On ordinal sums of partially ordered monoids: A unified approach to ordinal sum constructions of t-norms, t-conorms and uninorms☆
Introduction
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 Mostert-Shields theorem [30], it can be proved that each continuous t-norm on the unit interval can be represented as an ordinal sum of continuous Archimedean t-norms. Consequently, since any continuous Archimedean t-norm is isomorphic either to the Łukasiewicz or to the product t-norm, this result shows that each continuous t-norm on the unit interval is isomorphic to an ordinal sum of Łukasiewicz and product t-norms (see [21, Chapter 5.3]).
A lot of attention has been recently paid to ordinal sums of t-norms (t-conorms) on bounded lattices [35], [24], [11], [31], [12]. In [9], a new approach using interior (for t-norms) and closure (for t-conorms) operators has been introduced. This construction has been later generalized for bounded posets [10].
Other aggregation functions on bounded lattices have been also studied, notably uninorms [5], [6], [8], [19]. However, ordinal sums of uninorms have been analyzed merely for those defined on the unit interval [25], [28], [29], [36]. An ordinal sum construction of uninorms can be also recognized in the structure of n-uninorms [1], [2], [18], [26], [27], [37], where the resulting structure of the ordinal sum has so-called n-neutral element. Recently, Kalina studied uninorms and nullnorms on a more general structure, namely on a bounded poset [17]. He showed that on any bounded poset with at least three elements it is possible to define at least two uninorms with a neutral element different from the bottom and top elements.
In this paper, we investigate ordinal sums of partially ordered monoids (po-monoids). There is a close connection between po-monoids and uninorms. Since uninorms are commutative, associative and monotone operations with a neutral element, and monoidal operation on po-monoids is, by definition, associative, monotone and possesses a neutral element, it is possible to characterize uninorms simply as (monoidal operations on) commutative po-monoids.
The paper is structured as follows: In Section 2, we review necessary notions and results on posets, po-monoids and uninorms. We also define so-called e-operators on posets as generalization of interior and closure operators. Section 3 introduces a notion of compatibility of an e-operator with a monoidal operation from po-monoid. Compatible e-operators can be used to extend a monoidal operation from a po-monoid to its super-poset (Theorem 3.1). This construction is crucial for the generalized ordinal sums of po-monoids in Section 5.
In Section 4, we present our construction of ordinal sums of po-monoids. It turns out that it is necessary to introduce two special types (downward and upward) of partial ordering on a disjoint union of po-monoids that respect the role of the element that will be a neutral element of the resulting ordinal sum (Theorem 4.1, Theorem 4.2). Using these orderings, downward and upward ordinal sums of po-monoids are introduced (Theorem 4.4, Theorem 4.5). Since the ordinal sums are introduced for families of mutually disjoint po-monoids, a natural question arises whether this assumption can be relaxed, that is, whether the po-monoids can overlap for some of their elements, namely whether the top element of one po-monoid can be identical to the bottom element of other one. This question is partially answered with the help of a congruence on downward and upward ordinal sums of po-monoids (Theorem 4.6). In the closing part of this section, we provide a simple characterization of downward and upward ordinal sums in terms of initial objects in certain categories (Theorem 4.7, Theorem 4.8). A generalization of downward and upward ordinal sum constructions based on an e-operator is described in Section 5. Section 6 contains conclusions and directions of further research.
Section snippets
Preliminaries
We say that a poset [14] with the corresponding order ≤ is bounded [15], if there exist two elements such that for all it holds that . We call and the bottom and the top element, respectively, and write this bounded poset as . For a poset , we denote by the dual ordering on L, i.e., if and only if .
Now we define intervals on posets.
Definition 2.1 Let be a poset. Let be such that . The closed subinterval of L is a subposet of L
Extension of commutative po-monoids in posets
In this part, we provide a fundamental tool in the construction of new commutative po-monoids using so-called compatible e-operators. For the sake of simplicity, we will write po-monoids without the reference to their commutativity. Note that also non-commutative po-monoids can be considered, but all the rules have to be taken into account twice, which is uselessly complex and nothing new is obtained. Therefore, we restrict our study to the commutative case. Definition 3.1 Let be a poset, and let
Downward and upward ordinal sums of po-monoids
Ordinal sums are well-known constructions appearing in the theory of partially ordered sets (ordinal sums in the sense of Birkhoff [4]) and in the theory of semigroups (ordinal sums in the sense of Clifford [7]). 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
Generalized downward and upward ordinal sums of po-monoids
In this part, we extend the downward and upward ordinal sums of po-monoids to more general constructions by means of an h-extension that was introduced in Section 3. We are motivated by the construction of generalized ordinal sum of t-norms and t-conorms presented in [9], [10], where the interior and closure operators on lattices (posets) were considered. Theorem 5.1 Let be a poset, and assume that is a non-empty family of mutually disjoint po-monoids such that , has
Conclusion
In this paper, we introduced two types (downward and upward) of ordinal sums of po-monoids. These constructions provide a unified view on several known constructions of ordinal sums of t-norms and t-conorms on lattices and introduce a generalized ordinal sum construction of uninorms on lattices. These ordinal sums are determined by special partial orderings on the disjoint union of po-monoids. Further, we introduced the conditions under which an ordinal sum of bounded po-monoids can be obtained
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
We would like to thank the editor and anonymous referees for their valuable comments that significantly helped us to improve the paper.
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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 of the second and third author was also provided by the Czech Science Foundation through the project No. 18-06915S.