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We completely determine all commutative semigroup varieties that are uppermodular elements of the lattice of all semigroup varieties. It is verified that if a semigroup variety is an upper-modular element of this lattice and different from the variety of all semigroups then it is a periodic variety and every nilsemigroup in the variety is commutative and satisfies the identity x 2 y = xy 2 .

A variety of universal algebras is called a chain variety if its subvariety lattice is a chain. Non-group chain varieties of semigroups were completely classified by Sukhanov in 1982. Here we completely determine non-group chain varieties of monoids as algebras of type (2, 0).

Abstract. We survey results concerning special elements of nine types (modular, lower-modular, upper-modular, cancellable, distributive, codistributive, standard, costandard and neutral elements) in the lattice of all semigroup varieties and certain its sublattices, mainly in the lattices of all commutative varieties and of all overcommutative ones. The most part of results were published in 2005-2012, a few results are unpublished so far. Several open questions are formulated.