| Abstract: | In classical logic conjunction distributes over disjunction and disjunction distributes over conjunction.
Moreover, implication is left-distributive over conjunction and disjunction:
p ! (q ^ r) (p ! q) ^ (p ! r);
p ! (q _ r) (p ! q) _ (p ! r):
At the same time it is neither right-distributive over conjunction nor over disjunction. However, the following
two equalities, that are kind of right-distributivity of implications, hold:
(p ^ q) ! r (p ! r) _ (q ! r);
(p _ q) ! r (p ! r) ^ (q ! r):
We can rewrite the above four classical tautologies in fuzzy logic and obtain the following distributivity
equations:
I(x;C1(y; z)) = C2(I(x; y); I(x; z)); (D1)
I(x;D1(y; z)) = D2(I(x; y); I(x; z)); (D2)
I(C(x; y); z) = D(I(x; z); I(y; z)); (D3)
I(D(x; y); z) = C(I(x; z); I(y; z)); (D4)
that are satisfied for all x; y; z 2 [0; 1], where I is some generalization of classical implication, C, C1,
C2 are some generalizations of classical conjunction and D, D1, D2 are some generalizations of classical
disjunction. We can define and study those equations in any lattice L = (L;6L) instead of the unit
interval [0; 1] with regular order „6” on the real line, as well.
From the functional equation’s point of view J. Aczél was probably the one that studied rightdistributivity
first. He characterized solutions of the functional equation (D3) in the case of C = D,
among functions I there are bounded below and functions C that are continuous, increasing, associative
and have a neutral element. Part of the results presented in this thesis may be seen as a generalization
of J. Aczél’s theorem but with fewer assumptions on the functions F and G. As a generalization of
classical implication we consider here a fuzzy implication and as a generalization of classical conjunction
and disjunction - t-norms and t-conorms, respectively (or more general conjunctive and disjunctive uninorms).
We study the distributivity equations (D1) - (D4) for such operators defined on different lattices
with special focus on various functional equations that appear. In the first two sections necessary fuzzy logic concepts are introduced. The background and history of
studies on distributivity of fuzzy implications are outlined, as well. In Sections 3, 4 and 5 new results are
presented and among them solutions to the following functional equations (with different assumptions):
f(m1(x + y)) = m2(f(x) + f(y)); x; y 2 [0; r1];
g(u1 + v1; u2 + v2) = g(u1; u2) + g(v1; v2); (u1; u2); (v1; v2) 2 L1;
h(xc(y)) = h(x) + h(xy); x; y 2 (0;1);
k(min(j(y); 1)) = min(k(x) + k(xy); 1); x 2 [0; 1]; y 2 (0; 1];
where:
f : [0; r1] ! [0; r2], for some constants r1; r2 that may be finite or infinite, and for functions m2 that
may be injective or not;
g : L1 ! [1;1], for L1 = f(u1; u2) 2 [1;1]2 j u1 u2g (function g satisfies two-dimensional
Cauchy equation extended to the infinities);
h; c : (0;1) ! (0;1) and function h is continuous or is a bijection;
k : [0; 1] ! [0; 1], g : (0; 1] ! [1;1) and function k is continuous.
Most of the results in Sections 3, 4 and 5 are new and obtained by the author in collaboration with
M. Baczynski, R. Ger, M. E. Kuczma or T. Szostok. Part of them have been already published either
in scientific journals (see [5]) or in refereed papers in proceedings (see [4, 1, 2, 3]). |