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.
math.GR↗
arXiv subjects
D. V. Skokov
Publications and source records attributed to D. V. Skokov.
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.
math.GR↗
Chains and anti-chains in the lattice of epigroup varieties
Let $\mathcal E_n$ be the variety of all epigroups of index $\le n$. We prove that, for an arbitrary natural number $n$, the interval $[\mathcal E_n, \mathcal E_{n+1}]$ of the lattice of epigroup varieties contains a chain isomorphic to the chain of real numbers with the usual order and an anti-chain of the cardinality continuum.
math.GR↗