We introduce the concept of maximal directions of increasingness (resp. decreasingness) of an aggregation function. In the bivariate case, we derive these maximal directions with respect to points on the main diagonal of the unit square for a symmetric aggregation function that has either piecewise convex or piecewise concave level curves and is differentiable up to second order. With any bivariate aggregation function of the latter type we associate another bivariate aggregation function that has the same maximal directions of increasingness (resp. decreasingness) while having straight lines as level curves. We explore under which conditions the latter aggregation function is a semi-copula, a quasi-copula or a copula. As a by-product we establish a new construction method for aggregation functions with given diagonal section.

我们引入了聚合函数的最大递增方向（分别是递减方向）的概念。在双变量情况下，我们针对具有分段凸或分段凹水平曲线并且可微分到二阶的对称聚合函数，推导出这些关于单位正方形主对角线上的点的最大方向。对于后一种类型的任何二元聚合函数，我们将另一个二元聚合函数关联起来，该函数具有相同的最大递增方向（分别是递减方向），同时具有直线作为水平曲线。我们探讨后一种聚合函数在哪些条件下是半联结、准联结或联结。作为副产品，我们为给定对角线截面的聚合函数建立了一种新的构造方法。