Abstract
In this contribution, we propose an ordinal sum construction of t-norms on bounded lattices from a system of lattices (endowed by t-norms) whose index set forms a bounded lattice. The ordinal sum construction is determined by a lattice-based sum of bounded lattices and interior operators.
This research was partially supported from 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.
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- 1.
This is a consequence of the fact that \(0_L\) and \(1_L\) are nullary operations in the bounded lattice L.
- 2.
Note that if an index set P is a linearly ordered set of indices with the bottom and top elements, then P is clearly a bounded lattice.
- 3.
- 4.
The closure operator is defined dually to the interior operator, i.e., \(h:L\rightarrow L\) is a closure operator if 1. \(h(0_L)=0_L\), 2. \(h(h(x))=h(x)\), 3. \(h(x\vee y)=h(x)\vee h(y)\), 4. \(x\le h(x)\) holds for any \(x,y\in L\). For details, we refer to [6].
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Dvořák, A., Holčapek, M. (2020). Generalized Ordinal Sum Constructions of t-norms on Bounded Lattices. In: Ceberio, M., Kreinovich, V. (eds) Decision Making under Constraints. Studies in Systems, Decision and Control, vol 276. Springer, Cham. https://doi.org/10.1007/978-3-030-40814-5_9
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