Let E, F be Banach lattices, where E has the disjoint Riesz decomposition property. For a lattice homomorphism \(T:E\rightarrow F\) and a bounded subset A of E, we establish a necessary and sufficient condition under which TA is b-order bounded. Based on this, we study the b-order boundedness of subsets of E and obtain several characterizations of AM-spaces. Furthermore, we introduce and investigate a novel type of operators referred to as M-serially summing operator. The connections of this category of operators with classical notions of operators, such as majorizing operators, preregular operators and serially summing operators, are considered.
