Lower-modular elements of the lattice of semigroup varieties. III

We completely determine all lower-modular elements of the lattice of all semigroup varieties. As a corollary, we show that a lower-modular element of this lattice is modular.Comment: 10 pages, 1 figur

Cancellable elements of the lattices of varieties of semigroups and epigroups

We completely determine all semigroup [epigroup] varieties that are cancellable elements of the lattice of all semigroup [respectively epigroup] varieties.Comment: 17 pages, 3 figures. Compared with the previous version, we add Corollary 1.4 and Figure 1 and fix several typos. arXiv admin note: text overlap with arXiv:1806.0597

Special elements of the lattice of epigroup varieties

We study special elements of eight types (namely, neutral, standard, costandard, distributive, codistributive, modular, lower-modular and upper-modular elements) in the lattice EPI of all epigroup varieties. Neutral, standard, costandard, distributive and lower-modular elements are completely determined. A strong necessary condition and a sufficient condition for modular elements are found. Modular elements are completely classified within the class of commutative varieties, while codistributive and upper-modular elements are completely determined within the wider class of strongly permutative varieties. It is verified that an element of EPI is costandard if and only if it is neutral; is standard if and only if it is distributive; is modular whenever it is lower-modular; is neutral if and only if it is lower-modular and upper-modular simultaneously. We found also an application of results concerning neutral and lower-modular elements of EPI for studying of definable sets of epigroup varieties.Comment: In comparison with the previous version, we slightly optimize the proof of Theorem 1.1, eliminate a few typos and add Question 11.