (14) are usually generalized in fuzzy logic using t-norms (T) 8 , t-conorms (S) 9 and fuzzy negations (N) as: $T (x, y) = N (S (N (x), N (y)))$ (16) and $S (x, y) = N (T (N (x), N (y))) .$ (17)

Taking y = 1_L in Eq. (16), for all x ∈ L, we have: $\begin{matrix} N (N (x)) & = N (S (N (x), 0_{L})) \\ = N (S (N (x), N (1_{L}))) \\ = T (x, 1_{L}) = x, \end{matrix}$ that is, N should be a strong negation.

Usually, a De Morgan triple is a triple 〈T, S, N〉 where T is a t-norm, S is a t-conorm and N is a fuzzy negation (all on [0, 1]) satisfying Eq. (16) and Eq. (17). Nevertheless, there are other authors which consider the Eq. (15) instead of Eq. (14) (for more details see [24]). Palmeira and Bedregal generalized this second approach for an arbitrary bounded lattices L as follows.

Definition 5.1. [[18]] Given a t-norm T : L² → L, a t-conorm S : L² → L and a fuzzy negation N : L → L, 〈T, S, N〉 is a De Morgan triple if, for all x, y ∈ L, we have:

N (T (x, y)) = S (N (x) , N (y));

N (S (x, y)) = T (N (x) , N (y)).

De Morgan triples can be defined for other classes of operators, for example for quasi-overlap and quasi-grouping functions.

Definition 5.2. Given a quasi-overlap function O : L² → L, a quasi-grouping function G : L² → L and a fuzzy negation N : L → L, we say that 〈O, G, N〉 is a De Morgan triple on L if, for all x, y ∈ L, we have:

N (O (x, y)) = G (N (x) , N (y));

N (G (x, y)) = O (N (x) , N (y)).

Regarding the Definition 5.2, note that if N is a strong fuzzy negation, then (DM1) and (DM2) are equivalent. Furthermore, for some overlap functions, grouping functions, and fuzzy negations, only one of these items holds, as can be seen in the example below.

Example 5.3. Let L be the bounded lattice shown in Fig. 2. It is easy to check that the function N : L → L, given by setting $\begin{matrix} N (0_{L}) & = 1_{L}; & N (a) & = d; \\ N (b) & = N (c) = b; \\ N (d) & = a; & N (1_{L}) & = 0_{L}; \end{matrix}$ is a fuzzy negation. Table 1 shows N × N.

Fig. 2

Bounded lattice L.

Table 1

N × N

N × N	0_L	a	b	c	d	1_L
0_L	(1_L, 1_L)	(1_L, d)	(1_L, b)	(1_L, b)	(1_L, a)	(1_L, 0_L)
a	(d, 1_L)	(d, d)	(d, b)	(d, b)	(d, a)	(d, 0_L)
b	(b, 1_L)	(b, d)	(b, b)	(b, b)	(b, a)	(b, 0_L)
c	(b, 1_L)	(b, d)	(b, b)	(b, b)	(b, a)	(b, 0_L)
d	(a, 1_L)	(a, d)	(a, b)	(a, b)	(a, a)	(a, 0_L)
1_L	(0_L, 1_L)	(0_L, d)	(0_L, b)	(0_L, b)	(0_L, a)	(0_L, 0_L)

Let O : L² → L be the quasi-overlap and G : L² → L be the quasi-grouping defined in Table 2. Then, N ∘ O = G ∘ (N × N) (see Table 3). However, N ∘ G ¬ = O ∘ (N × N), as shown in Table 4.

Table 2

O : L² → L and G : L² → L

0_L

1_L

0_L

1_L

0_L

1_L

0_L

1_L

0_L

1_L

0_L

1_L

0_M

1_L

0_L

1_L

Table 3

N ∘ O and G ∘ (N × N)

N ∘ O

0_L

1_L

G ∘ (N × N)

0_L

1_L

0_L

1_L

0_L

1_L

0_L

1_L

0_L

Table 4

N ∘ G and O ∘ (N × N)

N ∘ G

0_L

1_L

O ∘ (N × N)

0_L

1_L

0_L

1_L

0_L

1_L

0_L

1_L

0_L

1_L

0_L

Taking into account such a possibility, we define the following relaxed notion of De Morgan triples.

Definition 5.4. Given a quasi-overlap function O : L² → L, a quasi-grouping function G : L² → L and a fuzzy negation N : L → L, we say that 〈O, G, N〉 is a De Morgan O-semitriple on L (resp. De Morgan G-semitriple on L) if, for all x, y ∈ L, (DM1) holds (resp. (DM2) holds).

