  This paper aims to establish the theoretical foundation for shift inclusion
in mathematical morphology. In this paper, we prove that the morphological
opening and closing concerning structuring elements of shift inclusion property
would preserve the ordering of images, while this property is important in
granulometric analysis and related image processing tasks. Furthermore, we
proposed a systematic way, called the decomposition theorem for shift
inclusion, to construct sequences of structuring elements with shift inclusion
property. Moreover, the influences of the image domain are discussed and the
condition named weak shift inclusion is defined, which is proved as an
equivalent condition for ensuring the order-preserving property.

Discrete Mathematics;Combinatorics