S1 ⊆ S1 b S1, and ≤L and ≤R are defined analogously. 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. The H and D relations The remaining relations are derived from L and R. Their intersection is H: a H b if and only if a L b and a R b. This is also an equivalence relation on S. The class Ha is the intersection of La and Ra. More generally, the intersection of any L-class with any R-class is either an H-class or the empty set. Green's Theorem states that for any -class H of a semigroup S either (i) or (ii) and H is a subgroup of S. An important corollary is that the equivalence class He, where e is an idempotent, is a subgroup of S (its identity is e, and all elements have inverses), and indeed is the largest subgroup of S containing e. No -class can contain more than one idempotent, thus is idempotent separating. In a monoid M, the class H1 is traditionally called the group of units. (Beware that unit does not mean identity in this context, i.e. in general there are non-identity elements in H1. The "unit" terminology comes from ring theory.) For example, in the transformation monoid on n elements, Tn, the group of units is the symmetric group Sn. Finally, D is defined: a D b if and only if there exists a c in S such that a L c and c R b. In the language of lattices, D is the join of L and R. (The join for equivalence relations is normally more difficult to define, but is simplified in this case by the fact that a L c and c R b for some c if