Proposition 5.5. Let O : L² → L be a quasi-overlap function, G : L² → L a quasi-grouping function and N : L → L an invertible fuzzy negation. Then, 〈O, G, N〉 is a De Morgan O-semitriple on L iff 〈O, G, N^-1〉 is a De Morgan G-semitriple on L.

Proof. Suppose 〈O, G, N〉 is a De Morgan O-semitriple on L. Then, for all x, y ∈ L, we have:

$\begin{matrix} N^{- 1} (G (x, y)) \\ = N^{- 1} (G (N (N^{- 1} (x)), N (N^{- 1} (y)))) \\ = N^{- 1} (N (O (N^{- 1} (x), N^{- 1} (y)))) \\ = O (N^{- 1} (x), N^{- 1} (y)) . \end{matrix}$ Thus, 〈O, G, N^-1〉 is a De Morgan G-semitriple on L. The reciprocal follows analogously.□

Theorem 5.6. Let O : M² → M be a quasi-overlap function, G : M² → M a quasi-grouping function and N : M → M a fuzzy negation. The following statements hold.

If 〈O, G, N〉 is a De Morgan O-semitriple on M, then 〈O^E, G^E, N^E〉 is a De Morgan O^E-semitriple on L.

If 〈O, G, N〉 is a De Morgan G-semitriple on M, then 〈O^E, G^E, N^E〉 is a De Morgan G^E-semitriple on L.

Proof.

If 〈O, G, N〉 is a De Morgan O-semitriple on M then, for all x, y ∈ L,

N^E (O^E (x, y))

= s (N (r (O^E (x, y)))) [ByEq . eqrefeq : extofneg .]

= s (N (r (s (O (r (x) , r (y)))))) [ByEq . eqrefeq : extensofquas3 .]

= s (N (O (r (x) , r (y))))

= s (G (N (r (x)) , N (r (y)))) [By (DM1) .]

= s (G (r (s (N (r (x)))) , r (s (N (r (y))))))

= G^E (N^E (x) , N^E (y)) . [ByEqs . eqrefeq : extensofqgroup, eqrefeq : extofneg .]

It follows similarly to the proof of (i). □

Corollary 5.7. Let O : M² → M be a quasi-overlap function, G : M² → M a quasi-grouping function and N : M → M a fuzzy negation. If 〈O, G, N〉 is a De Morgan triple on M, then 〈O^E, G^E, N^E〉 is a De Morgan triple on L.

Proof. Straightforward from Theorem 5.6.

□

6 Extensions and automorphisms

Definition 6.1. [[18]] A map ρ : L → L is said to be an automorphism if it is bijective and, for all x, y ∈ L, $x \leq_{L} y \Leftrightarrow ρ (x) \leq_{L} ρ (y) .$ (18)

Observe that, if ρ is an automorphism on L, then ρ^-1 is also an automorphism. More than that, the set Aut (L) of all automorphisms on L equipped with the composition operation is a group. It is also worth noting that $ρ (0_{L}) = 0_{L} \Leftrightarrow x = 0_{L}$ (19) and $ρ (1_{L}) = 1_{L} \Leftrightarrow x = 1_{L} .$ (20) Consequently, ρ^-1 (0_L) =0_L iff x = 0_L and ρ^-1 (1_L) =1_L iff x = 1_L.

Given a function F : Lⁿ → L and an automorphism ρ : L → L, the function F^ρ : Lⁿ → L, given by $F^{ρ} (x_{1}, \dots, x_{n}) = ρ^{- 1} (F (ρ (x_{1}), \dots, ρ (x_{n}))),$ (21) is called the conjugate of F by ρ.

Proposition 6.2. Given a quasi-overlap function O : L² → L, a quasi-grouping function G : L² → L, a fuzzy negation N : L → L and an automorphism ρ : L → L,

O^ρ : L² → L is a quasi-overlap function;

G^ρ : L² → L is a quasi-grouping function;

N^ρ : L → L is a fuzzy negation.

Proof.

Since O is commutative and increasing, and ρ and ρ^-1 satisfy Eq. (18), it follows that O^ρ is commutative and increasing. Further, For all (x, y) ∈ L², we have:

O^ρ (x, y) =0_L

⇔ρ^-1 (O (ρ (x) , ρ (y))) = 0_L [ByEq . eqrefeq : funcconjugate .]

⇔O (ρ (x) , ρ (y)) = 0_L [ByEq . eqrefeq : aut0L .]

⇔ρ (x) =0_L or ρ (y) =0_L [By (QO1) .]

