New construction of an ordinal sum of t-norms and t-conorms on bounded lattices

New construction of an ordinal sum of t-norms and t-conorms on bounded lattices

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New construction of an ordinal sum of t-norms and t-conorms on bounded lattices ´ Michal Holˇcapek Anton´ın Dvoˇrak, PII: DOI: Reference:

S0020-0255(19)31111-9 https://doi.org/10.1016/j.ins.2019.12.003 INS 15046

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Information Sciences

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8 November 2018 26 November 2019 2 December 2019

´ Please cite this article as: Anton´ın Dvoˇrak, Michal Holˇcapek, New construction of an ordinal sum of t-norms and t-conorms on bounded lattices, Information Sciences (2019), doi: https://doi.org/10.1016/j.ins.2019.12.003

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New construction of an ordinal sum of t-norms and t-conorms on bounded lattices✩ Anton´ın Dvoˇr´ ak, Michal Holˇcapek∗ Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, CE IT4Innovations, 30. dubna 22, 701 03 Ostrava, Czech Republic

Abstract This paper introduces a new ordinal sum construction of t-norms and t-conorms on bounded lattices based on interior and closure operators. The proposed method generalizes several known constructions and provides a simple tool to introduce new classes of t-norms and t-conorms. Keywords: t-norms, t-conorms, bounded lattices, ordinal sum

1. Introduction The operations of t-norms and t-conorms on the unit interval, originally introduced in the context of probabilistic metric spaces [18, 27], belong among the most popular operations used in the definitions of operations for a management of uncertainty. In fuzzy set theory and fuzzy logic, they serve as natural interpretations of logical connectives, namely of conjunction and disjunction, respectively.1 Because fuzzy logic gradually started to use more general structures of truth values than the real unit interval, following the seminal work of Goguen [12], and these structures fall under the concept of bounded lattices, it was quite natural to begin to study t-norms and t-conorms on bounded lattices [1, 9, 28].2 Ordinal sums are one of the most important constructions used in the theory of t-norms on the unit interval. For example, based on the famous MostertShields theorem [22], it can be proved that each continuous t-norm on the unit ✩ This research was partially supported by the ERDF/ESF project AI-Met4AI No. CZ.02.1.01/0.0/0.0/17 049/0008414. The additional support was also provided by the Czech Science Foundation through the project of No.18-06915S. ∗ Corresponding author. Tel.: +420 59 709 1406; fax: +420 596 120 478. Email addresses: [email protected] (Anton´ın Dvoˇr´ ak), [email protected] (Michal Holˇ capek) 1 Naturally, the role of t-norms and t-conorms is quite important in various applications of fuzzy logic, e.g., in multicriteria decision-making, fuzzy control, image processing, etc. 2 Since t-norms and t-conorms are dual concepts, to achieve better readability, the presentation in the following paragraphs is restricted to the case of t-norms.

Preprint submitted to Elsevier

December 3, 2019

interval can be represented as an ordinal sum of continuous Archimedean tnorms. Consequently, since any continuous Archimedean t-norm is isomorphic either to Lukasiewicz or to product t-norm, this result shows that each continuous t-norm on the unit interval is isomorphic to an ordinal sum of Lukasiewicz and product t-norms. (see [18, Chapter 5.3]). In addition, t-norms are often constructed by special unary functions called generators (usually additive or multiplicative). Various parametric constructions (Hamacher, Frank or SchweizerSklar families of t-norms) are also popular. Ordinal sums of t-norms on bounded lattices have been studied intensively in recent years. In [25], the definition of an ordinal sum of t-norms defined on a family of pair-wise disjoint subintervals of a bounded lattice is provided, and it is shown that this ordinal sum need not be a t-norm, in general. Conditions, under which the ordinal sum of any family of t-norms defined in this way is always a t-norm, are provided. In [20], a new characterization is given, concentrating on conditions that make possible to decide whether a specific ordinal sum, with respect to a particular family of t-norms, is a t-norm. In [26], it is investigated under which conditions the ordinal sum defined analogously as in [25], but this time on a sublattice of a lattice (this sublattice need not be an interval), can be extended to this lattice, and minimal and maximal such extensions are studied. Authors of [11] provide a modified definition of the ordinal sum from [25] and prove that this modified ordinal sum is always a t-norm. Further modifications are proposed in recent papers [3, 7] (see Examples 3.3-3.5). The main aim of this paper is to propose a new and more general definition of an ordinal sum of t-norms on bounded lattices. We show that several ordinal sum constructions already proposed in the literature, namely [3, 7, 11], are special cases of our new construction. We use the concept of lattice interior operator [24], which allows us to pick a sublattice (or sublattices) of a given bounded lattice appropriate for our construction. Using interior operators, we can define new classes of t-norms in a simple and, in our opinion, elegant way. In addition, the proposed approach can be used to simplify the proofs of previously established or potential newly proposed constructions. These proofs are typically designed as a long list of simple case-by-case verifications. Although our construction is quite general, it naturally cannot cover all possible t-norms on any bounded lattice (see Example 3.7). However, using various interior operators, infinitely many new classes of t-norms can be defined. Note that t-norms and t-conorms on the unit interval are examples of a broad class of aggregation operators [13]. A large class of aggregation operators (uninorms and nullnorms) can be constructed as combinations of t-norms and t-conorms. These combinations are related to ordinal sums [18, Chapter 10]. Ordinal sums are one of important construction methods also for other classes of aggregation operators, e.g., copulas [13, Chapter 6.7]. Aggregation operators on bounded lattices have been studied, too [10, 17, 19]. The paper is structured as follows. In Section 2, we recall notions of a bounded lattice, t-norms and t-conorms on bounded lattices, and we introduce interior operators on lattices. These operators are essential for our ordinal sum construction. Section 3 contains the main results: new constructions of t-norms 2

on bounded lattices using interior operators. First, we show that, if M is an image of a lattice L under an interior operator h, then we can, using h, extend a t-norm defined on M to a t-norm on L. Then we introduce a new type of ordinal sum of t-norms defined on certain subsets of a lattice L, again by means of an interior operator h, and call it a h-ordinal sum. We also show that an h-ordinal sum is, in some sense, uniquely determined by the interior operator h. Finally, let L be a bounded lattice and T be a t-norm on it. Suppose that a chain of intervals can be defined on L such that restrictions of T on these subintervals are t-norms. We prove that, under natural conditions, T is an h-ordinal sum of these t-norms for a suitable interior operator h. operator h. Throughout this section, constructions of t-norms are illustrated by examples. Several of these examples show that several published constructions of ordinal sums of t-norms on bounded lattices are special cases of our h-ordinal sum. In Section 4, closure operators on lattices are introduced, and dual results for ordinal sums of t-conorms as these for t-norms in the previous section are summarized. Section 5 contains conclusions and directions for further research. 2. Preliminaries We say that a lattice (L, ∧, ∨) [14] with the corresponding lattice order ≤ is bounded [15], if there exist two elements 0L , 1L ∈ L such that for all x ∈ L it holds that 0L ≤ x ≤ 1L . We call 0L and 1L the bottom and the top element, respectively, and write this bounded lattice as (L, ∧, ∨, 0L , 1L ). For our investigation of the constructions of t-norms on bounded lattices, we recognize that it is sufficient to consider bounded meet semilattices with bottom (L, ∧, 0L , 1L ) only.3 Indeed, a general idea of an ordinal sum of t-norms is that the lattice meet is used for these arguments, for which the result of the ordinal sum cannot be determined by the original t-norms. The join operation is not used at all. Therefore, in what follows, we investigate the ordinal sums on bounded meet semilattices with bottom. Note that each bounded lattice is a bounded meet semilattice.4 Hence, the results presented in Section 3 for bounded meet semilattices remain valid for bounded lattices, too. A formal definition is as follows. Definition 2.1. Let (L, ∧, 1L ) be a bounded meet semilattice with the corresponding lattice order ≤. (L, ∧, 1L ) is a bounded meet semilattice with bottom if there exists an element 0L ∈ L such that for all x ∈ L it holds that 0L ≤L x. This element 0L is called the bottom element. A bounded meet semilattice (L, ∧, 1L ) with a bottom element 0L is written as (L, ∧, 0L , 1L ). 3 Note that a bounded meet semilattice is standardly defined as a structure (L, ∧, 1 ), L where 1L is the top element. However, a bounded meet semilattice need not have a bottom element. 4 It is well known that if a bounded meet semilattice is complete, then it becomes a bounded lattice.

3

All meet semilattices considered in this paper will be bounded meet semilattices with bottom, hence, to save space, when we write “meet semilattice”, we actually have in mind a bounded meet semilattice with bottom.5 By ≤ we denote the lattice order of a (last-mentioned) lattice or semilattice. Recall that a sublattice of the lattice L is a non-empty subset of L that is a lattice with the same meet and join operations as L. Similarly, one can define a sub-semilattice of a semilattice. Now we define intervals on bounded lattices. Definitions of intervals on meet (join) semilattices are analogous. Definition 2.2. Let (L, ∧, ∨, 0L , 1L ) be a bounded lattice. Let a, b ∈ L be such that a ≤ b. The closed subinterval [a, b] of L is a sublattice of L defined as [a, b] = {x ∈ L | a ≤ x ≤ b}. Similarly, open subinterval (a, b) of L is defined as (a, b) = {x ∈ L | a < x < b}. Definitions of semi-open subintervals (a, b] and [a, b) are obvious. Next, let us provide the definition of an interior operator on a meet semilattice. Definition 2.3. Let L be a meet semilattice. A map h : L → L is said to be an interior operator on L if, for all x, y ∈ L, (i) h(1L ) = 1L , (ii) h(h(x)) = h(x), (iii) h(x ∧ y) = h(x) ∧ h(y), (iv) x ≥ h(x). Obviously, each interior operator is a meet-homomorphism of a meet semilattice to itself (i.e., a meet-endomorphism) and preserves lattice order. That is, h(x) ≤ h(y) whenever x ≤ y for all x, y ∈ L, which can be simply derived using (iii) from Definition 2.3. It is easy to see that the identity map idL is an interior operator on L. The following three examples show the interior operators that will be used later in the constructions of t-norms. Example 2.1. Let L be a meet semilattice, and let b ∈ L be arbitrary. Then, the map h∧ b : L → L defined by ( x, x ≥ b, ∧ hb (x) = (1) x ∧ b, otherwise, 5 Dual notion of a bounded join semilattice with top (L, ∨, 0 , 1 ) will be used for conL L structions of t-conorms in Section 4. Its definition is obvious. Again, all join semilattices considered in this paper will be bounded join semilattices with top.

