On Groupoid-Poset Structures

We introduce the notion of semi-ideals for ordered groupoids (magmas equipped with a partial order). The definition is based on the derived set [Formula: see text], which captures all products where the first argument is order-dominated by the second. Our main result shows that right compatibility of the operation with the order implies that [Formula: see text] is a right semi-ideal; we also discuss the conditions under which the converse holds, showing that the full equivalence requires the additional hypothesis of right admissibility. We prove that the family of right semi-ideals forms a closure system, hence a complete lattice, and that the associated closure operator yields a Galois insertion between the power set and the lattice of semi-ideals. A detailed inductive description of principal right semi-ideals is given. As an application, we show that in a standard BCK-algebra the sets [Formula: see text] and [Formula: see text] are right semi-ideals, and we exhibit the difference between right and left semi-ideals.