The lattice of varieties of monoids
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 Along with known results, some unpublished results are given with proofs.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.
math.GR↗
arXiv subjects
Boris M. Vernikov
Publications and source records attributed to Boris M. Vernikov.
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).
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Cancellable elements in the lattice of overcommutative semigroup varieties
We completely determine all cancellable elements in the lattice OC of overcommutative semigroup varieties. In particular, we prove that an overcommutative semigroup variety is a cancellable element of the lattice OC if and only if it is a neutral element of this lattice
math.GR↗
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).
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Cancellable elements of the lattice of semigroup varieties: varieties satisfying a permutational identity of length 3
We completely determine all semigroup varieties satisfiyng a permutational identity of length 3 that are cancellable elements of the lattice of all semigroup varieties. Using this result, we provide a series of new examples of semigroup varieties that are a modular but not cancellable elements of this lattice.
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On modular and cancellable elements of the lattice of semigroup varieties
We completely determine all semigroup varieties satysfiyng a permutational identity of length 3 that are modular elements of the lattice of all semigroup varieties. Using this result, we provide an example of a semigroup variety that is a modular but not cancellable element of this lattice.
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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↗