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An algebraic system is simple if the only congruences on it are and , the diagonal relation.
For example, let's find out what are the simple algebras in the class of groups. Let be a group that is simple in the sense defined above.
First, what are the congruences on ? A congruence on is a subgroup of and an equivalence relation on at the same time. As an equivalence relation, corresponds to a partition of in the following manner:
and
, where
for . Each of the is an equivalence class of . Let be the equivalence class containing . If , then
, so that
, or . In addition,
, so
. is a subgroup of . Furthermore, if ,
, so that
, is a normal subgroup of . Conversely, given a normal subgroup of , forming left (right) cosets of , and taking
gives us the congruence on .
Now, if is simple, then this says that the only congruences on are and , which corresponds to having and
as the only normal subgroups. So, as a simple algebra is just a simple group. Conversely, if is a simple group, then the only congruences on are those corresponding to and
, the only normal subgroups of . Therefore, a simple group is a simple algebra.
Remark. Any simple algebraic system is subdirectly irreducible.
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