Distributive Idempotent Uninorms

Abstract

A characterization of all idempotent uninorms satisfying the distributive property is given. The special cases of left-continuous and right-continuous idempotent uninorms are presented separately and it is also proved that all idempotent uninorms are autodistributive. Moreover, all distributive pairs of idempotent uninorms (pairs U\, U2 such that U\ is distributive over U2 and U2 is distributive over Ui) are also characterized.

Distributivity between uninorms and nullnorms

“…A new direction of investigations is mainly concerned of distributivity between triangular norms and triangular conorms [7, p. 17, 21]. Since a short time many authors deal with solution of distributivity equation for aggregation functions [3], fuzzy implications [2,18,19], uninorms and nullnorms [12,13,[15][16][17], which are generalization of triangular norms and conorms.…”

The distributive equations for idempotent uninorms and nullnorms

“…And the all new solutions to the equations are characterized as Theorems 3 and 4. Thus this paper has finished the characterization of them along with [17,19]. The results in Section 2 of this paper generalize the corresponding ones in [17], but results in Section 3 of this paper differ from the corresponding ones in [17].…”

“…First of all, let us prove that g(x) = 0 holds for any x > k. Suppose on the contrary that there exists some x 0 with 1 > x 0 > k and g(x 0 ) > 0 and then take z 0 and y 0 such that 0 < z 0 < g(x 0 ) e < k < x 0 < y 0 < 1 (19) and finally put x = x 0 ; y = y 0 ; z = z 0 in Eq. (11).…”

“…(1), i.e., Eq. (1) does not involve uninorms, this case has been discussed in detail in [17]; when both G 1 and G 2 are idempotent uninorms, this case has been discussed in detail in [19]. So it is sufficient to consider the remaining two cases: (i) G 1 is a nullnorm and G 2 an idempotent uninorm, this case will be discussed in Section 2 because of the preliminaries in Section 1; (ii) G 1 is an idempotent uninorm and G 2 a nullnorm, this case will be discussed in Section 3.…”

“…Symbols and concepts in this paper may be also traced back to [17,19]. In addition, uninorms, nullnorms and t-operators were investigated in [4,8,10,14,18,20].…”