On the Subsemigroups of a Finite Cyclic Semigroup

• Journal title : Kyungpook mathematical journal
• Volume 54, Issue 4,  2014, pp.607-617
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2014.54.4.607
Title & Authors
On the Subsemigroups of a Finite Cyclic Semigroup
Dobbs, David Earl; Latham, Brett Kathleen;

Abstract
Let S = C(r,m), the finite cyclic semigroup with index r and period m. Each subsemigroup of S is cyclic if and only if either r = 1; r = 2; or r = 3 with m odd. For $\small{r{\neq}1}$, the maximum value of the minimum number of elements in a (minimal) generating set of a subsemigroup of S is 1 if r = 3 and m is odd; 2 if r = 3 and m is even; (r-1)/2 if r is odd and unequal to 3; and r/2 if r is even. The number of cyclic subsemigroups of S is $\small{r-1+{\tau}(m)}$. Formulas are also given for the number of 2-generated subsemigroups of S and the total number of subsemigroups of S. The minimal generating sets of subsemigroups of S are characterized, and the problem of counting them is analyzed.
Keywords
Finite cyclic semigroup;subsemigroup;minimal generating set;index;period;greatest common divisor;Frobenius number;$\small{{\tau}(n)}$;$\small{{\varphi}(n)}$;
Language
English
Cited by
References
1.
R. Gilmer, Commutative Semigroup Rings, Univ. Chicago Press, Chicago, 1984.

2.