4

for any x ∈ L, is an interior operator on L. Indeed, h∧ b (1L ) = 1L , since 1L ≥ b. ∧ ∧ Let x, y ∈ L. If x ≥ b, then h∧ b (hb (x)) = hb (x) = x, and if x 6≥ b, then ∧ ∧ ∧ h∧ b (hb (x)) = hb (x ∧ b) = x ∧ b ∧ b = x ∧ b = hb (x), since b > x ∧ b, i.e., x ∧ b 6≥ b. Further, assume that x ≥ b and y ≥ b. Since x ∧ y ≥ b, we obtain ∧ ∧ h∧ b (x ∧ y) = x ∧ y = hb (x) ∧ hb (y). Let x ≥ b and y 6≥ b. Since x ∧ y 6≥ b, ∧ ∧ hb (x ∧ y) = (x ∧ y) ∧ b = x ∧ (y ∧ b) = h∧ b (x) ∧ hb (y). The same equality can be obtained for x 6≥ b and y ≥ b. If x 6≥ b and y 6≥ b, then x ∧ y 6≥ b, and ∧ ∧ h∧ b (x ∧ y) = (x ∧ y) ∧ b = (x ∧ b) ∧ (y ∧ b) = hb (x) ∧ hb (y). Finally, if x ≥ b, ∧ ∧ then hb (x) = x, and if x 6≥ b, then x ≥ x ∧ b = hb (x). Note that h∧ b = idL for b = 0L or b = 1L . Let us note that if we replace x ≥ b by x > b in (1), we obtain the same interior operator as in the above example, which follows from the fact that h∧ b (b) = b holds for both these types of inequalities. Example 2.2. Let L be a meet semilattice, and let b ∈ L be arbitrary. Then, the map h0b : L → L defined by ( x, x ≥ b, 0 hb (x) = (2) 0L , otherwise, for any x ∈ L, is an interior operator on L. Similarly as in Example 2.1, one can easily check that the map h0b is an interior operator on L. The interior operator h∧ b introduced in Example 2.1 can be extended as follows. Example 2.3. Let L be a meet semilattice, and let a, b ∈ L be arbitrary such that a ≤ b. Then, the map h∧ a,b : L → L defined by h∧ a,b (x)

( x, = x ∧ a,

x ≥ b, otherwise,

(3)

for any x ∈ L, is an interior operator on L. Similarly to Examples 2.1 and 2.2, ∧ ∧ h∧ a,b (1L ) = 1L , since 1L ≥ b. Let x, y ∈ L. If x ≥ b, then ha,b (ha,b (x)) = ∧ ∧ ∧ ∧ ∧ ha,b (x) = x, and if x 6≥ b, then ha,b (ha,b (x)) = ha,b (a∧x) = a∧x = ha,b (x), since a∧x 6≥ b. Further, assume that x∧y ≥ b. Then, x ≥ b and y ≥ b, and we obtain ∧ ∧ h∧ a,b (x ∧ y) = x ∧ y = ha,b (x) ∧ ha,b (y). If x ∧ y 6≥ b, then x 6≥ b or y 6≥ b. If x 6≥ b ∧ ∧ and y ≥ b, we find that ha,b (x ∧ y) = (x ∧ y) ∧ a = (x ∧ a) ∧ y = h∧ a,b (x) ∧ ha,b (y). Analogously one can prove the same equality for x ≥ b and y 6≥ b and x 6≥ b and ∧ y 6≥ b. Finally, if x ≥ b, then h∧ a,b (x) = x, and if x 6≥ b, then ha,b (x) = x∧a ≤ x. ∧ ∧ ∧ Obviously, ha,b = hb for a = b, where hb is defined by (1). Definition 2.4. Let h, g : L → L be maps on a meet (join) semilattice L. We say that h and g commute on L provided that h ◦ g = g ◦ h. The following example shows that certain previously defined interior operators commute. 5

Example 2.4. Let L be a meet semilattice and let a, b ∈ L \ {0L , 1L } such that a > b. ∧ (a) h∧ a and hb defined by (1) commute on L, that is, it holds that ∧ ∧ ∧ h∧ a ◦ hb = hb ◦ ha .

∧ ∧ ∧ ∧ If x ≥ a, then x ≥ b and h∧ a (hb (x)) = x = hb (x) = hb (ha (x)). If x 6≥ a ∧ ∧ ∧ ∧ ∧ and x ≥ b, then ha (hb (x)) = ha (x) = a ∧ x = hb (ha (x)), where we used ∧ ∧ x ∧ a ≥ b. Finally, if x 6≥ a and x 6≥ b, then h∧ a (hb (x)) = ha (x ∧ b) = ∧ (x ∧ b) ∧ a = (x ∧ a) ∧ b = h∧ (h (x)), where we used the fact that x ∧ a 6≥ b a b and x ∧ b 6≥ a.6

(b) h0a and h0b defined by (2) commute on L, too. If x ≥ a, then x ≥ b and h0a (h0b (x)) = x = h0b (h0a (x)). If x 6≥ a and x ≥ b, then h0a (h0b (x)) = h0a (x) = 0L = h0b (0L ) = h0b (h0a (x)). Finally, if x 6≥ a and x 6≥ b, then h0a (h0b (x)) = h0a (0L ) = 0L = h0b (0L ) = h0b (h0a (x)). Lemma 2.1. Let h and g be interior operators on a meet semilattice L that commute on L. Then h ◦ g is an interior operator on L. Proof. Put f = h ◦ g. Trivially, f (1L ) = 1L . From the commutativity of h and g and the associativity of the operation of composition of maps, we find that f ◦ f = (h ◦ g) ◦ (h ◦ g) = (h ◦ g) ◦ (g ◦ h) = h ◦ (g ◦ g) ◦ h = h ◦ (g ◦ h) = h ◦ (h ◦ g) = (h ◦ h) ◦ g = h ◦ g = f. Further, if x, y ∈ L, then f (x ∧ y) = h(g(x ∧ y)) = h(g(x) ∧ g(y)) = h(g(x)) ∧ h(g(y)) = f (x) ∧ f (y). Finally, if x ∈ L, then f (x) = h(g(x)) ≤ g(x) ≤ x.

2

Theorem 2.2. Let h1 , . . . , hn be mutually commutative interior operators on L, i.e., hi ◦ hj = hj ◦ hi for i, j = 1, . . . , n. Then h = h1 ◦ · · · ◦ hn is an interior operator on L. Proof. By induction, if h = h1 , we obtain that h is an interior operator on L. Let g = h1 ◦ · · · ◦ hn−1 , and let us assume that g is an interior operator on L. We will prove that h = g ◦ hn is an interior operator on L. According to Lemma 2.1, it is sufficient to show that g and hn commute on L. From the associativity of the composition of maps and the assumption on the commutativity of any pair of interior operators hi and hj , for i, j = 1, . . . , n, we obtain g ◦ hn = (h1 ◦ · · · ◦ hn−1 ) ◦ hn = (h1 ◦ · · · ◦ hn−2 ) ◦ (hn−1 ◦ hn )

= (h1 ◦ · · · ◦ hn−2 ) ◦ (hn ◦ hn−1 ) = (h1 ◦ · · · ◦ hn−2 ◦ hn ) ◦ hn−1 = · · · = (h1 ◦ hn ) ◦ (h2 ◦ · · · ◦ hn−1 ) = hn ◦ (h1 ◦ · · · ◦ hn−1 ) = hn ◦ g.

6 Note that the last equation can be also proved using the fact that an interior operator is ∧ ∧ ∧ ∧ a meet homomorphism. Indeed, h∧ a (hb (x)) = ha (x ∧ b) = ha (x) ∧ ha (b) = (x ∧ a) ∧ (a ∧ b) = ∧ ∧ (x ∧ a) ∧ b = h∧ b (x ∧ a) = hb (ha (x)).

6

Hence, h = g ◦ hn is an interior operator on L.

2

The definition of a t-norm and a t-conorm on a bounded lattice is as follows (cf. [25, Definition 3.1]). Definitions of t-norms (t-conorms) on meet (join) semilattices are completely analogous. Definition 2.5. An operation T : L2 → L (S : L2 → L) on a bounded lattice (L, ∧, ∨, 0L , 1L ) is a t-norm (t-conorm) if it is commutative, associative, nondecreasing with respect to both variables and 1L (0L ) is its neutral element. Remark 2.1. As a simple consequence of the monotonicity of a t-norm (tconorm) and the fact that 1L (0L ) is the neutral element, we find that T (x, y) ≤ x ∧ y (S(x, y) ≥ x ∨ y) for any x, y ∈ L (see, e.g., [18, Remark 1.5(i)]). 3. Ordinal sums of t-norms on meet semilattices Ordinal sums are well-known constructions appearing in the theory of partially ordered sets (ordinal sums in the sense of Birkhoff [2]) and in the theory of semigroups (ordinal sums in the sense of Clifford [6]). In principle, we have a family of pairwise disjoint sets, where each of them is equipped either with an order (for posets) or with an associative operation (for semigroups). This family is indexed by a linearly ordered index set. The ordinal sum is then the union of these sets equipped either with an appropriate partial order (for posets) or an appropriate associative operation (for semigroups). Since t-norms on unit intervals can be understood as special semigroups, the ordinal sum construction can be applied for them, too. For more details on these ordinal sum constructions, see [25, Section 2]. The following theorem provides a construction of t-norms on meet semilattices with the help of an interior operator. Note that this construction forms a core of the ordinal sum of t-norms as will be demonstrated later. Theorem 3.1. Let L be a meet semilattice, and let h : L → L be an interior operator on L. Let M denote the image of L under h, i.e., h(L) = M . Then, (i) M is a meet sub-semilattice of L with the bottom element 0L and the top element 1L , (ii) if V is a t-norm on M , then there exists its extension to a t-norm T on L as follows: ( V (h(x), h(y)), x, y ∈ L \ {1L }, T (x, y) = x ∧ y, otherwise. .

