J. M. Howie has shown that the finite monogenic semigroups are absolutely closed. We provide a new, simple and direct proof of Howie’s result. We also show that all the varieties defined by the identities ax = axa , axy = axay and axy = axyay are saturated if and only if they are epimorphically closed.

中文翻译：

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有限单基因半群和半群的饱和变体

JM Howie 证明了有限单基因半群是绝对闭的。我们提供了 Howie 结果的新的、简单的和直接的证明。我们还表明，由恒等式 ax = axa 、 axy = axay 和 axy = axyay 定义的所有变体都是饱和的，当且仅当它们是外胚闭的。