3,593 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
Special elements of the lattice of monoid varieties
We completely classify all neutral or costandard elements in the lattice
of all monoid varieties. Further, we prove that an arbitrary
upper-modular element of 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
. Thus, the problems of describing codistributive or
upper-modular elements of 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
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The lattice of varieties of implication semigroups
In 2012, the second author introduced and examined a new type of algebras as
a generalization of De Morgan algebras. These algebras are of type (2,0) with
one binary and one nullary operation satisfying two certain specific
identities. Such algebras are called implication zroupoids. They invesigated in
a number of articles by the second author and J.M.Cornejo. In these articles
several varieties of implication zroupoids satisfying the associative law
appeared. Implication zroupoids satisfying the associative law are called
implication semigroups. Here we completely describe the lattice of all
varieties of implication semigroups. It turns out that this lattice is
non-modular and consists of 16 elements.Comment: Compared with the previous version, we rewrite Section 3 and add
Appendixes A and
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