9,485 research outputs found
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
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