Types of varieties of recognizable \(\omega\)-languages and Eilenberg correspondences. (Q1431401)

A class \(\underline V\) of finite monoids is called an \(M\)-variety if it satisfies the condition: for any monoids \(M,N\in\underline V\) and any monoid \(P\), if \(P\prec M\times N\) then also \(P \in\underline V\). Here \(P\prec M\times N\) means that \(P\) is a homomorphic image of a submonoid of the direct product \(M\times N\). The author then defines \(\omega\)-varieties and \(L\)-varieties that are related to recognizable subsets and Boolean closures, and it is shown that the correspondences \(\underline V\Rightarrow V^\omega\) and \(\underline V\Rightarrow\vec V\) from the class of \(M\)-varieties to the class of all \(\omega\)-varieties and the class of \(L\)-varieties of recognizable \(\omega\)-languages, respectively, are one-to-one.