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Let be a semigroup. If has no ideals other than itself, then is said to be simple.
If has no left ideals [resp. right ideals] other than itself, then is said to be left simple [resp. right simple].
Right simple and left simple are stronger conditions than simple.
A semigroup is left simple if and only if for all . A semigroup is both left and right simple if and only if it is a group.
If has a zero element , then
is always an ideal of , so is not simple (unless it has only one element). So in studying semigroups with a zero, a slightly weaker definition is required.
Let be a semigroup with a zero. Then is zero simple, or 0-simple, if the following conditions hold:
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has no ideals except 0 and itself
The condition really only eliminates one semigroup: the 2-element null semigroup. Excluding this semigroup makes parts of the structure theory of semigroups cleaner.
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