In fuzzy logic, the problem of extending functions can be considered for lattice-valued fuzzy connectives (t-norms, t-conorms, negations, and others) since these connectives are functions, in particular. The pioneer work in this framework was put forward by Saminger-Platz et al. in [
1] which provides a method to extend a t-norm
T from a complete sublattice
M to a bounded lattice
L. Later, we have developed in [
2] an extension method to extend t-norms, t-conorms, and fuzzy negations that generalizes the method proposed in [
1] considering a modified notion of sublattice. Also, we have applied this method for fuzzy implications in [
3].
The class of QL-implications is the generalization for fuzzy logic of the implications of quantum logic which raised from the Garrett Birkhoff and John von Neumann conclusion that “propositional calculus of quantum mechanics has the same structure as an abstract projective geometry.” It opened the way for the development algebraic logic that have much weaker properties than Boolean algebras. Another interesting fact is that projective geometry is a non-distributive modular lattice.
In this work, we apply the extension method developed in [
2] for QL-implications. To do so, we recall some elementary concepts related to lattice theory in Section “
Background and literature review.” The extension method via retractions is presented in Section “
Research design and methodology,” for t-norms, t-conorms, fuzzy negations, and implications. Section “
Methods” is devoted to present the main results of this paper, namely the results concerning to the extension of QL-implications.
This is an extension of one of the best papers awarded in WEIT 2013 invited to be published at the Journal of the Brazilian Computer Society (JBCS).
Background and literature review
Lattice-valued fuzzy logic and related theories have been studied by many researchers since lattice provides a very good scenery for the real world issues. For example, in mathematical morphology, lattice appeals to integral geometry, stereolgy, and random set models; it is mainly its algebraic facet which has become popular. There are also many other applications for lattice in image processing. So it is essential to have a very consistent mathematical theory in order to provide a safe framework to deal with those issues (see [
4,
5]).
In this paper, we rise up a discussion on the lattice-valued QL-implications and its algebraic extension as a function. To do so, in which follows, we provide a review on some important definitions and results.
Bounded lattices: definition and related concepts
We consider here the algebraic notion of lattices the reasons for this choice will be clear from the context. But a discussion about the other approach to lattices (i.e., as posets) can be found in [
6‐
8].
If in 〈L,∧
L
,∨
L
〉 there are elements 0
L
and 1
L
such that, for all x∈L, x∧
L
1
L
=x and x∨
L
0
L
=x, then 〈L,∧
L
,∨
L
,0
L
,1
L
〉 is called a bounded lattice. Moreover, it is known that, given a lattice L, the relation x≤
L
y if and only if x∧
L
y=x defines a partial order on L. Recall also that a lattice L is called a complete lattice if every subset of it has a supremum and an infimum element1.
From now on, lattice homomorphisms will be called just homomorphisms for simplicity.
Let f:L
n
→L be a conjugate of g:L
n
→L. If f(x
1,…,x
n
)≤
L
g(x
1,…,x
n
) for each x
1,…,x
n
∈L then we denote it by f≤g.
Retracts and sublattices
In general, given a bounded lattice L and a nonempty subset M⊆L, it is said that M is a sublattice of L if, for all x,y∈M, the following conditions hold: x∧
L
y∈M and x∨
L
y∈M. In other words, M equipped with the restriction of the operations ∧
L
and ∨
L
inherits the lattice structure of L.
We would like to work in a generalized notion of sublattice in which the condition M⊆L is somewhat weakened.
The purpose of defining (r,s)-sublattices as done in Definition 5 is to provide a relaxed notion of this concept. It is done an identification of M with a subset K=s(M) of L in order to carry on some properties of M to K, including its lattice structure via retraction r. In this case, K works as an algebraic copy of M embedded into L since r is a homomorphism.
The main advantage behind the idea of using this relaxed version of sublattice is that it allows us to verify the validity for L of a property which is invariant under homomorphisms from a lattice M without requiring M be a subset of L.
Notice that both in Definitions 5 and 6, the pseudo-inverse s of a retraction r cannot be unique. This is an advantage of our notion of sublattice since if there exist more than one pseudo-inverse for the same retraction, it is possible to identify M with a subset of L in different ways what give us the possibility to choose the best one for our proposes. But we must be clear that when we say that M is a (lower, upper or neither) (r,s)-sublattice of L, we are considering the existence of at least one pseudo-inverse s and fixing it. No matter which pseudo-inverse is taken, every result presented here remains working.
It is worth noting that if M is a (r,s)-sublattice of L then there is a retraction r from L onto M, but it is not required to r to be a lower or an upper retraction. Nevertheless, as shown in the Remark above, there may be more than one retraction from L onto M with the same pseudo-inverse. This is a very useful particularity of Definition 5 and we would like to highlight it in a definition.
An immediate consequence of the definition of lower (upper) retraction is that if \(M \unlhd L\) then it follows that s∘r
1≤i
d
L
≤s∘r
2.
Fuzzy connectives
In which follows, we define some well-known interpretation of the classical connectives in lattice-valued fuzzy logic [
12
–
14].
Dually, it is possible to define the concept of t-conorms.
Moreover, the negation N is strong if it also satisfies the involution property, namely (N3) N(N(x))=x, for all x∈L
In case N satisfies (N4) N(x)∈{0
L
,1
L
} if and only if x=0
L
or x=1
L
,
it is called frontier. In addition, every element x∈L such that N(x)=x is said to be an equilibrium point of N.
From the point of view of lattice theory, a strong negation corresponds to what is known as involution (see [
6]).
Similarly, we can define a natural negation of a t-conorm S as follows.
Finally, we present the notion of fuzzy implication. There are some different interpretations of this fuzzy operator in the literature (see [
15‐
20]) since there is no consensus on the way to define it just that fuzzy implication have to behavior at least as in the crisp case. Here, we consider the notion presented in [
15] because we believe such a definition has the properties necessary for a fuzzy implication.
Consider also the following properties of an implication
I on
L:
(LB)
I(0
L
,y)=1
L
, for all y∈L
(RB)
I(x,1
L
)=1
L
, for all x∈L
(NP)
I(1
L
,y)=y for each y∈L (left neutrality principle)
(L-NP)
I(1
L
,y)≤
L
y for each y∈L
(EP)
I(x,I(y,z))=I(y,I(x,z)) for all x,y,z∈L (exchange principle)
(IP)
I(x,x)=1
L
for each x∈L (identity principle)
(OP)
I(x,y)=1
L
if and only if x≤y (ordering property)
(IBL)
I(x,I(x,y))=I(x,y) for all x,y,z∈L (iterative Boolean law)
(CP)
I(x,y)=I(N(y),N(x)) for each x,y∈L with N a fuzzy negation on L (law of contraposition)
(L-CP)
I(N(x),y)=I(N(y),x) (law of left contraposition)
(R-CP)
I(x,N(y))=I(y,N(x)) (law of right contraposition)
(P)
I(x,y)=0
L
if and only if x=1
L
and y=0
L
(Positivety)
(LEM)
S(N(x),x)=1
L
for each x∈L (law of excluded middle)
Note that, a special class of fuzzy implication can be naturally obtained by generalizing the implication operator from the quantum logic, namely p→q⇔¬p∨(p∧q). For bounded lattices, this implication is given as follows.
then the function
$$I_{T,S,N}(x,y) = \left\{ \begin{array}{rl} 1, & y=1;\\ y, & x=1;\\ N(x), & \text{otherwise}. \end{array} \right. $$
is not always a fuzzy implication, even if
S and
N satisfy (LEM).