Elsevier

Fuzzy Sets and Systems

Volume 382, 1 March 2020, Pages 158-164
Fuzzy Sets and Systems

A note on the lattice structure for subalgebras of the algebra of truth values of type-2 fuzzy sets

https://doi.org/10.1016/j.fss.2019.05.011Get rights and content

Abstract

The algebra of truth values of type-2 fuzzy sets is not a lattice, but some of its subalgebras may be lattices. In this paper, we give a necessary and sufficient condition under which subalgebras of the truth value algebra of type-2 fuzzy sets form lattices. Further, we prove that if a certain subalgebra is a lattice, then it is isomorphic to an appropriate subalgebra whose all elements are convex functions with the same height. Based on these results, if a certain subalgebra is a lattice, the equivalent characterizations of the partial order in it are given.

Section snippets

Introduction and preliminary

Type-2 fuzzy sets were introduced by Zadeh in 1975, as an extension of type-1 fuzzy sets, and have been heavily investigated both as a mathematical object and as a tool in applications (see, e.g. [8], [10], [11], [12]). The algebra of truth values of type-2 fuzzy sets is a set of all mappings from the unit interval into itself, with operations given by the various convolutions of the pointwise operations [10]. From [9] we know the truth value algebra of type-2 fuzzy sets does not form a lattice

Sublattices

In this section, we propose an equivalent characterization of sublattices and point out that any sublattice is isomorphic to a corresponding sublattice whose all elements are convex functions with the same height. Further, we give the equivalent characterizations of the partial order in each sublattice. As applications, the example of complete sublattices and non-complete sublattices containing non-convex functions is given, respectively.

Conclusions

In this paper, we give the equivalent characterization of a sublattice of M and the equivalent characterizations of the partial order in it, respectively, and prove that any sublattice A is isomorphic to Γ(A). In the future work, we will concentrate on the equivalent characterizations of complete sublattices.

Acknowledgements

The work described in this paper was supported by grants from the National Natural Science Foundation of China (Grant nos. 11571010 and 61179038).

References (13)

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