Some Order Properties of the Lattice of Varieties of Commutative Semigroups

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Abstract
The most complete work on the structure of the lattice of varieties of commutative semigroups available to this date is [12]. Nevertheless, it fails to give the structure of this lattice. In the positive direction, it shows in particular that the order structure of is determined by the order structure of well-known lattices of integers together with the sublattice of varieties of commutative nil semigroups.In the present work, we study from the point of view of order. Perkins [13] has shown that has no infinite descending chains and is countable. The underlying questions we consider here arose from the results of Almeida and Reilly [1] in connection with generalized varieties. There, it is observed that the best-known part of consisting of the varieties all of whose elements are abelian groups is in a sense very wide: it contains infinite subsets of mutually incomparable elements and allows the construction of uncountably many generalized varieties and infinite descending chains of generalized varieties.

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