Adv. Stud. Contemp. Math. (Kyungshang) 20 (2010), 139–135. FOUNDATIONS OF THE THEORY OF VECTOR RELATORS ´ Arp´ ad Sz´ az Abstract. A nonvoid family R of binary relations on a nonvoid set X is called a relator on X . In particular, a relator R on a vector space X is called a vector relator on X if (1) R ( x)= x + R (0) for all R ∈R and x ∈ X ; (2) R (0) is an absorbing balanced subset of X for all R ∈R ; (3) for each R ∈R there exists S ∈R such that S (0) + S (0) ⊂ R (0). Vector relators are more convenient means than vector topologies. They are mainly motivated by the fact that if P is a nonvoid family of preseminorms on X , then the collection R P of all surroundings B p r = { ( x, y ): p ( x - y ) <r } , where p ∈P and r> 0 , is a vector relator on X . Postulates (1) – (3) imply that R is a reflexive, symmetric, uniformly transitive and well-chained relator on X such that each member of R is a balanced translation relation. Moreover, it is also noteworthy that if in particular each member of P is a seminorm, then the members of R P are, in addition, convex. Therefore, before studying the most fundamental properties of vector relators, and the linearity properties of their induced basic tools, we shall briefly list some basic properties of translation, balanced and convex relations. Moreover, we shall greatly improve and supplement some relevant former results on relators. Contents Introduction 2 1. A few basic facts on relations 3 2. Translation relations on vector spaces 5 3. Balanced and convex relations 7 4. Relators and their induced basic tools 10 5. Fundamental properties of the basic tools 12 6. Set-valued functions and unary operations for relators 15 7. Some important unary operations for relators 18 8. Reflexive and topological relators 21 9. Proximal and well-chained relators 23 10. Transitive and filtered relators 26 11. Symmetric and separating relators 29 12. The definition of vector relators 32 1991 Mathematics Subject Classification. Primary 54E15; Secondary 46A19. Key words and phrases. Translation and balanced relations. Relators (families of relations) and their induced basic tools (interiors, open and fat sets, and convergences). Refinements and classifications of relators. Preseminorms and vector relators (alternatives for vector topologies). Typeset by A M S-T E X 1