Constructions for t-conorms and t-norms on interval-valued and interval-valued intuitionistic fuzzy sets by paving

Paving is a method for constructing new operations from a given one. Kalina and Kral in 2015 showed that on the real unit interval this method can be used to construct associative, commutative and monotone operations from particular given operations (from basic ‘paving stones’). In the present paper we modify the construction method for interval-valued fuzzy sets. From given (possibly representable) t-norms and t-conorms we construct new, non-representable operations. In the last section, we modify the presented construction method for interval-valued intuitionistic fuzzy sets.


Introduction
Intuitionistuc fuzzy sets are a special type of lattice-valued fuzzy sets, introduced by Goguen [8]. Important mile-stones in the theory of intuitionistic fuzzy sets, besides the monograph by Atanassov [1], are the papers by Atanassov and Gargov [3], and Deschrijver et al. [7]. In [3], Atanassov and Gargov have shown that the theory of intuitionistic fuzzy sets is equivalent to the theory of interval-valued fuzzy sets. In [6,7] Deschrijver et al. have shown that there exist t-norms which are not representable as a pair of a t-norm and a t-conorm. 1
2.2 Associative, commutative and monotone operations on [0, 1] and on L * Associative operations such as t-norms, t-conorms, or uninorms as their common generalization, play an important role in fuzzy logic, in decision making, fuzzy control, and so on. Among these operations, those which are strictly increasing on ]0, 1[ 2 , play an important role because of their cancelativity. A function N : [0, 1] → [0, 1] is said to be a negation if N is decreasing and N (0) = 1, N (1) = 0.
A negation N is said to be strong if it is involutive, i.e., if N (N (x)) = x for all x ∈ [0, 1].
Remark 1. Note that, for a strong negation N , the N -dual operation to a t-norm T defined by S(x, y) = N (T (N (x), N (y))) is called a t-conorm. For more information, see, e.g., [13].

Further, for increasing functions
where sup ∅ = a, inf ∅ = b.
For more details on t-norms and t-conorms we recommend the monographs [13,15]. For other types of aggregation functions we recommend the monograph [9].

Paving as construction method for operations on [0, 1]
The idea of paving (see [11], for further modifications [5] and [16]) is the following. We split the unit interval into countably many disjoint subintervals {I i } i∈I (in such a way we split the unit square into countably many disjoint sub-rectangles I i × I j ). Then we take an operation * : [0, 1] 2 → [0, 1], choose transformations f i : I i → [0, 1] and we 'pave' the whole unit square (see Fig. 1 for a graphical schema of paving) by i+j is a kind of inverse (could be pseudo-inverse or a quasi-inverse). Our intention is to construct an operation : [0, 1] 2 → [0, 1] which is (not necessarily strictly) increasing in both variables, commutative and associative. We split the unit interval [0, 1] into countably many sub-intervals by choosing a sequence of their end-points. For a technical reason we will need the set of indices to be closed under addition (or by any other associative, commutative and monotone operation). The set of indices will be denoted by J . We set 0 = a 0 < a 1 < a 2 < · · · < a n < · · · < a ∞ = 1 . We assume f i (a i−1 ) = 0 and f i (a i ) = 1. Moreover, we will assume continuity of all functions f i .
Because of uniqueness of the assignment we consider semi-open intervals. We introduce the following notation: The basic idea of paving is hidden in the following two formulae which will be used alternatively. We choose an operation * : [0, 1] 2 → [0, 1] which is isotone in both variables, commutative and associative. Then, for i, j ∈ J , x ∈ I < i and y ∈ I < j (x ∈ I > i and y ∈ I > j , alternatively) we define and Formulae (8) and (9) directly imply that both operations, * ∨ and * ∧ , are isotone and commutative (when properly defined on the border). We know already that, for continuous for all x ∈ I < i and y ∈ I < j , and the same condition has to be fulfilled also for the operation * ∧ (and analogically we could treat the situation with right-closed intervals). The reader can find more details in [11].

