An. S ¸t. Univ. Ovidius Constant ¸a Vol. 16(2), 2008, 31–38 Infimum and supremum completeness properties of ordered sets without axioms Zolt´ an BOROS * and ´ Arp´ ad SZ ´ AZ Abstract In this paper, by using the ideas of the second author, we establish several intimate connections among the most simple infimum and supre- mum completeness properties of a generalized ordered set. That is, an arbitrary set equipped with an arbitrary inequality relation. In particular, we obtain straightforward extensions of some basic theorems on partially ordered sets. Due to the equalities inf ( A)= sup ( lb ( A)) and sup( A) = inf ( ub ( A)) established first by the second author, the proofs given here are much shorter and more natural than the usual ones. 1 Prerequisites Throughout this paper, X will denote an arbitrary set equipped with an arbi- trary binary relation ≤ . Thus, X may be considered as a generalized ordered set, or an ordered set without axioms. For any A ⊂ X , the members of the families lb ( A)=  x ∈ X : ∀ a ∈ A : x ≤ a  and ub ( A)=  x ∈ X : ∀ a ∈ A : a ≤ x  are called the lower and upper bounds of A in X , respectively. And the members of the families Key Words: Generalized ordered sets, infimum and supremum completenesses Mathematics Subject Classification: 06A06, 06A23 Received: November, 2007 Accepted: August, 2008 * The research of the first author has been supported by the grant OTKA NK–68040. 31