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. (In the same way, the ideals of a field are a much less rich environment for study than the ideals of a ring.)

Instead of working directly with a semigroup S, we 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: . This is the same as, which is .
• The principal right ideal generated by a:, or equivalently .
• The principal two-sided ideal generated by a:, or .