Modification of paving for IVF-sets
When considering IVF-sets instead of the unit real interval (or a chain, in general) we have to overcome a new technical problem. Namely, on IVF-sets there are incomparable elements. We must pay attention to keeping monotonicity. We split the triangular area of L * as indicated in Fig. 2 into infinitely many rectangular and infinitely many triangular areas a 1 a 2 a 3 a n 1 Figure 2. Graphical schema of splitting L * into rectangular and triangular areas To preserve uniqueness of the partition of L * , we have two possibilities concerning the edges of rectangular and triangular areas: 1. We remove the upper and right edges from all rectangular areas and the upper edge from all triangular areas, these types of areas will be denoted I < (i,j) , i.e., for rectangular areas We remove the the lower and left edges from all rectangular areas and the left edge from all triangular areas, these types of areas will be denoted I > (i,j) , i.e., for recangular areas Now, we choose transforms of the rectangular and triangular areas into L * . We set transforms (i,j) : I (i,j) → L * , i < j, for the rectangular, and δ i : I (i.i) → L * for the triangular areas. Now, we define transforms for I (i,j) including all edges.
Alternatively, instead of the transforms (i,j) we can use¯ (i,j) : What happens to the coordinates of x = (x 1 , x 2 ) when using particular transforms, is the following: 1. We shift the area so that the left down vertex (a i , a j ) moved to (0, 0), we get I 1 (i,j) 2. We extend I 1 (i,j) to have (a i , a j+1 ) → (0, 1), (a i+1 , a j+1 ) → (1, 1). Now, rectangular areas are transformed to [0, 1] 2 (denoted by I 2 (i,j) ) and triangular to L * .
3. When using (i,j) , we transform the second coordinate in I 2 (i,j) so that the cut with the first coordinatex 1 is shrinked to 1 −x 1 , and then is moved upwardsx 1 . When using¯ (i,j) , we transfor the first coordinate in I 2 (i,j) so that the cut with the second coordinatex 2 is shrinked tox 2 .
Hence, we get that, when using transforms (i,j) , the right edges of rectangular areas are transformed to (1, 1), when using transforms¯ (i,j) , the lower edges are transformed to (0, 0). All other points are in one-one correspondence.
This reasoning leads to the conclusion that, for I < (i,j) transforms (i,j) are suitable and for I > transforms¯ (i,j) are suitable.

Construction of new t-conorms from a given one
In this part we describe in details the construction of a t-conorm by paving for the two mentioned possibilities, namely, using I < (i,j) and I > (i,j) areas.
The only case we have to consider is when x ∈ I < (0,0) and y ∈ I < (0, ) , or y ∈ I < (0,0) and x ∈ I < (0,j) . In all other cases the monotonicity is given by the choice of the triangular area I < ( x,y , x,y ) into which the result is transformed.
Let x ∈ I < (0,0) and y ∈ I < (0, ) . For keeping the monotonicity, the second coordinates are important. We have δ −1 (z 1 , z 2 ) = ((a +1 − a )z 1 + a , (a +1 − a )z 2 + a ). Then we get the following lower bound for the second coordinate of S 1 To prove the associativity, we have S 1 (x, S 1 (y, z)) = S 1 (S 1 (x, y), z) = 1 if and only if max(x 2 , y 2 , z 2 ) = 1. If max(x 2 , y 2 , z 2 ) < 1 and all x, y, z, are different form 0, we have S 1 (x, y) ∈ I < ( x,y , x,y ) for x ∈ I < (i,j) and y ∈ I < (k, ) and in that case, when computing S 1 (S 1 (x, y), z), δ −1 x,y is cancelled. This implies that the associativity of S 1 is a concequence of the associativity of S A new t-conorm can be constructed for a given one also when using areas of the type I > i,j) . Proposition 2. Let S : L 2 * → L * be an arbitrary t-conorm. Denote x,y = i + j + k + for (x, y) ∈ I (i−j) × I (k, ) and Then the following function is a t-conorm x ∨ y if x ∧ y ≤ (0, 1) and x ∨ y ≥ (0, 1), Proof. We give just an idea of the proof since it follows a similar idea as that of Proposition 1.
For the monotonicity just realize that, when x ∈ I > (0,0) and y ∈ I > (0, ) , the second coordinate of δ −1 is just the inverse transformation for the second coordinate of¯ (0, ) . In all other cases the monotonicity is directly due to the construction of S 2 .
For the associativity we can just re-write the proof of Proposition 1.
For a better understanding how the paving works, we sketch Fig. 3. The areas with equal shade of gray have values from the same triangular area. The darker the shade, the larger the values. This means, the values increase diagonally.

