Related papers: Varieties of aperiodic monoids with commuting idem…
We completely classify all varieties of aperiodic monoids with central idempotents whose subvariety lattice is distributive.
A monoid is aperiodic if all its subgroups are trivial. We completely classify all varieties of aperiodic monoids whose subvariety lattice is distributive.
A variety of algebras is called Cross if it is finitely based, finitely generated, and has finitely many subvarieties. In present article, we classify all Cross varieties of aperiodic monoids with commuting idempotents.
A variety of algebras is called limit if it is non-finitely based but all its proper subvarieties are finitely based. A monoid is aperiodic if all its subgroups are trivial. We classify all limit varieties of aperiodic monoids with…
We classify all varieties of aperiodic monoids with central idempotents whose subvariety lattice is finite or satisfies the descending chain condition or satisfies the ascending chain condition. It turns out that for varieties in this…
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…
We prove a representation theorem for totally ordered idempotent monoids via a nested sum construction. Using this representation theorem we obtain a characterization of the subdirectly irreducible members of the variety of semilinear…
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…
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…
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…
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…
Let M be a commutative monoid. We provide an explicit first-order formular that defines the variety generated by M in the lattice of commutative semigroup varieties.
In this paper subvarieties of pseudocomplemented distributive lattices are classified by their unification type. We determine the unification type of every particular unification problem in each subvariety of pseudocomplemented distributive…
An element $x$ of a lattice $L$ is modular if $L$ has no five-element sublattice isomorphic to the pentagon in which $x$ would correspond to the lonely midpoint. In the present work, we classify all modular elements of the lattice of all…
We study the equational theories and bases of meets and joins of several varieties of plactic-like monoids. Using those results, we construct sublattices of the lattice of varieties of monoids, generated by said varieties. We calculate the…
We survey results devoted to the lattice of varieties of monoids. Along with known results, some unpublished results are given with proofs. A number of open questions and problems are also formulated.
Finite monoids that generate monoid varieties with uncountably many subvarieties seem rare, and surprisingly, no finite monoid is known to generate a monoid variety with countably infinitely many subvarieties. In the present article, it is…
A rotational lattice is a structure (L;\vee,\wedge, g) where L=(L;\vee,\wedge) is a lattice and g is a lattice automorphism of finite order. We describe the subdirectly irreducible distributive rotational lattices. Using J\'onsson's lemma,…
We completely classify all standard elements in the lattice of all monoid varieties. In particular, we prove that an element of this lattice is standard if and only if it is neutral.
Divisibility monoids (resp. Garside monoids) are a natural algebraic generalization of Mazurkiewicz trace monoids (resp. spherical Artin monoids), namely monoids in which the distributivity of the underlying lattices (resp. the existence of…