On the ascending and descending chain conditions in the lattice of monoid varieties

In this work we consider monoids as algebras with an associative binary operation and the nullary operation that fixes the identity element. We found an example of two varieties of monoids with finite subvariety lattices such that their join covers one of them and has a continuum cardinality subvariety lattice that violates the ascending chain condition and the descending chain condition.Comment: 15 page

On the lattice of overcommutative varieties of monoids

It is unknown so far, whether the lattice of all varieties of monoids satisfies some non-trivial identity. The objective of this note is to give the negative answer to this question. Namely, we prove that any finite lattice is a homomorphic image of some sublattice of the lattice of overcommutative varieties of monoids (i.e., varieties that contain the variety of all commutative monoids). This implies that the lattice of overcommutative varieties of monoids and therefore, the lattice of all varieties of monoids does not satisfy any non-trivial identity.Comment: 5 page

A new example of a limit variety of monoids

A variety of universal algebras is called limit if it is non-finitely based but all its proper subvarieties are finitely based. Until recently, only two explicit examples of limit varieties of monoids, constructed by Jackson, were known. Recently Zhang and Luo found the third example of such a variety. In our work, one more example of a limit variety of monoids is given.Comment: 16 page

Special elements of the lattice of monoid varieties

We completely classify all neutral or costandard elements in the lattice $\mathbb{MON}$ of all monoid varieties. Further, we prove that an arbitrary upper-modular element of $\mathbb{MON}$ except the variety of all monoids is either a completely regular or a commutative variety. Finally, we verify that all commutative varieties of monoids are codistributive elements of $\mathbb{MON}$. Thus, the problems of describing codistributive or upper-modular elements of $\mathbb{MON}$ are completely reduced to the completely regular case.Comment: 12 page

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