Proof. To prove that M is a meet sub-semilattice of L, it is sufficient to show that if u, v ∈ M , then u ∧ v ∈ M , too. Because M = h(L), we know that u = h(x) and v = h(y) for some x, y ∈ L. Then, from (iii) of Definition 2.3 7

follows that u ∧ v = h(x) ∧ h(y) = h(x ∧ y), hence u ∧ v ∈ h(L) = M . From (i) of Definition 2.3 we have that h(1L ) = 1L , hence 1L ∈ M , and from (iv) of the same definition it follows that h(0L ) = 0L , hence 0L ∈ M . To prove the second statement, we verify that the map T satisfies the axioms of t-norms (Definition 2.5). The commutativity is trivial. Let x, y, z ∈ L be arbitrary. To prove the associativity of T , we can distinguish the following four cases: 1) x, y, z ∈ L \ {1L }, 2) x, y ∈ L \ {1L } and z = 1L , 3) x ∈ L \ {1L } and y = z = 1L , 4) x = y = z = 1L . ad 1) It is easy to see that h(x) = x for any x ∈ M and T (x, y) ∈ M for any x, y ∈ L \ {1L }. Indeed, if x ∈ M , then there exists y ∈ L such that h(y) = x. Hence, we obtain h(x) = h(h(y)) = h(y) = x. Moreover, T (x, y) = V (h(x), h(y)) ∈ M for any x, y ∈ L \ {1L }, which follows from the fact that V is a t-norm on M and h(x), h(y) ∈ M . Hence, using the associativity of V , T (x, T (y, z)) = V (h(x), h(T (y, z))) = V (h(x), T (y, z)) = V (h(x), V (h(y), h(z))) = V (V (h(x), h(y)), h(z)) = V (T (x, y), h(z)) = V (h(T (x, y)), h(z)) = T (T (x, y), z). ad 2) We have T (x, T (y, 1L )) = T (x, y ∧ 1L ) = T (x, y) ∧ 1L = T (T (x, y), 1L ). ad 3) We have T (x, T (1L , 1L )) = T (x, 1L ∧1L ) = T (x, 1L )∧1L = T (T (x, 1L ), 1L ). ad 4) Similarly as in case 3). To prove the monotonicity of T , it is sufficient to restrict ourselves to the monotonicity of T in the second argument and consider the previously formulated cases 1)-4) and one additional case: 5) x = 1L and y, z ∈ L \ {1L }. Let x, y, z ∈ L be such that y ≤ z. ad 1) Using the monotonicity of V and the fact that the interior operator h preserves the lattice order on L, we find that T (x, y) = V (h(x), h(y)) ≤ V (h(x), h(z)) = T (x, z). ad 2) Since V (a, b) ≤ a ∧ b holds for any a, b ∈ M , we find that T (x, y) = V (h(x), h(y)) ≤ h(x) ∧ h(y) ≤ h(x) ≤ x = x ∧ 1L = x ∧ z = T (x, z). ad 3) Trivially, T (x, y) = T (x, 1L ) = T (x, z). ad 4) Trivially, T (x, y) = T (1L , 1L ) = T (x, z). ad 5) We have T (x, y) = T (1L , y) = 1L ∧ y ≤ 1L ∧ z = T (1L , z) = T (x, z), and the proof of the monotonicity of T is finished. Finally, T (1L , x) = 1L ∧ x = x holds for any x ∈ L, and T is a t-norm on L. 2 8

The following theorem introduces the ordinal sum of t-norms on meet semilattices that can be partitioned into a chain of subintervals. Note that the claim of this theorem follows from Proposition 5.2 in [25]. We provide direct proof of this theorem, because, in our opinion, it can help the reader to understand better how t-norms on bounded semilattices work. Theorem 3.2. Let L be a meet semilattice, and let us assume Sn that there exist b1 , . . . , bn ∈ L \ {0L , 1L } such that b1 < · · · < bn and L = i=0 [bi , bi+1 ], where b0 = 0L and bn+1 = 1L . Then, (i) [bi , bi+1 ], i = 0, . . . , n, is a meet sub-semilattice of L, (ii) if Vi is a t-norm on [bi , bi+1 ] for i = 0, . . . , n, then the ordinal sum of t-norms V1 , . . . , Vn on L defined as follows: ( Vi (x, y), x, y ∈ [bi , bi+1 ), (4) T (x, y) = x ∧ y, otherwise, is a t-norm on L. Proof. (i) Let x, y ∈ [bi , bi+1 ] for a certain i ∈ {0, . . . , n}. Since bi ≤ x as well as bi ≤ y, we find that bi ≤ x ∧ y; therefore, x ∧ y ∈ [bi , bi+1 ]. (ii) The commutativity of T is obvious. Since the verification of the associativity of T for the case n = 1 can be obtained as a part of the proof for n ≥ 2, assume that n ≥ 2 and consider i, j, k ∈ {0, . . . , n} such that bi < bj < bk . Obviously, the associativity of T is satisfied, if it holds for the following five cases: 1) x, y, z ∈ [bi , bi+1 ), 2) x, y ∈ [bi , bi+1 ) and z ∈ [bj , bj+1 ), 3) x ∈ [bi , bi+1 ) and y, z ∈ [bj , bj+1 ), 4) x ∈ [bi , bi+1 ) and y ∈ [bj , bj+1 ) and z ∈ [bk , bk+1 ), 5) x = 1L or y = 1L or z = 1L . ad 1) The associativity of T immediately follows from the associativity of Vi . ad 2) We have T (x, T (y, z)) = T (x, y ∧ z) = T (x, y) = Vi (x, y) ∧ z = T (T (x, y), z), where we used Vi (x, y) < bi+1 ≤ z. ad 3) Since x < bi+1 ≤ Vj (y, z) ≤ y ∧ z and T (x, y) = x < bi+1 ≤ z, we obtain T (x, T (y, z)) = T (x, Vj (y, z)) = x ∧ Vj (y, z) = x

= (x ∧ y) ∧ z = T (x, y) ∧ z = T (T (x, y), z). 9

ad 4) Since x < y < z, we obtain T (x, T (y, z)) = T (x, y ∧ z) = T (x, y) = T (x, y) ∧ z = T (T (x, y), z). ad 5) Let x = 1L . Then, T (1L , T (y, z)) = 1L ∧ T (y, z) = T (y, z) = T (1L ∧ y, z) = T (T (1L , y), z). Similarly, one can prove the remaining possibilities in 5) and the proof of the associativity of T is finished. Let x, y, z ∈ L such that y ≤ z. To prove the monotonicity of T , it is sufficient to restrict ourselves to the monotonicity in the second argument and consider the cases 1)-5) formulated above. ad 1) The monotonicity of T follows from the monotonicity of Vi . ad 2) Since Vi (x, y) ≤ x∧y, we have T (x, y) = Vi (x, y) ≤ x∧y ≤ x∧z = T (x, z). ad 3) Similarly to 2), we have T (x, y) = x ∧ y ≤ x ∧ z = T (x, z). ad 4) We have T (x, y) = x ∧ y ≤ x ∧ z = T (x, z). ad 5) If x = 1L , then T (x, y) = 1L ∧ y ≤ 1L ∧ z = T (x, z). Let x 6= 1L . If y = 1L , then z = 1L and T (x, y) = x ∧ 1L = T (x, z). If y 6= 1L and z = 1L , then T (x, y) ≤ x ∧ y ≤ x ∧ 1L = T (x, z), where we used the fact that T (x, y) = V` (x, y) ≤ x ∧ y, if x, y ∈ [b` , b`+1 ), and T (x, y) = x ∧ y, if (x, y) ∈ [b` , b`+1 ) × [bm , bm+1 ) for `, m ∈ {0, . . . , n}, and the proof of the monotonicity is finished. Since 1L 6∈ [bn , 1L ), T (1L , x) = 1L ∧ x = x = x ∧ 1L = T (x, 1L ) holds for any x ∈ L. Hence, the ordinal sum T defines a t-norm on L. 2 According to the previous theorem, one can introduce a t-norm on L by the ordinal sum of t-norms on a meet semilattice L = [b0 , b1 ] ∪ · · · ∪ [bn , bn+1 ], where 0L = b0 < b1 < · · · < bn < bn+1 = 1L . In what follows, we extend this construction to a more general meet semilattices, where [b0 , b1 ]∪· · ·∪[bn , bn+1 ] ⊂ L. Theorem 3.3. Let L be a meet semilattice, and let us assume Sn that there exist b1 , . . . , bn ∈ L \ {0L , 1L } such that b1 < · · · < bn and M = i=0 [bi , bi+1 ] ⊂ L, where b0 = 0L and bn+1 = 1L . If h is an interior operator on L such that h(L) ⊆ M , bi is a fixed point of h and Vi is a t-norm on Ji+ = h(L) ∩ [bi , bi+1 ] for i = 0, . . . , n, then  2  Vi (h(x), h(y)), (h(x), h(y)) ∈ Ji , T (x, y) = h(x) ∧ h(y), (5) (h(x), h(y)) ∈ Ji × Jj , for i 6= j,   x ∧ y, otherwise, for any x, y ∈ L, where Ji = h(L) ∩ [bi , bi+1 ), is a t-norm on L, which is called the h-ordinal sum of t-norms V0 , . . . , Vn on L.

