What is the weakest idempotent Maltsev condition that implies that abelian tolerances generate abelian congruences?
We answer the question in the title. In the process, we correct an error in our AMS Memoir The Shape of Congruence Lattices.
math.LO↗
arXiv subjects
Emil W. Kiss
Publications and source records attributed to Emil W. Kiss.
Minimal abelian varieties of algebras, I
We show that any abelian variety that is not affine has a nontrivial strongly abelian subvariety. In later papers in this sequence we apply this result to the study of minimal abelian varieties.
math.LO↗
Varieties whose finitely generated members are free
We prove that a variety of algebras whose finitely generated members are free must be definitionally equivalent to the variety of sets, the variety of pointed sets, a variety of vector spaces over a division ring, or a variety of affine vector spaces over a division ring.
math.RA↗
Growth Rates of Algebras, III: Growth Rates of Solvable Algebras
We investigate how the behavior of the function d_A(n) that gives the size of a least size generating set for A^n, influences the structure of a finite solvable algebra A.
math.RA↗
Growth Rates of Algebras, II: Wiegold Dichotomy
We investigate the function $d_\mathbf{A}(n)$, which gives the size of a least size generating set for $\mathbf{A}^n$, in the case where $\mathbf{A}$ has a cube term. We show that if $\mathbf{A}$ has a $k$-cube term and $\mathbf{A}^k$ is finitely generated, then $d_\mathbf{A}(n) \in O(\log(n))$ if $\mathbf{A}$ is perfect and $d_\mathbf{A}(n) \in O(n)$ if $\mathbf{A}$ is imperfect. When $\mathbf{A}$ is finite, then one may replace "Big Oh" with "Big Theta" in these estimates.
math.RA↗
Growth rates of algebras, I: pointed cube terms
We investigate the function d_A(n), which gives the size of a least size generating set for A^n.
math.RA↗
Varieties whose tolerances are homomorphic images of their congruences
The homomorphic image of a congruence is always a tolerance (relation) but, within a given variety, a tolerance is not necessarily obtained this way. By a Maltsev-like condition, we characterize varieties whose tolerances are homomorphic images of their congruences (TImC). As corollaries, we prove that the variety of semilattices, all varieties of lattices, and all varieties of unary algebras have TImC. We show that a congruence n-permutable variety has TImC if and only if it is congruence permutable, and construct an idempotent variety with a majority term that fails TImC.
math.RA↗
Cubes of integral vectors in dimension four
A system of $m$ nonzero vectors in $\mathbb{Z}^n$ is called an $m$-icube if they are pairwise orthogonal and have the same length. The paper describes $m$-icubes in $\mathbb{Z}^4$ for $2\le m\le 4$ using Hurwitz integral quaternions, counts the number of them with given edge length, and proves that unlimited extension is possible in $\mathbb{Z}^4$.
math.NT↗
Sums of squares and orthogonal integral vectors
Two vectors in $\BZ^3$ are called \emph{twins} if they are orthogonal and have the same length. The paper describes twin pairs using cubic lattices, and counts the number of twin pairs with a given length. Integers $M$ with the property that each integral vector with length $\sqrt{M}$ has a twin are called twin-complete. They are completely characterized modulo a famous conjecture in number theory. The main tool is the decomposition theory of Hurwitz integral quaternions. Throughout the paper we made a concerted effort to keep the exposition as elementary as possible.
math.NT↗
Modularity prevents tails
We establish a direct correspondence between two congruence poroperties for finite algebras. The first property is that minimal sets of type i omit tails. The second property is that congruence lattices omit pentagons of type i.
math.RA↗