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On the Subsemigroups of a Finite Cyclic Semigroup

Dobbs, David Earl;Latham, Brett Kathleen

  • Received : 2012.08.04
  • Accepted : 2012.12.14
  • Published : 2014.12.23

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 $r{\neq}1$, 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 $r-1+{\tau}(m)$. 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;${\tau}(n)$;${\varphi}(n)$

References

  1. R. Gilmer, Commutative Semigroup Rings, Univ. Chicago Press, Chicago, 1984.
  2. W. J. LeVeque, Fundamentals of Number Theory, Addison-Wesley, Reading, Mass., 1977.