Add to ORKGAbstract. Let (S,=, 6=, ·, s) be an ordered semigroup under an antiorder s. If S is a subdirect product of the ordered semigroup {Si: i ∈ I}, then there exists a family {σi: i ∈ I} of quasi-antiorders on S which separates the elements of S. Conversely, if {σi: i ∈ I} is a family of quasi-antiorders on S which separates the elements if S, then S is a subdirect product of the ordered semigroups {S/(σi ∪ (σi)−1) : i ∈ I}. This investigation is in constructive algebra. Throughout this paper, S = (S,=, 6=, ·) always denotes a semigroup with apartness in the sense of the books [1], [3], [8], [9] and papers [4], [5,], [6] and [7]. The apartness 6 = on S is a binary relation with the following properties ([1], [9]): For every elements x, y and z in...