Authors: Eghosa Edeghagba, Elijah
Šešelja, Branimir
Tepavčević, Andreja 
Title: Ω-Lattices
Journal: Fuzzy Sets and Systems
Volume: 311
First page: 53
Last page: 69
Issue Date: 15-Mar-2017
Rank: M21a
ISSN: 0165-0114
DOI: 10.1016/j.fss.2016.10.011
In the framework of Ω-sets, where Ω is a complete lattice, we introduce Ω-lattices, both as algebraic and as order structures. An Ω-poset is an Ω-set equipped with an Ω-valued order which is antisymmetric with respect to the corresponding Ω-valued equality. Using a cut technique, we prove that the quotient cut-substructures can be naturally ordered. Introducing notions of pseudo-infimum and pseudo-supremum, we obtain a definition of an Ω-lattice as an ordering structure. An Ω-lattice as an algebra is a bi-groupoid equipped with an Ω-valued equality, fulfilling particular lattice-theoretic formulas. On an Ω-lattice we introduce an Ω-valued order, and we prove that particular quotient substructures are classical lattices. Assuming Axiom of Choice, we prove that the two approaches are equivalent.
Keywords: Complete lattice | Fuzzy congruence | Fuzzy equality | Fuzzy identity | Fuzzy lattice
Publisher: Elsevier
Project: Development of methods of computation and information processing: theory and applications 

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