Proof. Denote J = h(L). By the assumption, we have J ⊂ L. Since h is a meet-endomorphism, we find that J is a meet sub-semilattice of L with the 10

1L L Jn

x

bn Jn−1

h(x)

bn−1 Jn−2 T (x, y) = h(x) ∧ h(y)

bn−2 y b1 h(y) J0 0L

Figure 1: A general construction of the h-ordinal sum of t-norms V0 , . . . , Vn defined on the sub-semilattices J0 , . . . , Jn , respectively. Ovals (e.g., Jn−2 ) indicate sublattices of L. Curved lines with arrows (e.g., from y to h(y)) connect an element of L and its image under h. The curved line between h(x) and h(y) connects two elements used for the determination of T (x, y).

bottom element 0L and the top element 1L , where Ji+ = {x ∈ h(L) | bi ≤ x ≤ bi+1 } is a closed subinterval of J whose boundaries are bi and bi+1 (i.e., Ji+ = [bi , bi+1 ] in the meet semilattice J) for i = 0, . . . , n. SinceS b0 , . . . , bn+1 ∈ J n such that 0L = b0 < b1 < · · · < bn < bn+1 = 1L , J = i=0 Ji+ , where Ji+ = [bi , bi+1 ] in J, and Vi is a t-norm on Ji+ for i = 0, . . . , n, by Theorem 3.2, one can introduce the ordinal sum V of t-norms V0 , . . . , Vn on J as ( Vi (x, y), (x, y) ∈ Ji2 , V (x, y) = (6) x ∧ y, otherwise, for any x, y ∈ J. By a comparison of the formulas in (5) and (6), one can find that the map T can be expressed as ( V (h(x), h(y)), (x, y) ∈ L \ {1L }, T (x, y) = (7) x ∧ y, otherwise. Since V is a t-norm on h(L), it follows, using Theorem 3.1, that the map T is a t-norm on L. 2 In what follows, if we say that T is an h-ordinal sum of t-norms V0 , . . . , Vn on L, we assume that the assumptions of Theorem 3.3 are satisfied for certain 11

b1 , . . . , bn ∈ L \ {0L , 1L } such that b1 < · · · < bn and T is a t-norm constructed by formula (5). The previous theorem provides a general construction of a t-norm on a meet semilattice L as an ordinal sum of t-norms defined on closed subintervals of meet sub-semilattices from L with the help of an interior operator. This general construction of an ordinal sum of t-norms based on an interior operator h is demonstrated in Fig. 1. One can see that to determine T (x, y), first, the elements x and y are sent to sub-semilattices Jn−1 and J0 , respectively, by the interior operator h, and then the meet operation is applied to these images h(x) and h(y) (the second line of (5)), because the sub-semilattice Jn−1 is different from the sub-semilattice J0 . If images of x as well as of y belong to the same subsemilattice Ji , then the t-norm Vi defined on Ji is applied instead of the meet (the first line of (5)). It should be noted that in the assumptions of Theorem 3.3 we require that any bi (i = 1, . . . , n) has to be a fixed point of the interior operator h to guarantee the following condition: T (bi , bj ) = bi ∧ bj ,

i, j = 0, . . . , n + 1,

(8)

which seems to be quite natural for ordinal sum constructions of t-norms defined on meet sub-semilattices of L. An extension of Theorem 3.3, where (8) need not be satisfied, is likely possible. A detailed analysis of this issue will be a topic of our future research. Remark 3.1. Recall that a map T : L2 → L is called a t-subnorm, if T is a commutative, associative, non-decreasing with respect to both variables and T (x, y) ≤ x ∧ y for any x, y ∈ L. Obviously, each t-norm is a t-subnorm, but not vice-versa, because for t-subnorms T (x, 1L ) ≤ x only holds for any x ∈ L. Now, as a consequence of (5), one can show easily that if Wi denotes the restriction of an h-ordinal sum T of t-norms V0 , . . . , Vn to the interval [bi , bi+1 ] in L, then Wi is only a t-subnorm on [bi , bi+1 ] in general. Indeed, if x ∈ [bi , bi+1 ), then, using the second line of (5), Wi (x, bi+1 ) = h(x) ∧ h(bi+1 ) = h(x) ∧ bi+1 = h(x) ≤ x, since h is an interior operator on L. Remark 3.2. Let T be an h-ordinal sum of t-norms V0 , . . . , Vn on L, where Vi is a t-norm on the meet sub-semilattice [bi , bi+1 ] ∩ h(L) for i = 0, . . . , n. Let us define hi : [bi , bi+1 ] → [bi , bi+1 ] as hi (x) = h(x) for any x ∈ [bi , bi+1 ]. Since bi and bi+1 are fixed points of h, from the monotonicity of h we obtain bi = h(bi ) ≤ h(x) ≤ h(bi+1 ) = bi+1 . Hence, we simply obtain that hi is an interior operator on the sub-semilattice [bi , bi+1 ] and, by Theorem 3.1, the t-norm Vi on [bi , bi+1 ] ∩ h(L) = hi ([bi , bi+1 ]) can be extended to a t-norm Wi defined on [bi , bi+1 ]. It is easy to see that Wi (x, y) = Vi (x, y) for any x, y ∈ hi ([bi , bi+1 ]) for i = 0, . . . , n, and Theorem 3.3 can be reformulated in such a way that T is an h-ordinal sum of t-norms W0 , . . . , Wn on L, where Wi is a t-norm on [bi , bi+1 ] and its restriction to [bi , bi+1 ] ∩ h(L) = hi ([bi , bi+1 ]) is also a t-norm Vi for i = 0, . . . , n. A simple consequence of the previous theorem is the following generalized 12

ordinal sum construction. Compared with Theorem 3.3, we assume now that h(L) = M . Corollary 3.4. Let L be a meet semilattice, and let b1 , . . . , bn ∈ L \ {0L , 1L } Sn be such that b1 < · · · < bn and M = i=0 [bi , bi+1 ] ⊂ L, where b0 = 0L and bn+1 = 1L . If Vi is a t-norm on [bi , bi+1 ] for i = 0, . . . , n and h is an interior operator on L such that h(L) = M , then  2  Vi (h(x), h(y)), (h(x), h(y)) ∈ [bi , bi+1 ) , T (x, y) = h(x) ∧ h(y), (h(x), h(y)) ∈ [bi , bi+1 ) × [bj , bj+1 ), for i 6= j,   x ∧ y, otherwise, (9) for any x, y ∈ L, is an h-ordinal sum of t-norms V0 , . . . , Vn on L. Proof. Since h(L) = M , we have h(x) = x for any x ∈ M (see the proof of Theorem 3.1), hence, because bi ∈ M , bi is a fixed point of h for i = 1, . . . , n. The claim is a consequence of Theorem 3.3 and the fact that sets Ji used in Theorem 3.3 are equal to [bi , bi+1 ) for i = 0, . . . , n. 2 The following theorem shows that each h-ordinal sum of t-norms is, in some sense, uniquely determined by the interior operator h. Theorem 3.5. If T is an h-ordinal sum of V0 , . . . , Vn and simultaneously an h0 ordinal sum of V00 , . . . , Vn0 , where h and h0 coincide on [bn , 1L ], h(bi ) = h0 (bi ) = bi , Vi is a t-norm on [bi , bi+1 ] ∩ h(L) and Vi0 is a t-norm on [bi , bi+1 ] ∩ h0 (L) for i = 0, . . . , n, then h = h0 . Proof. Assume that h(x) 6= h0 (x) for a certain x ∈ L. By the assumption, x 6∈ {b0 , . . . , bn−1 } ∪ [bn , bn+1 ]. Let h(x) ∈ Ji = [bi , bi+1 ] ∩ h(L) and h0 (x) ∈ Jj = [bj , bj+1 ] ∩ h0 (L). If i = j, then i < n and according to the first line of (5), we have T (x, bi+1 ) = Vi (h(x), h(bi+1 )) = Vi (h(x), bi+1 ) = h(x) 6= h0 (x) = Vi0 (h0 (x), bi+1 ) = Vi0 (h0 (x), h0 (bi+1 )) = T (x, bi+1 ),

which is a contradiction. If bi+1 ≤ bj , then j < n (otherwise, x ≥ h0 (x) ≥ bn and thus x ∈ [bn , bn+1 ], which is impossible by the assumption) and T (x, bi+1 ) = h(x) ∧ h(bi+1 ) = h(x) ∧ bi+1 = h(x) ≤ bi+1 = Vi0 (bi+1 , bi+1 ) = T (h0 (x), bi+1 ) ≤ T (x, bi+1 ),

where we used the monotonicity of T , the fact that h0 (x) ≤ x, and the equality Vi0 (bi+1 , bi+1 ) = T (h0 (x), bi+1 ), which can be proved as follows: bi+1 = Vi0 (bi+1 , bi+1 ) = Vi0 (h0 (bi+1 ), h0 (bi+1 )) = T (bi+1 , bi+1 ) ≤ T (h0 (x), bi+1 ) ≤ h0 (x) ∧ bi+1 = bi+1 . 13

Hence, we obtain that T (x, bi+1 ) ≤ h(x) ≤ bi+1 ≤ T (x, bi+1 ), which implies that h(x) = bi+1 . Since h0 (x) 6= h(x), we find that h0 (x) > bi+1 . Since (h(x), h(bj+1 )) ∈ Ji × Jj for i 6= j, we find from the second line of (5) that 0

T (x, bj+1 ) = h(x) ∧ h(bj+1 ) = bi+1 ∧ bj+1 = bi+1 < h0 (x) =

h (x) ∧ bj+1 = h0 (x) ∧ h0 (bj+1 ) = Vj0 (h0 (x), h0 (bj+1 )) = T (x, bj+1 ), which is a contradiction (we used Remark 2.1). Since a similar contradiction can be found for bi+1 > bj , the proof of the theorem is finished. 2 One can see that each h-ordinal sum of t-norms is uniquely determined assuming the coincidence of interior operators h and h0 on the last interval [bn , 1L ] and the boundaries of the related intervals. The following example shows that the assumption of coincidence of interior operators on the interval [bn , 1L ] cannot be, in general, omitted. Example 3.1. Let T be an h-ordinal sum of V0 , . . . , Vn on L such that Vn is the drastic product on [bn , 1L ] ∩ h(L), i.e., ( min (x, y), x = 1L or y = 1L , Vn (x, y) = bn , otherwise, for any x, y ∈ [bn , 1L ] ∩ h(L), and assume that (bn , 1L ) is a non-empty open interval in L. Define two interior operators h1 , h2 : L → L such that ( 1L , x = 1L , h1 (x) = x and h2 (x) = bn , otherwise, for x ∈ [bn , 1L ], and h1 (x) = h(x) = h2 (x) for any x ∈ L \ [bn , 1L ]. Now, let T1 be the h1 -ordinal sum of V0 , . . . , Vn,1 and T2 the h2 -ordinal sum of V0 , . . . , Vn,2 on L, where Vn,1 and Vn,2 are drastic t-norms on [bn , 1L ] ∩ h1 (L) = [bn , 1L ] and [bn , 1L ] ∩ h2 (L) = {bn , 1L }, respectively. We will show that T1 = T2 . If (x, y) ∈ [bn , 1L )2 , by the first case in (5), we obtain that T1 (x, y) = Vn,1 (h1 (x), h1 (y)) = Vn,1 (x, y) = bn = Vn,2 (bn , bn ) = Vn,2 (h2 (x), h2 (y)) = T2 (x, y). Similarly, if (x, y) ∈ [bi , bi+1 )2 for i 6= n, then h1 (x) = h2 (x) and h1 (y) = h2 (y), which immediately implies T1 (x, y) = Vi (h1 (x), h1 (y)) = Vi (h2 (x), h2 (y)) = T2 (x, y). If (x, y) ∈ [bi , bi+1 ) × [bj , bj+1 ) for i < j (i.e., i < j ≤ n), then h1 (x) = h(x) = h2 (x) and, by the second case in (5), we obtain that T1 (x, y) = h1 (x) ∧ h1 (y) = h1 (x) = h(x) = h2 (x) ∧ h2 (y) = T2 (x, y). Similarly, we obtain T1 (x, y) = T2 (x, y) if (x, y) ∈ [bi , bi+1 ) × [bj , bj+1 ) for i > j. Finally, if x = 1L or y = 1L , then, by the third case in (5), we obtain that T1 (x, y) = x ∧ y = T2 (x, y). Hence, T1 (x, y) = T2 (x, y) for any x, y ∈ L, and one can see that the t-norm T1 is the h1 -ordinal sum as well as the h2 -ordinal sum of t-norms for different interior operators on L. The following example shows a general construction of a t-norm based on h-ordinal sum of t-norms, where h is defined in Example 2.1. 14

