Monogenic semigroup

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:

• am = an ⇒ m = n.
• There exist m ≠ n such that am = an. 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 is said to have infinite order. It is sometimes called the free monogenic semigroup because it is also a free semigroup with one generator.

In the latter case let m be the smallest positive integer such that am = ax for some positive integer x ≠ m, and let r be smallest positive integer such that am = 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 order of a is defined as m+r−1. The period and the index satisfy the following properties:

• am = 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 = {am, a**m+1, ... , a**m+r−1} is a cyclic subgroup and also an ideal of \langle a \rangle. It is called the kernel of a and it is the minimal ideal of the monogenic semigroup \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 exists 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.

The results in this section actually hold for any element a of an arbitrary semigroup and the monogenic subsemigroup \langle a \rangle it generates.

Related notions

A related notion is that of periodic semigroup (also called torsion semigroup), in which every element has finite order (or, equivalently, in which every monogenic subsemigroup is finite). A more general class is that of quasi-periodic semigroups (aka group-bound semigroups or epigroups) in which every element of the semigroup has a power that lies in a subgroup.

An aperiodic semigroup is one in which every monogenic subsemigroup has a period of 1.

References

References

1. Howie, J M. (1976). "An Introduction to Semigroup Theory". Academic Press.
2. A H Clifford. (1961). "The Algebraic Theory of Semigroups Vol.I". American Mathematical Society.
3. "Kernel of a semi-group - Encyclopedia of Mathematics".
4. "Minimal ideal - Encyclopedia of Mathematics".
5. "Periodic semi-group - Encyclopedia of Mathematics".
6. Peter M. Higgins. (1992). "Techniques of semigroup theory". Oxford University Press.