SEMIGROUPS SATISFYING CERTAIN REGULARITY CONDITIONS * STOJAN BOGDANOVI ´ C Faculty of Economics, University of Niˇ s, Trg Kralja Aleksandra 11, P. O. Box 121, 18000 Niˇ s, Serbia E-mail: sbogdan@pmf.ni.ac.yu MIROSLAV ´ CIRI ´ C Faculty of Sciences and Mathematics, University of Niˇ s, Viˇ segradska 33, P. O. Box 224, 18000 Niˇ s, Serbia E-mail: ciricm@bankerinter.net MELANIJA MITROVI ´ C Faculty of Mechanical Engineering, University of Niˇ s, Beogradska 14, 18000 Niˇ s, Serbia E-mail: meli@masfak.masfak.ni.ac.yu To the memory of Professor B. H. Neumann S. Bogdanovi´ c, M. ´ Ciri´ c, P. Stanimirovi´ c and T. Petkovi´ c in [4] proved that every regularity condition on semigroups is equivalent to one of the 14 conditions given here in Figure 1. Semigroups satisfying some of these conditions were studied only from the ideal-theoretical point of view, by S. Lajos and G. Sz´asz in [11]. The pur- pose of this paper is to describe the structure and give some new ideal-theoretical characterizations of these semigroups. 1. Introduction The notion of the regularity in semigroups and rings was introduced by J. von Neumann in [13], 1936, and his work initiated investigation of many other types of the regularity. Left, right and complete regularity were intro- duced and studied by A. H. Clifford in [6], 1941, and R. Croisot in [8], 1953, * This work is supported by grant MM-1227 of the Ministry of Science and Technologies of Republic of Serbia 46