Lattice of fuzzy subalgebras and closure systems in \(I^ X\). (English) Zbl 0731.08007

For a given universal algebra \(A=[X,F]\) with a specified set F of finitary operations on the set X, each operation is extended to a finitary operation on the lattice \(I^ X\) of fuzzy subsets of X over the unit interval of real numbers \(I=[0,1]\). It is shown that the lattice of fuzzy subalgebras of A is an algebraic closure system in \(I^ X\) according to an appropriate definition.


08A99 Algebraic structures
06B99 Lattices
Full Text: DOI


[1] Birkhoff, G., Lattice theory, () · Zbl 0126.03801
[2] Cohn, P.M., Universal algebra, (1981), Reidel Dordrecht-Boston · Zbl 0141.01002
[3] Gerla, G.; Tortora, R.R., Normalisation of fuzzy algebras, Fuzzy sets and systems, 17, 73-82, (1985) · Zbl 0589.08003
[4] Gratzer, G., Universal algebra, (1968), Van Nostrand New York · Zbl 0182.34201
[5] Katsaras, A.K.; Lui, D.B., Fuzzy vector spaces and fuzzy topological vector spaces, J. math. anal. appl., 58, 135-146, (1977) · Zbl 0358.46011
[6] Liu, W.J., Fuzzy ideals, Fuzzy sets and systems, 8, 133-139, (1982) · Zbl 0488.20005
[7] Murali, V.A., A category of fuzzy algebras, (1989), Preprint
[8] Rosenfeld, A., Fuzzy groups, J. math. anal. appl., 35, 512-517, (1971) · Zbl 0194.05501
[9] Scott, D.S.; Gierz, G., A compendium of continuous lattices, (1980), Springer-Verlag Berlin-New York · Zbl 0452.06001
[10] Zadeh, L.A., Fuzzy sets, Inform. and control, 8, 338-353, (1965) · Zbl 0139.24606
[11] Zadeh, L.A.; Zadeh, L.A., The concept of linguistic variable and its application to approximate reasoning, Inform. sci., Inform. sci., 8, 307-357, (1975) · Zbl 0404.68074
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.