# Mirko Navara presents Fuzzy logical operations and their generators

On 2019-04-25 12:00:00 at G205, Karlovo náměstí 13, Praha 2

As noticed already by Abel, an operation which is monotonic, commutative, and

associative is essentially the addition. This principle applies also to fuzzy

logical operations with values in the real unit interval; many of them can be

considered equal up to an isomorphism (i.e., increasing bijection of the set of

values). Although there are constructive proofs of these results, their

computation often becomes unfeasible. We proved that, in many cases, partial

derivatives allow to prove these results in a closed form, admitting also the

use of computer algebra.

References:

Navara, M., Petrík, M., Sarkoci, P.: Explicit formulas for generators of

triangular norms. Publ. Math. Debrecen 77 (2010), 171-191.

http://cmp.felk.cvut.cz/~navara/papers/PMD10.pdf

Navara, M.: Formulas for generators of R-implications. Fuzzy Sets Syst. 359

(2019), 80–89. DOI:10.1016/j.fss.2018.09.011

http://cmp.felk.cvut.cz/~navara/papers/gen_impl_FSS19.pdf

Navara, M., Petrík, M.: Generators of fuzzy logical operations. Submitted.

associative is essentially the addition. This principle applies also to fuzzy

logical operations with values in the real unit interval; many of them can be

considered equal up to an isomorphism (i.e., increasing bijection of the set of

values). Although there are constructive proofs of these results, their

computation often becomes unfeasible. We proved that, in many cases, partial

derivatives allow to prove these results in a closed form, admitting also the

use of computer algebra.

References:

Navara, M., Petrík, M., Sarkoci, P.: Explicit formulas for generators of

triangular norms. Publ. Math. Debrecen 77 (2010), 171-191.

http://cmp.felk.cvut.cz/~navara/papers/PMD10.pdf

Navara, M.: Formulas for generators of R-implications. Fuzzy Sets Syst. 359

(2019), 80–89. DOI:10.1016/j.fss.2018.09.011

http://cmp.felk.cvut.cz/~navara/papers/gen_impl_FSS19.pdf

Navara, M., Petrík, M.: Generators of fuzzy logical operations. Submitted.