Example 3.2. Let L be a meet semilattice, and let b1 , . . . , bn ∈ L \ {0L , 1L } such that b1 < · · · < bn . Assume that Vi is a t-norm on [bi , bi+1 ] for i = 0, . . . , n, where b0 = 0L and bn+1 = 1L on L. Put hi = hbi for i = 1, . . . , n, where hbi is defined in Example 2.1. According to Example 2.1, the map hi is an interior operator on L for i = 1, . . . , n. Moreover, h = h1 ◦ · · · ◦ hn is an interior operator on L, which follows from Theorem 2.2, where the commutativity of hi and hj Swas verified in Example 2.4. It is easy to see that n h(M ) = M , where M = i=0 [bi , bi+1 ]. Indeed, h(1L ) = 1L from the definition. If x ∈ M \ {1L }, then bi ≤ x < bi+1 for a certain i ∈ {0, . . . , n}. Hence, one can simply derive from (1) that hj (x) = x ∧ bj = x for j = i + 1, . . . , n and hj (x) = x for j = 0, . . . , i, which implies h(x) = x for any x ∈ M . Moreover, h(L) = M . Indeed, if x 6∈ M , then x 6≥ bn , therefore, hn (x) = x ∧ bn . If hn (x) ∈ [bn−1 , bn ), then hn (x) = hn−1 (hn (x)) = · · · = h1 (· · · hn−1 (hn (x))) = h(x), which implies that h(x) ∈ [bn−1 , bn ) ⊂ M . If hn (x) 6∈ [bn−1 , bn ), then hn−1 (hn (x)) = x ∧ bn ∧ bn−1 = x ∧ bn−1 . Repeating the procedure above, one can find i ∈ {1, . . . , n−1}, for which hi (· · · hn−1 (hn (x))) = hi (x) ∈ [bi−1 , bi ). Hence, we obtain hi (x) = hi (· · · hn−1 (hn (x))) = h1 (· · · hi (· · · hn−1 (hn (x))) = h(x) and thus h(x) ∈ [bi−1 , bi ) ⊂ M ; therefore, h(L) = M . Moreover, we also proved that h(x) ∈ [bi−1 , bi ) if and only if hi (x) = x ∧ bi ∈ [bi−1 , bi ) for i = 1, . . . , n. By Corollary 3.4, the h-ordinal sum of t-norms V0 , . . . , Vn on L can be expressed as:  Vi (x ∧ bi+1 , y ∧ bi+1 ), (x ∧ bi+1 , y ∧ bi+1 ) ∈ [bi , bi+1 )2 ,    x ∧ y ∧ b (x ∧ bi+1 , y ∧ bj+1 ) ∈ [bi , bi+1 ) × [bj , bj+1 ), i+1 ∧ bj+1 , T (x, y) =  for i 6= j,    x ∧ y, otherwise. (10) Note that h(x) = x for x ∈ [bn , 1L ), nevertheless, we use h(x) = x ∧ 1L in (10) to get a more compact formula for this ordinal sum of t-norms. As a special case of the previous construction, one can obtain the class of t-norms on bounded lattices provided by Ertuˇ grul et al. in [11]. Example 3.3. Let L be a meet semilattice, and let b ∈ L \ {0L , 1L }. Let V be a t-norm on [b, 1L ], let ∧ be a t-norm on [0L , b], and let h∧ b be defined by (1) in Example 2.1. Then, using (10), the h∧ b -ordinal sum of t-norms ∧ and V on L is a t-norm expressed as   (x, y) ∈ [b, 1L )2 , V (x, y), T (x, y) = x ∧ y (11) x = 1L or y = 1L ,   x ∧ y ∧ b, otherwise.

Indeed, put V1 = ∧ and V2 = V , and consider the general construction of the ordinal sum of t-norms described in (10). Let x, y ∈ L be arbitrary. If (x, y) = (x ∧ 1L , y ∧ 1L ) ∈ [b, 1L )2 , then T (x, y) = V2 (x ∧ 1L , y ∧ 1L ) = V2 (x, y) = V (x, y). 15

If (x ∧ b, y ∧ b) ∈ [0L , b)2 , we find that T (x, y) = V1 (x ∧ b, y ∧ b) = (x ∧ b) ∧ (y ∧ b) = x ∧ y ∧ b. If (x ∧ b, y ∧ 1L ) ∈ [0L , b) × [b, 1L ), then T (x, y) = x ∧ y ∧ b ∧ 1L = x ∧ y ∧ b, and the same result can be obtained for (x ∧ 1L , y ∧ b) × [b, 1L ) × [0L , b). If x = 1L or y = 1L , then T (x, y) = x ∧ y. Gathering together these partial expressions of the t-norm T , we obtain the h∧ b -ordinal sum of t-norms ∧ and V displayed in (11). An extension of the construction from [11] is proposed in [7]. There, the t-norm V is defined on an arbitrary interval [a, b], a, b ∈ L \ {0L , 1L }. Also this construction is a special case of the h-ordinal sum of t-norms on L given in Example 3.2, as the next example shows. Example 3.4. Let L be a meet semilattice, and let a, b ∈ L \ {0L , 1L } such that a < b. Let V be a t-norm on [a, b], let ∧1 = ∧ be a t-norm on [0L , a] and ∧2 = ∧ ∧ ∧ ∧ be a t-norm on [b, 1L ], and let h = h∧ a ◦ hb , where ha and hb have been defined by (1) in Example 2.1. Then the h-ordinal sum of t-norms ∧1 , V and ∧2 on L is a t-norm expressed as  V (x ∧ b, y ∧ b), (x ∧ b, y ∧ b) ∈ [a, b)2 ,      x ∧ y ∧ a, (x ∧ a, y ∧ a) ∈ [0L , a)2 ,      (x, y ∧ b) ∈ [b, 1L ] × [a, b), y ∧ b, (12) T (x, y) = x ∧ b, (x ∧ b, y) ∈ [a, b) × [b, 1L ],   y ∧ a, (x ∧ b, y ∧ a) ∈ [a, 1 ) × [0 , a),  L L     x ∧ a, (x ∧ a, y ∧ b) ∈ [0 , a) × [a, 1L ),  L   x ∧ y, otherwise. It can easily be shown that this t-norm coincides with the t-norm T from [7, Theorem 3.3].

The construction in the previous example shows that the following statement is true: For any bounded lattice L and any sub-interval [a, b] of L (assume that a, b ∈ L \ {0L , 1L }), on which a t-norm is defined, there exists an interior operator h : L → L such that the t-norm on [a, b] can be extended to a t-norm on L using h. The next example shows that a class of t-norms on a bounded lattice L recently proposed by C ¸ ayli in [3] is a special case of the h-ordinal sum. Example 3.5. Let L be a meet semilattice, and let b ∈ L \ {0L , 1L }. Let V be a t-norm on [b, 1L ], ∧ be a t-norm on [0L , b], and let h0b be defined by (2) in Example 2.2. Then the h0b -ordinal sum of t-norms V and ∧ on L is a t-norm expressed as  2  V (x, y), (x, y) ∈ [b, 1L ) , T (x, y) = x ∧ y (13) x = 1L or y = 1L ,   0L , otherwise. 16

A verification of the correctness of the previous expression of the t-norm T can be provided by similar arguments as in Example 3.3. However, now we use the construction of the h-ordinal sum from Theorem 3.3. Note that h0b (x) = 0L for any x ∈ L \ [b, 1L ]. Hence, h0b (L) = {0L , b} ∪ [b, 1L ] ⊂ M = [0L , b] ∪ [b, 1L ]. Since the restriction of ∧ defined on [0L , b] to the sub-semilattice {0L , b} is a t-norm on {0L , b}, the assumptions of Theorem 3.3 are satisfied and (13) can be simply derived from (5). Previous three examples show that our h-ordinal sum construction is able to cover recently introduced classes of t-norms on meet semilattices in a simple and elegant way using a suitable interior operator. It is easy to see that, using various interior operators h, infinitely many new classes of t-norms can be introduced. Although our construction is quite general, we do not claim that it covers all possible t-norms on any bounded lattice, as Example 3.7 below shows. An analysis of such “non-standard” constructions suggests a basic question asking which constructions can still be accepted as ordinal sums of t-norms (and tconorms). It will be one of the topics of our future research. The following interesting question naturally arises from previous considerations. Let L be any t-norm T on a meet semilattice L, for which there exists a finite chain 0L = b0 < b1 < · · · < bn < bn+1 = 1L in L such that T (bi , bi ) = bi , and the restriction of T to a meet sub-semilattice Ji+ of [bi , bi+1 ] with the bottom element bi and the top element bi+1 is a t-norm on Ji+ for i = 0, . . . , n.7 Is it true that T is an h-ordinal sum of V0 , . . . , Vn for a suitable interior operator h? In other words, we are interested in the validity of a statement opposite to Theorem 3.3. Notice that if T is an h-ordinal sum of t-norms V0 , . . . , Vn , where Vi is a t-norm on Ji+ for i = 0, . . . , n, then T (x, y) ∈

n [

x, y ∈ L \ {1L },

Ji ,

i=0

(14)

where Ji = Ji+ \ {bi+1 } (cf. (5)). Therefore, we restrict ourselves to a characterization of t-norms on L for which condition (14) holds. Moreover, an analysis of the interior operators and t-norms on L shows us that we are unable (or, we do not see a solution how) to determine interior operators from t-norms in a systematic way whenever Jn+ is a strict subset of [bn , 1L ], i.e., Snthere are x ∈ Jn+ and y ∈ [bn , 1L ] \ Jn+ such that T (x, y) ∈ Jn+ . Since y 6∈ i=0 Ji , we should have T (x, y) = Vn (x, h(y)) for a suitable interior operator h on L, but the open problem for us is how to define h to ensure that T is an h-ordinal sum of V0 , . . . , Vn . Therefore, we assume in the following theorem that Jn+ = [bn , 1L ]. Theorem 3.6. Let T be a t-norm on a meet semilattice L, and let 0L = b0 < b1 < · · · < bn < bn+1 = 1L be elements of L such that T (bi , bi ) = bi for 7 Note that if the restriction of T to J i+ is a t-norm, then T need not be a t-norm on [bi , bi+1 ].

