Meet-continuous codomain lattice and fuzzy set inequations
Само за регистроване кориснике
2023
Конференцијски прилог (Рецензирана верзија)
Метаподаци
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Extremal problems in fuzzy set and fuzzy relational equations and inequations - both existential and constructional - strongly depend on the choice of the codomain lattice, and the way we define the composition of fuzzy relations. A complete lattice is meet-continuous if infimum commutes with the supremum of chains. For such a lattice, I prove that some inequations and equations that are often studied have a maximal solution. If the existing maximal solution is unique, it is also the greatest solution. Solutions to some inequations are connected to the notions of fuzzy relations closed under composition with a given fuzzy relations, thus we prove the existence of maximal such relations. The property of being transitive relation may also be expressed in terms of being a solution to a fuzzy relational inequation, thus I also prove the existence of a maximal transitive relation contained in a given arbitrary relation.
In case of complete codomain lattice, there exists the least solution... to some fuzzy relational inequations. If such complete codomain lattice is also meet-continuous, there exist algorithms giving such least solutions in at most countably many sets. In particular, there exists an algorithm for finding the transitive closure of an arbitrary fuzzy relation in at most countably many steps. Even if the number of steps is infinite, in some cases, by inductive reasoning, we shall be able to determine the transitive closure in real time, as an example illustrates.
Кључне речи:
fuzzy relation / fuzzy relational inequations / maximal solution / the least solution algorithmsИзвор:
The 5th International Conference on Machine Learning and Intelligent Systems (MLIS 2023), Macao, Kina, 2023Институција/група
Poljoprivredni fakultetTY - CONF AU - Stepanović, Vanja PY - 2023 UR - http://aspace.agrif.bg.ac.rs/handle/123456789/6798 AB - Extremal problems in fuzzy set and fuzzy relational equations and inequations - both existential and constructional - strongly depend on the choice of the codomain lattice, and the way we define the composition of fuzzy relations. A complete lattice is meet-continuous if infimum commutes with the supremum of chains. For such a lattice, I prove that some inequations and equations that are often studied have a maximal solution. If the existing maximal solution is unique, it is also the greatest solution. Solutions to some inequations are connected to the notions of fuzzy relations closed under composition with a given fuzzy relations, thus we prove the existence of maximal such relations. The property of being transitive relation may also be expressed in terms of being a solution to a fuzzy relational inequation, thus I also prove the existence of a maximal transitive relation contained in a given arbitrary relation. In case of complete codomain lattice, there exists the least solution to some fuzzy relational inequations. If such complete codomain lattice is also meet-continuous, there exist algorithms giving such least solutions in at most countably many sets. In particular, there exists an algorithm for finding the transitive closure of an arbitrary fuzzy relation in at most countably many steps. Even if the number of steps is infinite, in some cases, by inductive reasoning, we shall be able to determine the transitive closure in real time, as an example illustrates. C3 - The 5th International Conference on Machine Learning and Intelligent Systems (MLIS 2023), Macao, Kina T1 - Meet-continuous codomain lattice and fuzzy set inequations UR - https://hdl.handle.net/21.15107/rcub_agrospace_6798 ER -
@conference{ author = "Stepanović, Vanja", year = "2023", abstract = "Extremal problems in fuzzy set and fuzzy relational equations and inequations - both existential and constructional - strongly depend on the choice of the codomain lattice, and the way we define the composition of fuzzy relations. A complete lattice is meet-continuous if infimum commutes with the supremum of chains. For such a lattice, I prove that some inequations and equations that are often studied have a maximal solution. If the existing maximal solution is unique, it is also the greatest solution. Solutions to some inequations are connected to the notions of fuzzy relations closed under composition with a given fuzzy relations, thus we prove the existence of maximal such relations. The property of being transitive relation may also be expressed in terms of being a solution to a fuzzy relational inequation, thus I also prove the existence of a maximal transitive relation contained in a given arbitrary relation. In case of complete codomain lattice, there exists the least solution to some fuzzy relational inequations. If such complete codomain lattice is also meet-continuous, there exist algorithms giving such least solutions in at most countably many sets. In particular, there exists an algorithm for finding the transitive closure of an arbitrary fuzzy relation in at most countably many steps. Even if the number of steps is infinite, in some cases, by inductive reasoning, we shall be able to determine the transitive closure in real time, as an example illustrates.", journal = "The 5th International Conference on Machine Learning and Intelligent Systems (MLIS 2023), Macao, Kina", title = "Meet-continuous codomain lattice and fuzzy set inequations", url = "https://hdl.handle.net/21.15107/rcub_agrospace_6798" }
Stepanović, V.. (2023). Meet-continuous codomain lattice and fuzzy set inequations. in The 5th International Conference on Machine Learning and Intelligent Systems (MLIS 2023), Macao, Kina. https://hdl.handle.net/21.15107/rcub_agrospace_6798
Stepanović V. Meet-continuous codomain lattice and fuzzy set inequations. in The 5th International Conference on Machine Learning and Intelligent Systems (MLIS 2023), Macao, Kina. 2023;. https://hdl.handle.net/21.15107/rcub_agrospace_6798 .
Stepanović, Vanja, "Meet-continuous codomain lattice and fuzzy set inequations" in The 5th International Conference on Machine Learning and Intelligent Systems (MLIS 2023), Macao, Kina (2023), https://hdl.handle.net/21.15107/rcub_agrospace_6798 .