Narrow Results

For a finite lattice L, the congruence lattice Con L of L can be easily computed from the partially ordered set J(L) of join-irreducible elements of L and the join-dependency relation D_L on J(L). We establish a similar version of this result for the dimension monoid Dim L of L, a natural precursor of Con L. For L join-semidistributive, this result takes the following form:

Theorem 1. Let L be a finite join-semidistributive lattice. Then Dim L is isomorphic to the commutative monoid defined by generators Δ(p), for p ∈ J(L), and relations

As a consequence of this, we obtain the following results:

Theorem 2. Let L be a finite join-semidistributive lattice. Then L is a lower bounded homomorphic image of a free lattice iff Dim L is strongly separative, iff it satisfies the axiom

Theorem 3. Let A and B be finite join-semidistributive lattices. Then the box product A □ B of A and B is join-semidistributive, and the following isomorphism holds:

A variety of algebras is called limit if it is nonfinitely-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 commuting idempotents.

Since topoi were introduced, there have been efforts putting mathematics into the context of topoi. Amongst known topoi, the topoi of sheaves or presheaves over a small category are of special interest. We have here as the base topos that of sheaves over a monoid $M$ as a one object category. By means of closure operators we then obtain categories of sheaves related to the right ideals of $M$. These categories have already been studied but we give these categories a more thorough treatment and reveal some additional properties. Namely, for a weak topology determined by a right ideal $I$ of $M$, we show that the category of sheaves associated to this topology is a subtopos of $MSet$ (the presheaves over $M$) and determine the Lawvere–Tierney topology yielding the same subtopos, which is the Lawvere–Tierney topology associated to the idempotent hull of the (not necessarily idempotent) closure operator associated to $I$. We will then find conditions under which the subcategory of separated objects turns out to be a topos, and in the last section, we find conditions under which the category of sheaves becomes a De Morgan topos.

We study the endomorphisms and derivations of an infinite-dimensional cyclic Leibniz algebra. Among others it was found that if $L$ is a cyclic infinite-dimensional Leibniz algebra over a field $F$, then the group of all automorphisms of $L$ is isomorphic to a multiplicative group of the field $F$. The description of an algebra of derivations of a cyclic infinite-dimensional Leibniz algebra has been obtained.

Factorizations of monoids are studied. Two necessary and sufficient conditions in terms of the so-called descent 1-cocycles for a monoid to be factorized through two submonoids are found. A full classification of those factorizations of a monoid whose one factor is a subgroup of the monoid is obtained. The relationship between monoid factorizations and non-abelian cohomology of monoids is analyzed. Some applications of semi-direct product of monoids are given.