Other possibilities for construction of new t-conorms
The t-conorms constructed in Propositions 1 and 2 and also their modifications mentioned in Remark 4 have been constructed from some t-conorms S or t-super-conormsS, respectively (mentioned in Remark 4). These operations have one important property, namely S(x, y) ≥ x∨y and alsoS(x, y) ≥ x ∨ y. All other binary functions are either below or incomparable with the lattice disjunction. For this reason, for the next constructions of t-conorms, we change the enumeration of the splitting L * into {I (i,j) } i∈I,j∈J for the index-sets We will refer to this enumeration as of type (B).
Proposition 3. Let V : L 2 * → L * be an arbitrary associative, commutative and increasing binary function. Let us have families of areas I < = {I < (i,j) } i∈I,j∈J and I > = {I > (i,j) } i∈I,j∈J and the enumeration of type (B). Denote Θ (i.j) and Ψ (i,j) the transforms of I (i,j) defined by formulae (13) and (15), respectively. Denote x,y = i + j + k + for (x, y) ∈ I (i,j) × I (k, ) . Assume V be such that V(x, y) = (z 1 , z 2 ) with z 2 = 1 whenever x 2 = 1 and y 2 = 1. Then the following function is a t-conorm Assume V be such that V(x, y) = (z 1 , z 2 ) with z 1 = 0 whenever x 1 = 0 and y 1 = 0. Then the following function is a t-conorm x ∨ y if x ∧ y ≤ (0, 1) and x ∨ y ≥ (0, 1), Proof. The constructions of S 3 and S 4 are modifications of S 1 and S 2 , introduced by (14) and (16), respectively. Just in the cases of S 3 and S 4 we use a general binary increasing associative and commutative function V. This means, we have to check the monotonicity of the resulting functions. Since by the enumeration of type (B) we have x,y > max(i, j, k, ) and moreover, the monotonicity is guaranteed.

Some modifications of paving
Other possibilities for constructing new t-conorms we get when splitting the lattice L * into finitely many rectangular and triangular areas {I (i,j) } i∈I,j∈J . This gives enumerations of types (C) and In these cases, formulae (14), (16), (17) and (18) remain valid with one change, namely, if x,y ≥ n in expressions δ −1 x,y = 1. Another possibility to modify the constructions is to change addition for computing the new index in x,y for another operation ⊗ that is strictly increasing, associative and commutative, e.g., for multiplication. When using ⊗, we have to modify the index-set J (see [5,16] such that j 1 ⊗ j 2 ∈ J for all j 1 , j 2 ∈ J .
When constructing t-norms, we can just use the duality between t-norms and t-conorms.

A note on the transform of the results into interval-valued IF-sets
All the results achieved in this paper for IVF-sets are directly transformable into IF-sets just starting with an IF-operation and changing the enumeration of splitting [0, 1] in the second coordinate into 1 = a 0 > a 1 > a 2 > · · · > a n > · · · > a ∞ = 0.
Interval-valued intuitionistic fuzzy sets (IVIF-sets) were introduced in [3] and are well described in the recent monograph [2]. IVIF-sets are a generalization of the IF-sets. This means, they are also a special kind of lattice-valued fuzzy sets [8]. The membership grades of IVIF-sets are given by quadruples (x 1 , x 2 , x 3 , x 4 ) ∈ [0, 1] 4 such that The interval [x 1 , x 2 ] represents the membership of an element ξ of a universe and the interval [x 3 , x 4 ] represents the non-membership of ξ. The set of all membership grades of IVIF-sets will be denoted by L [ * ] . The order of elements in L [ * ] is given by There are several possibilities how to adopt the construction for t-conorms (t-norms) by paving on IVIF-sets. We show now one of the constructions for t-conorms.
The proof of Proposition 4 is just a modification of the proof of Proposition 1. That is why it is omitted.

Conclusions
We discussed some construction possibilities for t-conorms by paving. Particularly, we provided explicitly four formulea showing the construction of new t-conorms from given increasing associative and commutative operations. Finally, we gave some modifications of these construction possibilities. All the constructed t-conorms are non-representable in the sence of [6,7]. New t-norms can be constructed just by their duality with t-conorms. In Section 4, first, we made a comment on how the results for interval-valued fuzzy sets can be transformed to intuitionistic fuzzy sets. Finally, we extendeded the results to the case of interval-valued intuitionistic fuzzy sets where, in Proposition 4, we constructed a t-conorm by paving.