Green equivalence relations
on a semi-group
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have a similar meaning after "left" has been replaced by "right" and "two-sided" , respectively;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 [1].
References
[1] | J. Green, "On the structure of semigroups" Ann. of Math. , 54 (1951) pp. 163–172 |
[2] | E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian) |
[3] | A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967) |
[4] | , The algebraic theory of automata, languages and semi-groups , Moscow (1975) (In Russian; translated from English) |
[5] | K.H. Hofmann, P.S. Mostert, "Elements of compact semigroups" , C.E. Merrill (1966) |
Green equivalence relations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Green_equivalence_relations&oldid=16826