# Linear Inequations and Regularity Conditions on Ordered Semigroups

 Received: 27.II.2015, Accepted: 2.IX.2015 Abstract. As in the results proved by Bogdanovi et al. [S. Bogdanovic, M. Ciric, P. Stanimirovic and T. Petkovic, Linear Equations and Regularity Conditions on Semigroups, Semigroup Forum, 69, 2004, 63-74], all types of the regularity including regularity, $(m,n)$-regularity, and $(p,q,r)$-regularity of elements of an ordered semigroup defined by inequations, called linear inequations are determined. We prove that there are 16 types of the regularity of ordered semigroups defined by such inequations. Keywords: semigroup, ordered semigroup, free semigroup, linear inequation, regular, $(m,n)$-regular, $(p,q,r)$-regular Mathematics Subject Classification (2010): 06F05 Authors: Theerayoot Phochai, Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand Thawhat Changphas, Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand