# Green equivalence relations

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Binary relations , , , , defined as follows: means that and generate identical left principal ideals (cf. Principal ideal); and have a similar meaning after "left" has been replaced by "right" and "two-sided" , respectively; (union in the lattice of equivalence relations); . The relations and are commutative in the sense of multiplication of binary relations, so that coincides with their product. The relation is a right congruence, i.e. is stable from the right: implies for all ; the relation is a left congruence (stable from the left). An -class and an -class intersect if and only if they are contained in the same -class. All -classes in the same -class are equipotent. If a -class contains a regular element, then all elements in are regular and contains with some given element all elements inverse to it; such a -class is said to be regular. In a regular -class each -class and each -class contains an idempotent. Let be an arbitrary -class; then either is a group (which is the case if and only if is a maximal subgroup of the given semi-group), or else . All group -classes of the same -class are isomorphic groups. In the general case , but if, for example, some power of each element of the semi-group belongs to a subgroup (in particular, if is a periodic semi-group), then . The inclusion of principal left ideals defines in a natural manner a partial order relation on the set of -classes; similar considerations are valid for -classes and -classes. These relations were introduced by J. Green .