6 Subgroups of Semigroups
In this chapter, we define the notion of a subgroup within a semigroup. Unlike the standard definition in group theory where subgroups are subsets of groups, here we consider subgroups as subsets of semigroups that themselves form groups under the inherited operation.
Let \(S\) be a semigroup. A subgroup of \(S\) is a subset \(H \subseteq S\) equipped with:
A carrier set \(H \subseteq S\);
Closure under multiplication: For all \(a, b \in H\), we have \(a \cdot b \in H\);
An identity element \(e \in H\) such that:
\(e \in H\) (the identity is in the carrier);
For all \(x \in H\), \(e \cdot x = x\) (left identity);
For all \(x \in H\), \(x \cdot e = x\) (right identity);
An inverse function \(\mathrm{inv} : S \to S\) such that:
For all \(x \in H\), \(\mathrm{inv}(x) \in H\) (closure under inverses);
For all \(x \in H\), \(\mathrm{inv}(x) \cdot x = e\) (left inverse);
For all \(x \in H\), \(x \cdot \mathrm{inv}(x) = e\) (right inverse).
With these properties, the carrier \(H\) forms a group under the multiplication inherited from \(S\).
Note that the identity element \(e\) is uniquely determined by the carrier set: if \(e_1\) and \(e_2\) are both identities for the same carrier \(H\), then \(e_1 = e_1 \cdot e_2 = e_2\). Similarly, the inverse function is uniquely determined by the carrier and identity.
A subgroup \(H\) of a semigroup \(S\) is called maximal if it is not properly contained in any other subgroup of \(S\). That is, \(H\) is maximal if for every subgroup \(K\) of \(S\) with \(H \subseteq K\), we have \(H = K\).