MAT-KOL (Banja Luka) ISSN: 0354-6969 (p), ISSN: 1986-5228 (o) http://www.imvibl.org/dmbl/dmbl.htm Vol. XXIII (1)(2017), 21-25 DOI: 10.7251/1701021R FINITELY DUAL QUASI-NORMAL RELATION Daniel Abraham Romano Abstract. In this paper, following Jiang Guanghao and Xu Luoshan’s con- cepts of finitely conjugative and finitely dual normal relations on sets, the concept of finitely dual quasi-normal relations is introduced. A characteriza- tion of that relations is obtained. 1. Introduction and Preliminaries In this article, following concepts of finitely conjugative relations ([1], Jiang Guanghao and Xu Luoshan), finitely dual normal relations ([2], Jiang Guanghao and Xu Luoshan) and finitely quasi-conjugative relations ([5], D.A.Romano and M.Vinˇ ci´ c) introduced in their articles, we introduce and analyze notion of finitely dual quasi-normal relations on sets For a set X, we call ρ a binary relation on X, if ρ ⊆ X × X. Let B(X) be denote the set of all binary relations on X. For α, β ∈B(X), define β ◦ α = {(x, z) ∈ X × X :(∃y ∈ X)((x, y) ∈ α ∧ (y,z) ∈ β)}. The relation β ◦ α is called the composition of α and β. It is well known that (B(X), ◦) is a semigroup. The latter family, with the composition, is not only a semigroup, but also a monoiod. Namely, Id X = {(x, x): x ∈ X} is its identity element. For a binary relation α on a set X, define α -1 = {(x, y) ∈ X × X :(y,x) ∈ α} and α C =(X × X)  α. Let A and B be subsets of X. For α ∈B(X), set Aα = {y ∈ X :(∃a ∈ A)((a, y) ∈ α)}, αB = {x ∈ X :(∃b ∈ B)((x, b) ∈ α)}. It is easy to see that Aα = α -1 A holds and (α C ) -1 =(α -1 ) C . Specially, we put aα instead of {a}α and αb instead of α{b}. The following classes of elements in the semigroup B(X) have been investigated: – dually normal ([2]) if there exists a relation β ∈B(X) such that 2010 Mathematics Subject Classification. 20M20, 03E02, 06A11. Key words and phrases. relation on set, dually quasi-normal relations, finitely dual quasi- normal relations. 21