On minimal simple lattices
The equational (pre)ordering of lattices \(L_ 0\), \(L_ 1\) is defined by \(L_ 0\leq L_ 1\) iff \(L_ 0\in \text{HSP}(L_ 1)\) and \(L_ 0\equiv L_ 1\) iff \(\text{HSP}(L_ 0)= \text{HSP}(L_ 1)\). Here, \(\text{HSP}(L_ 1)\) is the equational class (variety) generated by \(L_ 1\). \(S\) \((^ b S)\) are all simple lattices \(L\) with \(| L|> 2\) (which have bounds 0, 1, \(^ b S\subseteq S\)). \(S_{\min}\), \(^ b S_{\min}\) are the subsets of \(S\), \(^ b S\) of minimal elements under the above preordering. There exists an infinite member \(L\) of \(S_{\min}\), without 0, 1, which is (up to isomorphism) the only member of \(S\cap \text{HSP}(L)\). There exists \(L\in S\) with \(\text{HSP}(L)\cap{^ b S}=\emptyset\) and \(\text{HSP}(L)\cap S_{\min}= \emptyset\). There exists an infinite five-generated lattice in \(S_{\min}\) (and hence in \(^ b S_{\min}\)). If \(L\in S\) is \(n\)- generated (\(n\)-finite), then there exists an \(n\)-generated \(L'\in S\) with \(L'\leq L\) and \(L'\) is \(\leq\)-minimal in the class of all \(n\)-generated members of \(S\). There exists a seven-generated \(L\in {^ b S}\) with \(^ b S_{\min}\cap \text{HSP}(L)= \emptyset\). The variety generated by the class of finite members of \(S_{\min}\) is the class of all lattices. Some open questions are: Is \(^ b S_{\min}- S_{\min}\) non-empty? Does there exist a member of \(^ b S\) that dominates no member of \(S_{\min}\) in the equational ordering?.
scientific article; zbMATH DE number 679610