2014/08/02 by V. Yu. Shaprynskiǐ, Shaprynskii, V. Yu., Dmitry V. Skokov +3 · 1 citation
Computer Science · Mathematics · #semigroups and automata theory #Advanced Algebra and Logic #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1408.0356
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.