International Mathematical Forum, 4, 2009, no. 5, 227 - 232 A Theorem on Semilattice-Ordered Semigroup 1 Melanija Mitrovi´ c Faculty of Mechanical Engineering, University of Niˇ s 14, Aleksandra Medvedeva, 18000 Niˇ s, Serbia meli@junis.ni.ac.yu Daniel Abraham Romano 2 Department of Mathematics and Informatics, Banja Luka University 2, Mladen Stojanovic Street, 78000 Banja Luka, Bosnia and Herzegovina bato49@hotmail.com Milovan Vinˇ ci´ c Faculty of Mechanical Engineering, Banja Luka University 71, Stepe Stepanovi´ ca,78000 Banja Luka, Bosnia and Herzegovina Abstract Semilattice-ordered semigroup is important algebraic structure. It is equipped with the natural defined positive quasi-antiorder relation. Mathematical Subject Classification: 03F65; 06F05, 20M10 Keywords: Constructive mathematics, semigroups with apartness, semilattice- ordered semigroup, anti-order relation, quasi-antiorder relation, anti-ideal, pos- itive quasi-antiorder relation 1 Introduction and preliminaries This investigation is in Bishop’s constructive algebra in sense of papers [9]-[12] and books [7] and [13] (Chapter 8: Algebra). Let (S, =, =) be a constructive set (i.e. it is a relational system with the relation ”= ”). The diversity relation ”=” ([10]) is a binary relation on S , which satisfies the following properties: 1 Partially supported by the Ministry of science and technology of Srpska, Banja Luka, Bosnia and Herzegovina. 2 Corresponding author