论文标题
半群是有限的，并且仅当它是链条和抗抗素的时
A semigroup is finite if and only if it is chain-finite and antichain-finite
论文作者
论文摘要
如果任何（不同的）元素$ x，y \ in s $，则$ xy \ in \ xy \ in \ {x，y \} $（$ xy \ notin \ {x，y \} $）的子集$ a $称为$链$（$ antichain $）。如果$ s $包含无限（抗）链，则$ s $ $ s $称为（$ anti $）$链条$ - $有限$。 We prove that each antichain-finite semigroup $S$ is periodic and for every idempotent $e$ of $S$ the set $\sqrt[\infty]{e}=\{x\in S:\exists n\in\mathbb N\;\;(x^n=e)\}$ is finite.抗菌素半群的这种特性用于证明半群是有限的，并且仅当它是链条和抗抗细胞时。另外，我们提供了一个敌人半层次的示例，这不是有限的链条的结合。
A subset $A$ of a semigroup $S$ is called a $chain$ ($antichain$) if $xy\in\{x,y\}$ ($xy\notin\{x,y\}$) for any (distinct) elements $x,y\in S$. A semigroup $S$ is called ($anti$)$chain$-$finite$ if $S$ contains no infinite (anti)chains. We prove that each antichain-finite semigroup $S$ is periodic and for every idempotent $e$ of $S$ the set $\sqrt[\infty]{e}=\{x\in S:\exists n\in\mathbb N\;\;(x^n=e)\}$ is finite. This property of antichain-finite semigroups is used to prove that a semigroup is finite if and only if it is chain-finite and antichain-finite. Also we present an example of an antichain-finite semilattice that is not a union of finitely many chains.
