# Green equivalence relations

Binary relations $\mathcal L$, $\mathcal R$, $\mathcal J$, $\mathcal D$, $\mathcal H$ defined as follows: $x\mathcal Ly$ means that $x$ and $y$ generate identical left principal ideals (cf. Principal ideal); $x\mathcal Ry$ and $x\mathcal Jy$ have a similar meaning after "left" has been replaced by "right" and "two-sided", respectively; $\mathcal D=\mathcal{L}\lor\mathcal R$ (union in the lattice of equivalence relations); $\mathcal H=\mathcal L\cap\mathcal R$. The relations $\mathcal L$ and $\mathcal R$ are commutative in the sense of multiplication of binary relations, so that $\mathcal D$ coincides with their product. The relation $\mathcal L$ is a right congruence, i.e. is stable from the right: $a\mathcal Lb$ implies $ac\mathcal Lbc$ for all $c$; the relation $\mathcal R$ is a left congruence (stable from the left). An $\mathcal L$-class and an $\mathcal R$-class intersect if and only if they are contained in the same $\mathcal D$-class. All $\mathcal H$-classes in the same $\mathcal R$-class are equipotent. If a $\mathcal D$-class $D$ contains a regular element, then all elements in $D$ are regular and $D$ contains with some given element all elements inverse to it; such a $\mathcal D$-class is said to be regular. In a regular $\mathcal D$-class each $\mathcal L$-class and each $\mathcal R$-class contains an idempotent. Let $H$ be an arbitrary $\mathcal H$-class; then either $H$ is a group (which is the case if and only if $H$ is a maximal subgroup of the given semi-group), or else $H\cap H^2=\emptyset$. All group $\mathcal H$-classes of the same $\mathcal D$-class are isomorphic groups. In the general case $\mathcal D\neq\mathcal J$, but if, for example, some power of each element of the semi-group $S$ belongs to a subgroup (in particular, if $S$ is a periodic semi-group), then $\mathcal D=\mathcal J$. The inclusion of principal left ideals defines in a natural manner a partial order relation on the set of $\mathcal L$-classes; similar considerations are valid for $\mathcal R$-classes and $\mathcal J$-classes. These relations were introduced by J. Green .