⇔x = 0_L or y = 0_L . [ByEq . eqrefeq : aut0L .]

Analogously, for all (x, y) ∈ L², we have O^ρ (x, y) =1_L iff x = 1_L and y = 1_L.

It follows analogously to the proof of (i).

N^ρ (0_L) = ρ^-1 (N (ρ (0_L))) = ρ^-1 (N (0_L)) = ρ^-1 (1_L) =1_L, and N^ρ (1_L) = ρ^-1 (N (ρ (1_L))) = ρ^-1 (N (1_L)) = ρ^-1 (0_L) =0_L. Additionally, for all x, y ∈ L, we have: $\begin{matrix} x \leq y & \Rightarrow ρ (x) \leq ρ (y) \\ \Rightarrow N (ρ (y)) \leq N (ρ (x)) \\ \Rightarrow ρ^{- 1} (N (ρ (y))) \leq ρ^{- 1} (N (ρ (x))) \\ \Rightarrow N^{ρ} (y) \leq N^{ρ} (x) . \end{matrix}$

□

Proposition 6.3. Let O : L² → L be a quasi-overlap function, G : L² → L a quasi-grouping function, N : L → L a fuzzy negation and ρ : L → L an automorphism. If 〈O, G, N〉 is a De Morgan O-semitriple (resp. De Morgan G-semitriple) on L, then 〈O^ρ, G^ρ, N^ρ〉 is a De Morgan O-semitriple (resp. De Morgan G-semitriple) on L.

Proof. Suppose 〈O, G, N〉 is a De Morgan O-semitriple on L. Then, for all x, y ∈ L, we have:

N^ρ (O^ρ (x, y))

= ρ^-1 (N (ρ (O^ρ (x, y)))) [ByEq . eqrefeq : funcconjugate .]

= ρ^-1 (N (ρ (ρ^-1 (O (ρ (x) , ρ (y)))))) [ByEq . eqrefeq : funcconjugate .]

= ρ^-1 (N (O (ρ (x) , ρ (y))))

= ρ^-1 (G (N (ρ (x)) , N (ρ (y)))) [From (DM1) .]

= ρ^-1 (G (ρ (ρ^-1 (N (ρ (x)))) , ρ (ρ^-1 (N (ρ (y))))))

= G^ρ (N^ρ (x) , N^ρ (y)) [ByEq . eqrefeq : funcconjugate .]

Analogously, if 〈O, G, N〉 is a De Morgan G-semitriple, then 〈O^ρ, G^ρ, N^ρ〉 is a De Morgan G-semitriple.□

Corollary 6.4. Let O : L² → L be a quasi-overlap function, G : L² → L a quasi-grouping function, N : L → L a fuzzy negation and ρ : L → L an automorphism. If 〈O, G, N〉 is a De Morgan triple on L, then 〈O^ρ, G^ρ, N^ρ〉 is a De Morgan triple on L.

Proof. It follows straightforward from Proposition 6.3. □

In what follows we prove that, for sublattices in Palmeira and Bedregal’s sense (see Definition 2.8), the conjugate of an extension is equal to the extension of the conjugate for quasi-overlap functions, quasi-grouping functions and fuzzy negations. First, let us demonstrate a result that will be necessary for our proof.

Lemma 6.5. If (r, s) is a LM-retraction and ψ : L → L is an automorphism which extends an automorphism ρ on M then ψ^-1 extends ρ^-1.

Proof. For all x ∈ M, we have:

(ψ^-1 ∘ s) (x)

= (ψ^-1 ∘ s ∘ ρ ∘ ρ^-1) (x) [Since ρ ∘ ρ^-1 = Id_M .]

= (ψ^-1 ∘ ψ ∘ s ∘ ρ^-1) (x) [Since ψ extends ρ .]

= (s ∘ ρ^-1) (x) [Since ψ^-1 ∘ ψ = Id_L .]

and similarly, for all x ∈ L, we have (r ∘ ψ^-1) (x) = (ρ^-1 ∘ r) (x). Therefore, ψ^-1 extends ρ^-1.□

Theorem 6.6. Let ψ : L → L be an automorphism which extends an automorphism ρ on M. If O : M² → M is a quasi-overlap function, G : M² → M is a quasi-grouping function, N : M → M is a fuzzy negation and (r, s) is a LM-retraction satisfying Eq. (5) and Eq. (6), then

(O^E) ^ψ = (O^ρ) ^E;

(G^E) ^ψ = (G^ρ) ^E;

(N^E) ^ψ = (N^ρ) ^E.

