IDA: Interactive Document on Algebra

If D is a subset of the monoid M, then <D> is defined to be the set of elements of M that are products of elements of D.

The empty product is (by definition) the unit e of M; in this way, <D> is a submonoid of M. This submonoid is called the submonoid generated by D. The elements of D are called the generators of <D>.

Here is an abstract characterization of <D>.

Theorem

If D is a subset of the monoid M then

<D> is the smallest submonoid of M containing D;
<D> = _{H
C} H, where C is the collection of all submonoids of M containing D.

A monoid that can be generated by a single element is called cyclic. Let k, n N with n > 0. An example of a cyclic monoid is the monoid C_k,n = {e, c, ..., c^k+n-1} (with k + n elements) with multiplication defined as follows.

c^j * cⁱ = c^j+i, if j + i < k + n;
c^j * cⁱ = c^{k +
((j+i-k) mod n)}, if j + i > k + n - 1;
c⁰ = e is the unit.

Clearly, C_k,n is cyclic with generator c.

Characterization of cyclic monoids

Every cyclic monoid is isomorphic with either C_k,l for certain k, l N or (N,+,0).