arXiv subjects

Sergey V. Gusev

Publications and source records attributed to Sergey V. Gusev.

25 records · Page 2Linked to original sources

arXiv2022

Limit varieties of monoids satisfying a certain identity

A limit variety is a variety that is minimal with respect to being non-finitely based. Since the turn of the millennium, much attention has been given to the classification of limit varieties of aperiodic monoids. Seven explicit examples have so far been found, and the task of locating other examples has recently been reduced to two subproblems, one of which is concerned with monoids that satisfy the identity $xsxt \approx xsxtx$. In the present article, we provide a complete solution to this subproblem by showing that there are precisely two limit varieties that satisfy this identity. One of them turns out to be the first example having infinitely many subvarieties. It is also deduced that the variety generated by any monoid of order five or less contains at most countably many subvarieties.

math.GR↗

arXiv2022

Distributive and lower-modular elements of the lattice of monoid varieties

The sets of all neutral, distributive and lower-modular elements of the lattice of semigroup varieties are finite, countably infinite and uncountably infinite, respectively. In 2018, we established that there are precisely three neutral elements of the lattice of monoid varieties. In the present work, it is shown that the neutrality, distributivity and lower-modularity coincide in the lattice of monoid varieties. Thus, there are precisely three distributive and lower-modular elements of this lattice.

math.GR↗

arXiv2021

Cancellable elements of the lattice of monoid varieties

The set of all cancellable elements of the lattice of semigroup varieties has recently been shown to be countably infinite. But the description of all cancellable elements of the lattice $\mathbb{MON}$ of monoid varieties remains unknown. This problem is addressed in the present article. The first example of a monoid variety with modular but non-distributive subvariety lattice is first exhibited. Then a necessary condition of the modularity of an element in $\mathbb{MON}$ is established. These results play a crucial role in the complete description of all cancellable elements of the lattice $\mathbb{MON}$. It turns out that there are precisely five such elements.

math.GR↗

arXiv2020

Two weaker variants of congruence permutability for monoid varieties

We completely determine all varieties of monoids on whose free objects all fully invariant congruences or all fully invariant congruences contained in the least semilattice congruence permute. Along the way, we find several new monoid varieties with the distributive subvariety lattice (only a few examples of varieties with such a property are known so far).

math.GR↗

arXiv2020

Varieties of monoids with complex lattices of subvarieties

A variety is finitely universal if its lattice of subvarieties contains an isomorphic copy of every finite lattice. Examples of finitely universal varieties of semigroups have been available since the early 1970s, but it is unknown if there exists a finitely universal variety of monoids. The main objective of the present article is to exhibit the first examples of finitely universal varieties of monoids. The finite universality of these varieties is established by showing that the lattice of equivalence relations on every sufficiently large finite set is anti-isomorphic to some subinterval of the lattice of subvarieties.

math.GR↗

arXiv2018

Chain varieties of monoids

A variety of universal algebras is called a chain variety if its subvariety lattice is a chain. Non-group chain varieties of semigroups were completely classified by Sukhanov in 1982. Here we completely determine non-group chain varieties of monoids as algebras of tyoe (2,0).

math.GR↗

arXiv2017

Cancellable elements of the lattice of semigroup varieties

We completely determine all commutative semigroup varieties that are cancellable elements of the lattice SEM of all semigroup varieties. In particular, we prove that, for commutative varieties, the properties of being cancellable and modular elements of SEM are equivalent.

math.GR↗