Combinatorial results for zero-divisors regarding right zero elements of order-preserving transformations

For any positive integer [Formula: see text], let [Formula: see text] be the semigroup of all order-preserving full transformations on [Formula: see text]. For any [Formula: see text], let [Formula: see text] be the constant map defined by [Formula: see text] for all [Formula: see text]. In this paper, we introduce and study the sets of left, right, and two-sided zero-divisors of [Formula: see text]: [Formula: see text] [Formula: see text] We determine the structures and cardinalities of [Formula: see text], [Formula: see text] and [Formula: see text] for each [Formula: see text]. Furthermore, we compute the ranks of [Formula: see text],[Formula: see text][Formula: see text],[Formula: see text][Formula: see text],[Formula: see text][Formula: see text] and [Formula: see text] for each [Formula: see text], because these are significant subsemigroups of [Formula: see text].