生成非Cross簇的最小幺半群 [The smallest monoid that generates a non-Cross variety]

一个幺半群簇是一个在同态像、子幺半群和任意直积运算下封闭的幺半群类。一个有限生成的、有限基的且包含有限多个子簇的幺半群簇称作Cross幺半群簇。证明了5阶幺半群M₅生成一个非Cross幺半群簇。通过逐个验证，在同构和反同构意义下，除M₅外的所有阶数小于等于5的幺半群都生成Cross幺半群簇。在同构和反同构意义下，M₅是生成非Cross幺半群簇的唯一最小幺半群。

A monoid variety is a class of monoids that is closed under the formation of homomorphic images, submonoids, and arbitrary direct products. A finitely generated, finitely based variety of monoids that contains only finitely many subvarieties is a Cross monoid variety. The 5-element monoid M₅ is first shown to generate a non-Cross monoid variety. Then it is shown that, up to isomorphism and anti-isomorphism, any monoid of order five or less different from M₅ generates a Cross monoid variety. Therefore up to isomorphism and anti-isomorphism, M₅ is the only smallest monoid that generates a non-Cross monoid variety.