Green's relations

In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951. John Mackintosh Howie, a prominent semigroup theorist, described this work as "so all-pervading that, on encountering a new semigroup, almost the first question one asks is 'What are the Green relations like?'" (Howie 2002). The relations are useful for understanding the nature of divisibility in a semigroup; they are also valid for groups, but in this case tell us nothing useful, because groups always have divisibility.

Instead of working directly with a semigroup S, it is convenient to define Green's relations over the monoid S¹. (S¹ is "S with an identity adjoined if necessary"; if S is not already a monoid, a new element is adjoined and defined to be an identity.) This ensures that principal ideals generated by some semigroup element do indeed contain that element. For an element a of S, the relevant ideals are:

• The principal left ideal generated by a: ${\displaystyle S^{1}a=\{sa\mid s\in S^{1}\}}$. This is the same as ${\displaystyle \{sa\mid s\in S\}\cup \{a\}}$, which is ${\displaystyle Sa\cup \{a\}}$.
• The principal right ideal generated by a: ${\displaystyle aS^{1}=\{as\mid s\in S^{1}\}}$, or equivalently ${\displaystyle aS\cup \{a\}}$.
• The principal two-sided ideal generated by a: ${\displaystyle S^{1}aS^{1}}$, or ${\displaystyle SaS\cup aS\cup Sa\cup \{a\}}$.