Monogenic semigroup

In mathematics, a monogenic semigroup is a semigroup generated by a set containing only a single element.^[1] Monogenic semigroups are also called cyclic semigroups.^[2]

Structure

The monogenic semigroup generated by the singleton set { a } is denoted by $\langle a \rangle$. The set of elements of $\langle a \rangle$ is { a, a², a³, ... }. There are two possibilities for the monogenic semigroup $\langle a \rangle$:

• a ^m = a ⁿ ⇒ m = n.
• There exist m ≠ n such that a ^m = a ⁿ.

In the former case $\langle a \rangle$ is isomorphic to the semigroup ( {1, 2, ... }, + ) of natural numbers under addition. In such a case, $\langle a \rangle$ is an infinite monogenic semigroup and the element a has infinite order. In the latter case let m be the smallest positive integer such that a ^m = a ^x for some positive integer x ≠ m, and let r be smallest positive integer such that a ^m = a ^{m + r}. The positive integer m is referred to as the index and the positive integer r as the period of the monogenic semigroup $\langle a \rangle$. The period and the index satisfy the following properties:

• a ^m = a ^{m + r}
• a ^{m + x} = a ^{m + y} if and only if m + x ≡ m + y ( mod r )
• $\langle a \rangle$ = { a, a², ... , a ^{m + r − 1} }
• K_a = { a^m, a ^{m + 1}, ... , a ^{m + r − 1} } is a cyclic subgroup of $\langle a \rangle$.

The pair ( m, r ) of positive integers determine the structure of monogenic semigroups. For every pair ( m, r ) of positive integers, there does exist a monogenic semigroup having index m and period r. The monogenic semigroup having index m and period r is denoted by M ( m, r ). The monogenic semigroup M ( 1, r ) is the cyclic group of order r.

See also

• Special classes of semigroups

References

1. ^ Howie, J M (1976). An Introduction to Semigroup Theory. L.M.S. Monographs. 7. Academic Press. pp. 7–11. ISBN 0123569508. 
2. ^ A H Clifford; G B Preston (1961). The Algebraic Theory of Semigroups Vol.I. Mathematical Surveys. 7. American Mathematical Society. pp. 19–20. ISBN ISBN 978-0821802724.