Proof. For all x, y ∈ L, we have:

(O^E) ^ψ (x, y)

= ψ^-1 (O^E (ψ (x) , ψ (y))) [ByEq . eqrefeq : funcconjugate .]

= ψ^-1 (s (O (r (ψ (x)) , r (ψ (y))))) [ByEq . eqrefeq : extensofquas3 .]

= ψ^-1 (s (O (ρ (r (x)) , ρ (r (y))))) [ByEq . eqrefeq : extensionr .]

= ρ^-1 (O (ρ (r (x)) , ρ (r (y)))) [ByEq . eqrefeq : extensions .]

= O^ρ (r (x) , r (y)) [ByEq . eqrefeq : funcconjugate .]

= (O^ρ) ^E (x, y) [ByEq . eqrefeq : extensofquas3 .]

The proofs of (ii) and (iii) follow analogously.□

Corollary 6.7. Let ψ : L → L be an automorphism which extends an automorphism ρ on M. If O : M² → M is a quasi-overlap function, G : M² → M is a quasi-grouping function, N : M → M is a fuzzy negation and (r, s) is a LM-retraction, then (O^E) ^ψ = (O^ρ) ^E, (G^E) ^ψ = (G^ρ) ^E and (N^E) ^ψ = (N^ρ) ^E if one of the following conditions is satisfied.

(r, s) is a lower LM-retraction satisfying Eq. (5);

(r, s) is an upper LM-retraction satisfying Eq. (6).

Proof. It follows from Proposition 2.9 and Theorem 6.6.□

Corollary 6.8. Let ψ : L → L be an automorphism which extends an automorphism ρ on M. If O : M² → M is a quasi-overlap function, G : M² → M is a quasi-grouping function, N : M → M is a fuzzy negation and (r, s) a LM-retraction satisfying Eq. (5) and Eq. (6). Then, 〈 (O^E) ^ρ, (G^E) ^ρ, (N^E) ^ρ〉 = 〈 (O^ρ) ^E, (G^ρ) ^E, (N^ρ) ^E〉.

Proof. It follows straightforward from Theorem 6.6. □

7 Conclusion

Saminger-Platz, Klement, and Mesiar [17] introduced a method for expanding a t-norm from a complete sublattice to the corresponding lattice. Their approach adhered to the conventional definition of a sublattice, wherein the sublattice must necessarily be a subset of the lattice. To broaden this constraint, Palmeira and Bedregal presented an alternative approach in their work [18]. Instead of considering only the strict relationship of subsets, they proposed a more comprehensive definition of sublattices by using the concept of retractions. In their groundbreaking work Palmeira and Bedregal extended various important operators, including t-norms, t-conorms, fuzzy negations, and automorphisms. Furthermore, they introduced the concept of De Morgan triples (semi-triples) for the categories of t-norms, t-conorms, and fuzzy negations, also providing their extensions. On the other hand, the generalizations: quasi-overlap and quasi-grouping have been widely investigated recently and a natural question that arises is whether the extension mechanism using retractions can be used for these operators.

In this paper we have investigated how to extend important fuzzy functions from a bounded lattice to a superlattice. We have extended: quasi-overlap functions, quasi-grouping functions, De Morgan (semi) triples and Conjugated functions. The investigation of some important properties is also provided.

As a future work we intend to study the extension of further operators as well further properties.

Acknowledgments

This work has been supported by the National Council for Scientific and Technological Development (CNPq-Brazil) within the projects 311429/2020-3, 312899/2021-1 and 403167/2022-1.

Footnotes

Observe that this concept encompasses the ordinary notion of sublattices as a special case. This occurs when s is the identity function.

A⁽ⁿ⁾ (x) is as in Eq. (2).

Observe that, once aggregations functions are increasing, if a is a zero divisor of an aggregation on X, then all b ∈ X with b ≤ a is also a zero divisor.

A is known as Łukasiewicz t-norm.

A non-unit element m of a lattice L is meet-irreducible if, for all x, y ∈ L, m = x ∧ y implies m = x or m = y

Um elemento q ∈ L is said to be co-prime iff x ∨ y ≥ q implies x ≥ q or y ≥ q.

For example, fixed ξ ∈ L, the function $N_{ξ} (x) = {\begin{matrix} 1, & if x = 0_{L}, \\ 0, & if x = 1_{L}, \\ ξ, & otherwise, \end{matrix}$ is a frontier fuzzy negation such that N (ξ) = ξ.

A function T : L² → L is said to be a t-norm if it is commutative, associative, increasing in each argument and 1_L is its identity element.

A function S : L² → L is said to be a t-conorm if it is commutative, associative, increasing in each argument and 0_L is its identity element.

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