Covering relation for equational theories of commutative semigroups (Q5925810)

The purpose of this paper is to give an effective description of the covering relation in \(L(\text{Com})\), the lattice of equational theories of commutative semigroups which is dually isomorphic to the lattice of varieties of commutative semigroups. In an earlier paper the second author characterized such equational theories by a quadruple consisting of one positive integer \(r\) and one nonnegative integer \(m\), a non-empty filter \(J\) contained in the filter generated by \(m\), and an equivalence relation \(\pi\) on the set of finite sequences of positive integers that are not in \(J\). If a theory \(E_2\) covers \(E_1\) then \(E_1\) is a dual cover for \(E_2\).NEWLINENEWLINENEWLINEThere are two main results; the first shows that \(E_1\) is a dual cover for \(E\) iff the quadruple for \(E_1\) is of one of four types with respect to that of \(E\) while the second theorem is a similar result for covers. Some examples are given and it transpires that all 16 possibility types for covering really occur.