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Let $S$ be a (commutative) semigroup with distinguished element 0 such that $ab=0$ for $a,b\in S.$ Of course this is a very simple family of semigroups, defined only by their cardinality. Does it have a name? Would it be reasonable to call these nilpotent (of order 1)?

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If $S$ is a semigroup, an element $0\in S$ such that $0x=x0=0$ for all $x\in S$ is called a "zero element." If $S$ is a semigroup with a zero, then $x\in S$ is said to be "nilpotent" if and only if there exists a natural number $n$ such that $x^n=0$.

The usual name for a semigroup in which $ab=0$ for all $a,b\in S$ is that $S$ is a "zero semigroup", or a "semigroup with zero multiplication".

"Nilpotent semigroup" is not a good name, because that is a semigroup $S$ (necessarily with $0$) for which there exists an $n\gt 0$ such that $S^n = \{0\}$, where $$S^n = \{ s_1\cdots s_n\mid s_i\in S\}$$ is the set of all products with $n$ factors, not necessarily distinct. Calling $S$ a "nilpotent semigroup of order $1$" would be interpreted as a rather roundabout way of saying that $S$ is a semigroup with one element (since it would imply that $|S|=1$ and $S$ is "nilpotent", or perhaps that $S^1=\{s\mid s\in S\} = \{0\}$, which in fact amounts to the same thing).

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That's just what I was looking for, thanks. –  Charles Mar 28 '11 at 18:07
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