Regularity and Green’s relations for semigroups of transformations preserving a zigzag order and an equivalence

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Consider a finite ordered set [Formula: see text] whose order forms a zigzag. For an equivalence [Formula: see text] on [Formula: see text], let [Formula: see text] be the set of all order homomorphism preserving [Formula: see text]. In this paper, we completely describe the regularity of elements and Green’s relations for [Formula: see text] under a certain condition.