On α-Cross-Migrativity of Overlap (0-Overlap) Functions

Overlap functions, as one kind of special binary aggregation functions, have been continuously concerned and studied by many scholars and widely applied in classification, decision making, image processing, and fuzzy community detection problems. Meanwhile, the migrativity as a vital, and particularly, interesting property for binary aggregation functions, has been studied in the literature since it was proposed. This article continues to consider this research topic and mainly focuses on the
$\alpha$
-cross-migrativity for overlap functions. First, we introduce the concept of the
$\alpha$
-cross-migrativity for overlap functions and show some vital properties of
$\alpha$
-cross-migrative overlap functions. Second, we study the overlap functions that are
$\alpha$
-cross-migrative with respect to the minimum overlap function
$O_M$
and obtain an equivalent characterization of them by the ordinal sum of overlap functions. Third, we prove that an overlap function is
$\alpha$
-cross-migrative with respect to the
$p$
-product overlap function
$O_p$
only if
$p$
is equal to 1 and give an equivalent characterization of the overlap functions that satisfy the
$\alpha$
-cross-migrativity with respect to the 1-product overlap function
$O_1$
also by the ordinal sum of overlap functions. Finally, we extend the
$\alpha$
-cross-migrativity of overlap functions to the 0-overlap functions status and get two equivalent characterizations of the 0-overlap functions that are
$\alpha$
-cross-migrative with respect to
$O_M$
and an equivalent characterization of the 0-overlap functions that satisfy the
$\alpha$
-cross-migrativity with respect to
$O_1$
, respectively.