Asian-European Journal of Mathematics Vol. 3, No. 4 (2010) 531–544 c  World Scientific Publishing Company DOI: 10.1142/S1793557110000416 A NEW EXAMPLE FOR MINIMALITY OF MONOIDS Firat Ate¸ s, Eylem G. Karpuz Balikesir University, Faculty of Art and Science, Mathematics Department Balikesir, 10145, Turkey firat@balikesir.edu.tr, eguzel@balikesir.edu.tr A. Dilek G¨ ung¨or, A. Sinan C ¸ evik ∗ Sel¸cuk University, Faculty of Science, Mathematics Department Konya, 42075, Turkey agungor@selcuk.edu.tr, sinan.cevik@selcuk.edu.tr Communicated by M. Arslanov Received September 10, 2009 Revised February 24, 2010 By considering the split extension of a free abelian monoid having finite rank by a finite monogenic monoid, the main purposes of this paper are to present examples of efficient monoids and, also, minimal but inefficient monoids. Although results presented in this paper seem as in the branch of pure mathemat- ics, they are actually related to applications of Combinatorial and Geometric Group- Semigroup Theory, especially computer science, network systems, cryptography and physics etc., which will not be handled here. Keywords : Efficiency; minimality; monogenic monoids; split extensions. AMS Subject Classification: 20L05, 20M05, 20M15, 20M50, 20M99 1. Introduction and Preliminaries For a finite presentation P =[x ; s] of a monoid M , the Euler characteristic is defined by χ(P )=1 −|x| + |s| and an upper bound of M is defined by δ(M )=1 − rk Z (H 1 (M ))+d(H 2 (M )). It is well known that χ(P ) ≥ δ(M ). With this background, we say that P is efficient if χ(P )= δ(M ) and M is efficient if it has an efficient presentation. Moreover a presentation, say P 0 , for M is called minimal if χ(P 0 ) ≤ χ(P ), for all presentations P of M . There is also interest in finding inefficient finitely presented monoids since if we can find a minimal presentation P 0 for M such that P 0 is not efficient then we have χ(P ) ≥ χ(P 0 ) >δ(M ), for all presentations P defining ∗ Corresponding author 531