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.