On fuzzy ideals and fuzzy filters of fuzzy lattices /
In the literature there are several proposals of fuzzification of lattices and ideals concepts. Chon in (Korean J. Math 17 (2009), No. 4, 361-374), using the notion of fuzzy order relation defined by Zadeh, introduced a new notion of fuzzy lattice and studied the level sets of fuzzy lattices, but di...
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| Formato: | Tese |
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| Endereço do item: | https://repositorio.ufrn.br/jspui/bitstream/123456789/18692/1/IvanM_TESE.pdf |
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| Resumo: | In the literature there are several proposals of fuzzification of lattices and ideals concepts. Chon in (Korean J. Math 17 (2009), No. 4, 361-374), using the notion of fuzzy order relation defined by Zadeh, introduced a new notion of fuzzy lattice and studied the level sets of fuzzy lattices, but did not define a notion of fuzzy ideals for this type of fuzzy lattice. In this thesis, using the fuzzy lattices defined by Chon, we define fuzzy homomorphism between fuzzy lattices, the operations of product, collapsed sum, lifting, opposite, interval and intuitionistic on bounded fuzzy lattices. They are conceived as extensions of their analogous operations on the classical theory by using this definition of fuzzy lattices and introduce new results from these operators. In addition, we define ideals and filters of fuzzy lattices and concepts in the same way as in their characterization in terms of level and support sets. One of the results found here is the connection among ideals, supports and level sets. The reader will also find the definition of some kinds of ideals and filters as well as some results with respect to the intersection among their families. Moreover, we introduce a new notion of fuzzy ideals and fuzzy filters for fuzzy lattices defined by Chon. We define types of fuzzy ideals and fuzzy filters that generalize usual types of ideals and filters of lattices, such as principal ideals, proper ideals, prime ideals and maximal ideals. The main idea is verifying that analogous properties in the classical theory on lattices are maintained in this new theory of fuzzy ideals. We also define, a fuzzy homomorphism h from fuzzy lattices L and M and prove some results involving fuzzy homomorphism and fuzzy ideals as if h is a fuzzy monomorphism and the fuzzy image of a fuzzy set ~h(I) is a fuzzy ideal, then I is a fuzzy ideal. Similarly, we prove for proper, prime and maximal fuzzy ideals. Finally, we prove that h is a fuzzy homomorphism from fuzzy lattices L into M if the inverse image of all principal fuzzy ideals of M is a fuzzy ideal of L. Lastly, we introduce the notion of α-ideals and α-filters of fuzzy lattices and characterize it by using its support and its level set. Moreover, we prove some similar properties in the classical theory of α- ideals and α-filters, such as, the class of α-ideals and α-filters are closed under intersection. We also define fuzzy α-ideals of fuzzy lattices, some properties analogous to the classical theory are also proved and characterize a fuzzy α-ideal on operation of product between bounded fuzzy lattices L and M and prove some results. |
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