A note on commutative semigroups
Robert Schwabauer · Proceedings of the American Mathematical Society · 1969
From Theorem 19.1 of [l] it follows that the lattice of varieties (equational classes) of Abelian groups is distributive.We show that the lattice of varieties of commutative semigroups is not modular by considering an example from the dual lattice of closed commutative semigroup laws.We will assume and use without comment that any given set of laws contains the commutative and associative laws.The binary operation is denoted by juxtaposition.By (a, b), where a and b are positive integers, we will mean the term xayb.Let the sets A', B', and C of laws be given by the following table.