Green's Relations - The L, R, and J Relations

The L, R, and J Relations

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 L_a (and similarly for the other relations).

Green used the lowercase blackletter, and for these relations, and wrote for a L b (and likewise for R and J). Mathematicians today tend to use script letters such as instead, and replace Green's modular arithmetic-style notation with the infix style used here. Ordinary letters are used for the equivalence classes.

The L and R relations are left-right dual to one another; theorems concerning one can be translated into similar statements about the other. For example, L is right-compatible: if a L b and c is another element of S, then ac L bc. Dually, R is left-compatible: if a R b, then ca R cb.

If S is commutative, then L, R and J coincide.