17

i = 0, . . . , n + 1. Let J = {T (x, y) | x, y ∈ L \ {1L }} ∪ [bn , 1L ]

(15)

and put Ji+ =S J ∩ [bi , bi+1 ] for i = 0, . . . , n. If J is a meet sub-semilattice of L n such that J ⊆ i=0 [bi , bi+1 ] and the restriction Vi = T  Ji+ is a t-norm on Ji+ for i = 0, . . . , n, then there exists an interior operator h on L such that T is an h-ordinal sum of V0 , . . . , Vn . Sn Proof. Denote M = i=0 [bi , bi+1 ] and define the map h : L → L as ( x, x ∈ J, h(x) = (16) T (x, bn ), otherwise, for any x ∈ L. Put B = {bi | i = 0, . . . , n + 1}. Before we prove the statement of the theorem, we verify the following useful claims: T (bi , bj ) = bi ∧ bj for any bi , bj ∈ B and T (bi , x) = x for any bi ∈ B and x ∈ J such that bi ≥ x. To show the first claim, let bi , bj ∈ B be arbitrary. From Remark 2.1, we know that T (bi , bj ) ≤ bi ∧ bj , and, by the assumption, we have T (bi , bi ) = bi for any bi ∈ B. Since the elements of B are linearly ordered, bi ≤ bj or bj < bi . If bi ≤ bj , then bi ∧ bj = bi = T (bi , bi ) ≤ T (bi , bj ), which follows from the monotonicity of T ; therefore, T (bi , bj ) = bi ∧ bj . Similarly, if bj < bi , then bi ∧ bj = bj = T (bj , bj ) ≤ T (bi , bj ); therefore, T (bi , bj ) = bi ∧ bj , and the first claim is proved. Further, let bi ∈ B and x ∈ J be arbitrary such that bi ≥ x. If bi = x, then T (bi , x) = T (bi , bi ) = bi = x as was proved above. Let bi > x. Since x ∈ J ⊆ M , there exists j ∈ {0, . . . , n − 1} such that x ∈ Jj+ . Obviously, bi ≥ bj+1 , otherwise, x 6∈ [bj , bj+1 ]. Moreover, T (bj+1 , x) = x, which follows from the fact that x ∈ Jj+ and bj+1 is the neutral element of the t-norm Vj defined on Jj+ . Hence, we obtain x ≥ T (bi , x) ≥ T (bj+1 , T (bi , x)) = T (T (bj+1 , bi ), x) = T (bj+1 ∧ bi , x) = T (bj+1 , x) = x,

where we used the associativity of T , Remark 2.1 and T (bj+1 , bi ) = bj+1 ∧ bi , which was proved above. Hence, the second claim is proved too. To prove the theorem, we first verify that h is an interior operator on L. Trivially h(1L ) = 1L , since 1L ∈ J. Let x ∈ L be arbitrary. If x ∈ J, then h(h(x)) = x = h(x). If x 6∈ J, then h(x) = T (x, bn ) ∈ J. Hence, h(h(x)) = h(T (x, bn )) = T (x, bn ) = h(x); therefore, h(h(x)) = h(x) for any x ∈ L. Further, let us show that h preserves the meet operation on L. Let x, y ∈ L be arbitrary. If x, y ∈ J, then x ∧ y ∈ J, since J is a meet subsemilattice of L. Hence, we simply find that h(x ∧ y) = x ∧ y = h(x) ∧ h(y). Let x ∈ J and y 6∈ J. We can distinguish two cases, namely, x ∧ y ∈ J and x ∧ y 6∈ J. Let x ∧ y ∈ J. It is easy to see that bn > x ∧ y, otherwise, y ∈ Jn+ ⊂ J (recall that Jn+ = [bn , 1L ]), which is a contradiction with y 6∈ J. Then, we have h(x ∧ y) = x ∧ y = x ∧ (y ∧ bn ) ≥ x ∧ T (y, bn ) = h(x) ∧ h(y), 18

where Remark 2.1 is used. Since bn > x∧y and x∧y ∈ J, we have T (x∧y, bn ) = x ∧ y. Hence, using also the monotonicity of T , we obtain h(x ∧ y) = x ∧ y = T (x ∧ y, bn ) ≤ T (x, bn ) ∧ T (y, bn ) ≤ x ∧ T (y, bn ) = h(x) ∧ h(y).

Let x ∧ y 6∈ J. Since T (y, bn ) ∈ J, we obtain x ∧ T (y, bn ) ∈ J as was shown above. Moreover, we have bn > x ∧ T (y, bn ), otherwise, y ∈ Jn+ ⊂ J, which is a contradiction with y 6∈ J. Then h(x ∧ y) = T (x ∧ y, bn ) ≥ T (x ∧ T (y, bn ), bn ) = x ∧ T (y, bn ) = h(x) ∧ h(y), where we used the monotonicity of T and the fact that y ≥ T (y, bn ) according to Remark 2.1. On the other hand, using the monotonicity of T , we obtain h(x ∧ y) = T (x ∧ y, bn ) ≤ T (x, bn ) ∧ T (y, bn ) ≤ x ∧ T (y, bn ) = h(x) ∧ h(y). Hence, h(x ∧ y) = h(x) ∧ h(y) for any x ∈ J and y 6∈ J. From the commutativity of ∧, we find that h(x ∧ y) = h(x) ∧ h(y) for any x 6∈ J and y ∈ J. Let x, y 6∈ J. Again, we can distinguish two cases, namely, x ∧ y ∈ J and x ∧ y 6∈ J. Let x ∧ y ∈ J. Similarly to the previous case, bn > x ∧ y, otherwise, x, y ∈ Jn+ ⊂ J, which is a contradiction with the assumption x, y 6∈ J. Hence, we obtain h(x ∧ y) = x ∧ y = (x ∧ bn ) ∧ (y ∧ bn ) ≥ T (x, bn ) ∧ T (y, bn ) = h(x) ∧ h(y), where Remark 2.1 is used. We proved above that T (x, bi ) = x for any x ∈ J and bi ∈ B such that bi ≥ x. Hence, using the monotonicity of T , we find that h(x ∧ y) = x ∧ y = T (x ∧ y, bn ) ≤ T (x, bn ) ∧ T (y, bn ) = h(x) ∧ h(y). Let x ∧ y 6∈ J. We know that T (x, bn ), T (y, bn ) ∈ J; therefore, T (x, bn ) ∧ T (y, bn ) ∈ J, since J is a meet sub-semilattice of L. Moreover, T (x, bn ) ∧ T (y, bn ) < bn , otherwise, x, y ∈ Jn+ ⊂ J, which is a contradiction with x, y 6∈ J. Hence, we find that T (T (x, bn ) ∧ T (y, bn ), bn ) = T (x, bn ) ∧ T (y, bn ) and, using the monotonicity of T and Remark 2.1, we obtain h(x ∧ y) = T (x ∧ y, bn ) ≥ T (T (x, bn ) ∧ T (y, bn ), bn ) = T (x, bn ) ∧ T (y, bn ) = h(x) ∧ h(y),

On the other hand, we have h(x ∧ y) = T (x ∧ y, bn ) ≤ T (x, bn ) ∧ T (y, bn ) = h(x) ∧ h(y), where the monotonicity of T is used. Hence, h(x ∧ y) = h(x) ∧ h(y) holds for any x, y ∈ L, and h preserves the meet operation on L. Finally, let x ∈ L be arbitrary. Obviously, h(x) = x for x ∈ J, and h(x) = T (x, bn ) ≤ x for x 6∈ J; therefore, h(x) ≤ x for any x ∈ L, which completes the proof of the claim that h is an interior operator on L. 19

Let V be the h-ordinal sum of V0 , . . . , Vn , i.e.,  2  Vi (h(x), h(y)), (h(x), h(y)) ∈ Ji , V (x, y) = h(x) ∧ h(y), (h(x), h(y)) ∈ Ji × Jj , for i 6= j,   x ∧ y, otherwise,

(17)

for any x, y ∈ L, where Ji = [bi , bi+1 ) ∩ J. Let us prove that T = V . First, we show that V (x, y) = T (x, y) for any x, y ∈ J. By the definition of the interior operator h, we have h(x) = x for any x ∈ J. Therefore, we can distinguish three cases: 1) (x, y) ∈ Ji2 for a certain i ∈ {0, . . . , n}, 2) (x, y) ∈ Ji × Jj for certain i, j ∈ {0, . . . , n} such that i 6= j, 3) x = 1L or y = 1L . ad 1) We simply have V (x, y) = Vi (x, y) = T (x, y). ad 2) Without loss of generality, let us assume that bi < bj . Note that T (bj , y) = bj , since bj is the annihilator for the t-norm Vj on Jj+ .8 Then, V (x, y) = x ∧ y = x = Vi (x, bi+1 ) = Vi (x, bi+1 ∧ bj ) = T (x, T (bi+1 , bj )) = Vi (x, T (bi+1 , T (bj , y))) = Vi (x, T (T (bi+1 , bj ), y)) = Vi (x, T (bi+1 , y)) = T (x, T (bi+1 , y)) = T (T (x, bi+1 ), y) = T (x, y), where we used the associativity of T and the fact that T (bi , bj ) = bi ∧ bj for any bi , bj ∈ B as was proved above. ad 3) If x = 1L , then V (1L , y) = 1L ∧ y = y = T (1L , y), and similarly we have V (x, 1L ) = T (x, 1L ) for y = 1L . Hence, T and V coincide on J. Moreover, one can easily see that the restriction of V on J is an ordinal sum of V0 , . . . , Vn in the sense of (4) of Theorem 3.2, i.e. ( Vi (x, y), x, y ∈ Ji2 , V (x, y) = (18) x ∧ y, otherwise, for any x, y ∈ J (recall that h(x) = x for x ∈ J). From the coincidence of V and T on J, we obtain ( Vi (x, y), x, y ∈ Ji2 , T (x, y) = (19) x ∧ y, otherwise, for any x, y ∈ J, and T restricted to J is an ordinal sum of V0 , . . . , Vn on J. By the comparison of (17) and (19), one can find that the h-ordinal sum V of 8 It

means that Vj (bj , x) = bj for any x ∈ Jj+ . For details, we refer to [18].

