PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 93 (107) (2013), 127–132 DOI: 10.2298/PIM1307127R QUASI-REGULAR RELATIONS – A NEW CLASS OF RELATIONS ON SETS Daniel A. Romano Communicated by Siniša Crvenković Abstract. Following Jiang Guanghao and Xu Luoshan’s concept of conjuga- tive, dually conjugative, normal and dually normal relations on sets, the con- cept of quasi-regular relations is introduced. Characterizations of quasi-regular relations are obtained and it is shown when an anti-order relation is quasi- regular. Some nontrivial examples of quasi-regular relations are given. At the end we introduce dually quasi-regular relations and give a connection between these two types of relations. 1. Introduction and Preliminaries The regularity of binary relations was first characterized by Zareckiˇ ı[9]. Fur- ther criteria for regularity were given by Hardy and Petrich [3], Markowsky [7], Schein [8] and Xu Xiao-quan and Liu Yingming [11] (see also [1] and [2]). The concepts of conjugative relations, dually conjugative relations and dually normal relations were introduced by Guanghao Jiang and Luoshan Xu [4], [5], and a char- acterization of normal relations was introduced and analyzed by Jiang Guanghao, Xu Luoshan, Cai Jin and Han Guiwen [6]. In this paper, we introduce and analyze the so-called quasi-regular relations on sets. Notions and notations which are not explicitly exposed but are used in this article, readers can find e.g., in [3] and [11]. For a set X , we call ρ a binary relation on X , if ρ ⊆ X × X . Let B(X ) 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. 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. 2010 Mathematics Subject Classification: Primary 06A11; 06B35. 127