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Commutative zeropotent semigroups with few prime ideals. (English) Zbl 1249.20061

This paper is a contribution to the open problem: Is there an infinite commutative semigroup with a finite endomorphism monoid? Let \(S\) be a commutative semigroup satisfying the identity \(x+x=y+y+y\), a subset \(I\subseteq S\) is an ideal containing \(0\) (\(0\) is a zero of \(S\), hence \(0=x+x\) for some \(x\in S\)) and \(x+y\in I\) whenever \(x\in I\) or \(y\in I\). An ideal \(I\) is prime whenever \(x+y\in I\) for \(x,y\in S\) implies \(x+y=0\) or \(x\in I\) or \(y\in I\). If \(I\subseteq S\) is a prime ideal then a mapping \(\varphi\colon S\to S\) such that \(\varphi(x)=0\) for \(x\in I\) and \(\varphi(x)=x\) for \(x\in S\setminus I\) is an endomorphism of \(S\). The paper presents an infinite commutative semigroup satisfying the identity \(x+x=y+y+y\) and containing exactly two prime ideals.

MSC:

20M14 Commutative semigroups
20M12 Ideal theory for semigroups
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