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Concept
Green's relations
Summary
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: . 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 .
For elements a and b of S, Green's relations L, R and J are defined by
a L b if and only if S1 a = S1 b.
a R b if and only if a S1 = b S1.
a J b if and only if S1 a S1 = S1 b S1.
That is, a and b are L-related if they generate the same left ideal; R-related if they generate the same right ideal; and J-related if they generate the same two-sided ideal. These are equivalence relations on S, so each of them yields a partition of S into equivalence classes. The L-class of a is denoted La (and similarly for the other relations). The L-classes and R-classes can be equivalently understood as the strongly connected components of the left and right Cayley graphs of S1. Further, the L, R, and J relations define three preorders ≤L, ≤R, and ≤J, where a ≤J b holds for two elements a and b of S if the ideal generated by a is included in that of b, i.
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