20

V0 , . . . , Vn on L can be equivalently expressed as follows: ( T (h(x), h(y)), x, y ∈ L \ {1L }, V (x, y) = x ∧ y, otherwise,

(20)

for any x, y ∈ L. Let x, y ∈ L \ {1L }. Since T (x, y) ∈ J, there exists i ∈ {0, . . . , n} such that T (x, y) ∈ Ji . Note that x ≥ bi and y ≥ bi . We can distinguish the following four cases: 1) x, y ∈ Ji , 2) x ∈ Ji and y 6∈ Ji , 3) x 6∈ Ji and y ∈ Ji , 4) x, y 6∈ Ji . ad 1) It was already proved as x, y ∈ J. ad 2) If y ∈ J, then x, y ∈ J and V (x, y) = T (x, y), which was proved above. If y 6∈ J, then T (x, y) < bn , otherwise, y ∈ Jn ⊂ J, which is a contradiction with y 6∈ J. Moreover, T (x, y) ∈ J, which implies T (T (x, y), bn ) = T (x, y). Hence, we find that V (x, y) = T (h(x), h(y)) = T (x, T (y, bn )) = T (T (x, y), bn ) = T (x, y). ad 3) This can be proved by similar arguments as 2). ad 4) First, assume that x, y ∈ J. Since x, y 6∈ Ji and T (x, y) ∈ Ji , we obtain x, y ≥ bi+1 . Hence, we obtain T (x, y) ≥ T (bi+1 , bi+1 ) = bi+1 , which is a contradiction with T (x, y) ∈ Ji . Therefore, it is impossible that x, y ∈ J. Let x ∈ J and y 6∈ J. Then T (x, y) < bn , otherwise, y ∈ Jn ⊂ J, which is a contradiction with y 6∈ J. Hence, we obtain V (x, y) = T (h(x), h(y)) = T (x, T (y, bn )) = T (T (x, y), bn ) = T (x, y). Similarly, one can find the equality for x 6∈ J and y ∈ J. Let x, y 6∈ J. Then T (x, y) < bn , otherwise, x, y ∈ Jn ⊂ J, which is a contradiction with x, y 6∈ J. Hence, we find that V (x, y) = T (h(x), h(y)) = T (T (x, bn ), T (y, bn )) = T (T (T (x, bn ), y), bn ) = T (T (T (x, y), bn ), bn ) = T (T (x, y), T (bn , bn )) = T (T (x, y), bn ) = T (x, y), where we used the commutativity and associativity of T and the fact that T (T (x, y), bn ) = T (x, y), which holds for T (x, y) ≤ bn as was proved above. Hence, V (x, y) = T (x, y) for any x ∈ L \ {1L }. Finally, let x = 1L and y ∈ L. Then T (1L , y) = y = y ∧ 1L = V (1L , y), and similarly for x ∈ L and y = 1L . Hence, we obtain that T and V also coincide on L; therefore, T is an h-ordinal sum of V0 , . . . , Vn on L. 2

21

The previous theorem shows conditions under which a t-norm on L is an h-ordinal sum of t-norms defined on meet sub-semilattices of L. One can see that this theorem is, in some sense, opposite to Theorem 3.3. More precisely, each t-norm T on L, for which T (bi , bi ) = bi holds for a chain 0L = b0 < b1 < · · · < bn < bn+1 = 1L in L and T restricted to [bi , bi+1 ] ∩ J is a t-norm on [bi , bi+1 ] ∩ J for i = 0, . . . , n, where J is defined by (15), can be determined as an h-ordinal sum of t-norms for a suitable interior operator h on L. Example 3.6. Let us consider a t-norm T from Example 3.5, which is an h0b ordinal sum of t-norms V and ∧ on L for h0b defined by (2) . Let us demonstrate that the interior operator h0b is determined from the t-norm T by (16). Consider ( x, x ∈ J, h(x) = T (x, b), otherwise, where J = {T (x, y) | x, y ∈ L \ {1L }} ∪ [b, 1L ]. One can easily see that J = [b, 1L ] ∪ {0L }, since {V (x, y) | (x, y) ∈ [b, 1L )2 } ⊆ [b, 1L ]. Hence, J is a meet sub-semilattice of L and h is well-defined. If x ∈ J, then h(x) = x, and if x 6∈ J, then h(x) = T (x, b) = 0L , which can be rewritten as ( x, x ≥ b, h(x) = 0L , otherwise. Hence, h coincides with h0b . The following example shows a non-trivial t-norm (i.e., different from the meet operation) on a bounded lattice Sn that does not satisfy the assumptions of Theorem 3.6, namely, that J ⊆ i=0 [bi , bi+1 ] need not be true. Therefore, this t-norm cannot be determined as an h-ordinal sum of t-norms for a suitable interior operator h in the sense of (5) in Theorem 3.3. Example 3.7. Let L be a bounded lattice L = {0L , a, b, c, d, e, 1L } displayed in Fig. 2. Put b0 = 0L , b1 = b and b2 = 1L , and consider t-norms V1 and V2 on the intervals [b0 , b1 ] and [b1 , b2 ] to be drastic products, i.e., ( x ∧ y, if x = bi+1 or y = bi+1 , Vi (x, y) = bi , otherwise, for any x, y ∈ [bi , bi+1 ]. Further, consider a t-norm on L defined as follows: ( Vi (x, y), (x, y) ∈ [bi , bi+1 ]2 , T (x, y) = (21) x ∧ y, otherwise, for any x, y ∈ L. One can simply check that T is indeed a t-norm. Moreover, we have J = {T (x, y) | x, y ∈ L \ {1L }} ∪ [b, 1L ] = L. Hence, we obtain that S1 J 6⊆ i=0 [bi , bi+1 ]. Thus, there exist non-trivial t-norms on L for which the 22

1L c d

b

e

a 0L

Figure 2: The bounded L lattice from Example 3.7.

Sn condition J ⊆ i=0 [bi , bi+1 ] used in Theorem 3.6 is not fulfilled. One can see S1 that there is no interior operator h : L → L such that h(L) ⊆ i=0 [bi , bi+1 ] is satisfied and simultaneously, the t-norm T is determined as an h-ordinal sum of V1 and V2 . Indeed, for example, if h(e) = a, then, by (5) of Theorem 3.3, the t-norm T has to satisfy T (c, e) = h(c)∧h(e) = h(c)∧a ≤ a, but T (c, e) = c∧e = e > a according to (21). Now, a natural question arises whether such cases of t-norms are generalized ordinal sums of other t-norms. A serious answer to this question will be a topic of our future research. 4. Ordinal sums of t-conorms on join semilattices The investigation of ordinal sums of t-conorms on join semilattices is dual to our previous considerations. The aim of this section is to summarize the results for t-conorms in the same order as for t-norms. Since the proofs can be formulated by dual arguments to those used for t-norms, we omit them. Similarly to t-norms, we restrict our investigation to t-conorms defined in join semilattices. We start with the definition of a closure operator, which is conceptually dual to the interior operator intensively used in our ordinal sum constructions of t-norms. Definition 4.1. Let L be a join semilattice. A map g : L → L is said to be a closure operator on L if, for all x, y ∈ L, 1. g(0L ) = 0L , 2. g(g(x)) = g(x), 3. g(x ∨ y) = g(x) ∨ g(y), 4. x ≤ g(x). It is easy to see that the identity map 1L is a closure operator on L. 23

Example 4.1. Let L be a join semilattice, and let b ∈ L be arbitrary. Then, the map gb∨ : L → L defined by ( x, x ≤ b, ∨ gb (x) = x ∨ b, otherwise, for any x ∈ L, is a closure operator on L. Example 4.2. Let L be a join semilattice, and let b ∈ L be arbitrary. Then, the map gb1 : L → L defined by ( x, x ≤ b, 1 gb (x) = 1L , otherwise, for any x ∈ L, is a closure operator on L. Lemma 4.1. Let h and g be closure operators on a join semilattice L that commute on L. Then h ◦ g is a closure operator on L. Theorem 4.2. Let g1 , . . . , gn be mutually commutative closure operators on L, i.e., gi ◦gj = gj ◦gi for i, j = 1, . . . , n. Then g = g1 ◦· · ·◦gn is a closure operator on L. The following theorem provides a construction for t-conorms on the join semilattices with the help of a closure operator, dually to the construction of t-norms in Theorem 3.1. Theorem 4.3. Let L be a join semilattice, and let g : L → L be a closure operator on L. Let M denote the image of L under g, i.e., g(L) = M . Then, (i) M is a join sub-semilattice of L with the bottom element 0L and the top element 1L , (ii) if W is a t-conorm on M , then there exists its extension to a t-conorm S on L as follows: ( W (g(x), g(y)), x, y ∈ L \ {0L }, S(x, y) = x ∨ y, otherwise. In the next theorem, the ordinal sum of t-conorms on join semilattices that can be partitioned into a chain of subintervals is introduced. Theorem 4.4. Let L be a join semilattice, and let us assume Sn that there exist b1 , . . . , bn ∈ L \ {0L , 1L } such that b1 > · · · > bn and L = i=0 [bi+1 , bi ], where bn+1 = 0L and b0 = 1L . Then, (i) [bi+1 , bi ], i = 0, . . . , n + 1, is a join sub-semilattice of L,

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(ii) if Wi is a t-conorm on [bi+1 , bi ] for i = 0, . . . , n, then the ordinal sum of t-conorms W1 , . . . , Wn defined as follows: ( Wi (x, y), x, y ∈ (bi , bi+1 ], S(x, y) = x ∨ y, otherwise, is a t-conorm on L. The construction introduced in the previous theorem will be now extended by means of a closure operator to more general join semilattices. Theorem 4.5. Let L be a join semilattice, and let us assume Sn that there exist b1 , . . . , bn ∈ L \ {0L , 1L } such that b1 > · · · > bn and M = i=0 [bi+1 , bi ] ⊂ L, where bn+1 = 0L and b0 = 1L . If g is a closure operator on L such that g(L) ⊆ M , bi is a fixed point of g and Wi is a t-conorm on Ji+ = g(L)∩[bi+1 , bi ] for i = 0, . . . , n, then  2  Wi (g(x), g(y)), (g(x), g(y)) ∈ Ji , (22) S(x, y) = g(x) ∨ g(y), (g(x), g(y)) ∈ Ji × Jj , for i 6= j,   x ∨ y, otherwise, for any x, y ∈ L, where Ji = g(L) ∩ (bi+1 , bi ], is a t-conorm on L, which is called the g-ordinal sum of t-conorms W0 , . . . , Wn on L.

