On the Subsemigroups of a Finite Cyclic Semigroup
David E. Dobbs, Brett Kathleen Latham · Kyungpook mathematical journal · 2014
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 = 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+τ (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.