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.Comment: 5 pages, 2 figure

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

Endomorphisms of the lattice of epigroup varieties

We examine varieties of epigroups as unary semigroups, that is semigroups equipped with an additional unary operation of pseudoinversion. The article contains two main results. The first of them indicates a countably infinite family of injective endomorphisms of the lattice of all epigroup varieties. An epigroup variety is said to be a variety of finite degree if all its nilsemigroups are nilpotent. The second result of the article provides a characterization of epigroup varieties of finite degree in a language of identities and in terms of minimal forbidden subvarieties. Note that the first result is essentially used in the proof of the second one.Comment: In comparison with the previous version, we eliminate a few typos onl

On congruences of GG-sets

summary:We describe $G$-sets whose congruences satisfy some natural lattice or multiplicative restrictions. In particular, we determine $G$-sets with distributive, arguesian, modular, upper or lower semimodular congruence lattice as well as congruence $n$-permutable $G$-sets for $n=2,2.5,3$

The lattice of varieties of implication semigroups

In 2012, the second author introduced and examined a new type of algebras as a generalization of De Morgan algebras. These algebras are of type (2,0) with one binary and one nullary operation satisfying two certain specific identities. Such algebras are called implication zroupoids. They invesigated in a number of articles by the second author and J.M.Cornejo. In these articles several varieties of implication zroupoids satisfying the associative law appeared. Implication zroupoids satisfying the associative law are called implication semigroups. Here we completely describe the lattice of all varieties of implication semigroups. It turns out that this lattice is non-modular and consists of 16 elements.Comment: Compared with the previous version, we rewrite Section 3 and add Appendixes A and

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.