In what follows, if we say that S is a g-ordinal sum of t-conorms W0 , . . . , Wn on L, we assume that the assumptions of Theorem 4.5 are satisfied for certain b1 , . . . , bn ∈ L\{0L , 1L } such that b1 > · · · > bn and S is a t-conorm constructed by formula (22). Corollary 4.6. Let L be a join semilattice, and let b1 , . . . , bn ∈ L \ {0L , 1L } Sn such that b1 > · · · > bn and M = i=0 [bi , bi+1 ] ⊂ L, where b0 = 1L and bn+1 = 0L . If Wi is a t-conorm on [bi+1 , bi ] for i = 0, . . . , n and g is a closure operator on L such that g(L) = M , then  2  Wi (g(x), g(y)), (g(x), g(y)) ∈ (bi+1 , bi ] , S(x, y) = g(x) ∨ g(y), (g(x), g(y)) ∈ (bi+1 , bi ] × [bj+1 , bj ), for i 6= j,   x ∨ y, otherwise,

for any x, y ∈ L, is a g-ordinal sum of W0 , . . . , Wn on L.

The following theorem shows that each g-ordinal sum of t-conorms is, in some sense, uniquely determined by the closure operator g. Theorem 4.7. If S is a g-ordinal sum of W0 , . . . , Wn and simultaneously a g 0 -ordinal sum of W00 , . . . , Wn0 , where g(bi ) = g 0 (bi ) = bi , Wi is a t-conorm on [bi+1 , bi ] ∩ g(L) and Wi0 is a t-conorm on [bi+1 , bi ] ∩ g 0 (L) for i = 0, . . . , n, then g = g0 . 25

Finally, the result dual to Theorem 3.6 is provided, stating that, under certain conditions, each t-conorm on a join semilattice L is a g-ordinal sum of t-conorms W0 , . . . , Wn for a suitable closure operator g. Theorem 4.8. Let S be a t-conorm on a join semilattice L, and let 0L = bn+1 > bn > · · · > b1 > b0 = 1L be elements of L such that S(bi , bi ) = bi for i = 0, . . . , n + 1. Let J = {S(x, y) | x, y ∈ L \ {0L }} ∪ [0L , bn ] and put Ji+ = S J ∩ [bi+1 , bi ] for i = 0, . . . , n. If J is a join sub-semilattice of L n such that J ⊆ i=0 [bi+1 , bi ] and the restriction Wi = S  Ji+ is a t-conorm on Ji+ for i = 0, . . . , n, then there exists a closure operator g on L such that S is a g-ordinal sum of W0 , . . . , Wn . 5. Conclusion In this paper, we introduced a new construction of t-norms on bounded lattices, called h-ordinal sum, that generalizes previous ordinal sum approaches using the concept of lattice interior operator. Analogous results for t-conorms were also presented (without proofs). Our construction assumes an existence of a chain of subintervals of the given lattice (containing its top and bottom element), on which t-norms are partially defined, see the formulation of Theorem 3.3. The generalization of results of [26], that is, the case where t-norms are defined on some sublattices (not necessarily subintervals) of a given lattice L, and we search for their extension on L, is one of the topics of our further research. We also plan to study similar constructions for other classes of aggregation operators. Particularly, ordinal sums of uninorms have been analyzed merely for those defined on the unit interval [21]. For uninorms on bounded lattices (see, e.g., [4, 5, 8, 16]), it is an open problem. Motivated among others by recent paper [23], where interior and closure operators are used to construct uninorms on bounded lattices, we plan to solve it using a construction generalizing the one used in this paper. Note that each uninorm on a bounded lattice possesses the properties of a t-norm or a t-conorm on specific subintervals of this bounded lattice (see [16]). The interior operators (and dually for the closure operators) together with the meet operation are used in the mentioned paper [23] to extend a t-conorm defined in the subinterval [e, 1L ] of a given bounded lattice L to a uninorm by introducing, roughly speaking, a t-norm like operation on the complementary part L \ (e, 1L ] which becomes a t-norm on the subinterval [0L , e]. Comparing it with our approach, the extension procedure provided in [23] and used in the definition of new uninorms is applied only to the meet and join operations,9 whereas the interior (closure) operators in our case extend an arbitrary t-norm (t-conorm) defined on sublattices of a given lattice to the whole 9 Using our notation, for example, a t-norm like operation T ∗ on L \ (e, 1 ], which forms L the essential part of the definition of a uninorm in [23, Theorem 5.1], is an extension of

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lattice. Nevertheless, merging our ideas and the ideas presented in [23], it is an interesting task to find an appropriate map h : L → L, which behaves partially as an interior and partially as a closure operator. This map allows us to define a general extension of uninorms from bounded sublattices to bounded lattices. A general solution of ordinal sums of uninorms on bounded lattices would bring a deeper insight into the ordinal sum constructions not only for uninorms but also for other aggregation operators on these lattices. Acknowledgement We would like to thank the editor and anonymous referees for their valuable comments that significantly helped us to improve the paper. References [1] Bernard De Baets and Radko Mesiar. Triangular norms on product lattices. Fuzzy Sets and Systems, 104:61–75, 1999. [2] Garrett Birkhoff. Lattice Theory. American Mathematical Society, Providence, 1973. [3] G¨ ul Deniz C ¸ aylı. On a new class of t-norms and t-conorms on bounded lattices. Fuzzy Sets and Systems, 332:129–143, 2018. [4] G¨ ul Deniz C ¸ aylı. Uninorms on bounded lattices with the underlying t-norms and t-conorms. Fuzzy Sets and Systems, 2019. doi: 10.1016/j.fss.2019.06.005. [5] G¨ ul Deniz C ¸ aylı, Funda Kara¸cal, and Radko Mesiar. On a new class of uninorms on bounded lattices. Information Sciences, 367–368:221–231, 2016. [6] Alfred H. Clifford. Naturally totally ordered commutative semigroups. American Journal of Mathematics, 76:631–646, 1954. [7] Yexing Dan, Bao Qing Hu, and Junsheng Qiao. New construction of tnorms and t-conorms on bounded lattices. Fuzzy Sets and Systems, 2019. doi: 10.1016/j.fss.2019.05.017. [8] Yexing Dan, Bao Qing Hu, and Junsheng Qiao. New constructions of uninorms on bounded lattices. International Journal of Approximate Reasoning, 110:185–209, 2019. the meet operation on a sub-semilattice of the semilattice L \ (e, 1L ] by an interior operator h : L \ (e, 1L ] → L \ (e, 1L ]. More precisely, T ∗ is defined as T ∗ (x, y) = h(x) ∧ h(y) for h(x), h(y) ∈ h(L \ (e, 1L ]), or T ∗ (x, y) = x for y = e, and T ∗ (x, y) = y for x = e. In contrast to our work, the semilatice L \ (e, 1L ] is not bounded from above in general, but there is no principal difficulty to generalize our approach to non-bounded semilattices (lattices).

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[9] Gerd de Cooman and Etienne E. Kerre. Order norms on bounded partially ordered sets. Journal of Fuzzy Mathematics, 2:281–310, 1994. [10] Mustafa Demirci. Aggregation operators on partially ordered sets and their categorical foundations. Kybernetika, 42:261–277, 2006. ¨ [11] Umit Ertuˇ grul, Funda Kara¸cal, and Radko Mesiar. Modified ordinal sums of triangular norms and triangular conorms on bounded lattices. International Journal of Intelligent Systems, 30:807–817, 2015. [12] Joseph A. Goguen. L-fuzzy sets. Journal of Mathematics, Analysis and Applications, 18:145–174, 1967. [13] Michel Grabisch, Jean-Luc Marichal, Radko Mesiar, and Endre Pap. Aggregation Functions. Cambridge University Press, Cambridge, 2009. [14] George Gr¨ atzer. General Lattice Theory. Academic Press, New York – London, 1978. [15] George Gr¨ atzer. Lattice Theory: Foundation. Birkh¨ auser, Basel, 2011. [16] Funda Kara¸cal and Radko Mesiar. Uninorms on bounded lattices. Fuzzy Sets and Systems, 261:33–43, 2015. [17] Funda Kara¸cal and Radko Mesiar. Aggregation functions on bounded lattices. International Journal of General Systems, 46:37–51, 2017. [18] Erich Petr Klement, Radko Mesiar, and Endre Pap. Triangular Norms, volume 8 of Trends in Logic. Kluwer, Dordrecht, 2000. [19] Magda Komorn´ıkov´ a and Radko Mesiar. Aggregation functions on bounded partially ordered sets and their classification. Fuzzy Sets and Systems, 175, 2011. [20] Jes´ us Medina. Characterizing when an ordinal sum of t-norms is a t-norm on bounded lattices. Fuzzy Sets and Systems, 202:75–88, 2012. [21] Andrea Mesiarov´ a-Zem´ ankov´ a. Ordinal sum construction for uninorms and generalized uninorms. International Journal of Approximate Reasoning, 76:1–17, 2016. [22] Paul S. Mostert and Allen L. Shields. On the structure of semigroups on a compact manifold with boundary. Annals of Mathematics Second Series, 65:117–143, 1957. [23] Yao Ouyang and Hua-Peng Zhang. Constructing uninorms via closure operators on a bounded lattice. Fuzzy Sets and Systems, 2019. doi: 10.1016/j.fss.2019.05.006. [24] Daniel E. Rutherford. Introduction to Lattice Theory. Oliver & Boyd, Edinburgh and London, 1965. 28

[25] Susanne Saminger. On ordinal sums of triangular norms on bounded lattices. Fuzzy Sets and Systems, 157:1403–1416, 2006. [26] Susanne Saminger-Platz, Erich Peter Klement, and Radko Mesiar. On extensions of triangular norms on bounded lattices. Indagationes Mathematicae-New Series, 19:135–150, 2008. [27] Berthold Schweizer and Abe Sklar. Probabilistic Metric Spaces. NorthHolland, New York, 1983. [28] Dexue Zhang. Triangular norms on partially ordered sets. Fuzzy Sets and Systems, 153:195–209, 2005.

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