arXiv subjects
Publications and source records attributed to Erhard Aichinger.
For a polynomial map $\mathbf{f} : k^n \to k^m$ ($k$ a field), we investigate those polynomials $g \in k[t_1,\ldots, t_n]$ that can be written as a composition $g = h \circ \mathbf{f}$, where $h: k^m \to k$ is an arbitrary function. In the case that $k$ is algebraically closed of characteristic $0$ and $\mathbf{f}$ is surjective, we will show that $g = h \circ \mathbf{f}$ implies that $h$ is a polynomial.
We investigate when the clone of congruence preserving functions is finitely generated. We obtain a full description for all finite $p$-groups, and for all finite algebras with Mal'cev term and simple congruence lattice. The characterization for $p$-groups allows a generalization to a large class of expansions of groups.
Two varieties $V, W$ of the same type are independent if there is a binary term $t(x,y)$ such that $V \models t(x,y) \approx x$ and $W \models t(x,y) \approx y$. In this note, we give necessary and sufficient conditions for two finite algebras with a Mal'cev term (or, more generally, with an edge term) to generate independent varieties. In particular we show that the independence of finitely generated varieties with edge term can be decided by a polynomial time algorithm.
Classes of algebraic structures that are defined by equational laws are called varieties or equational classes. A variety is finitely generated if it is defined by the laws that hold in some fixed finite algebra. We show that every subvariety of a finitely generated congruence permutable variety is finitely generated; in fact, we prove the more general result that if a finitely generated variety has an edge term, then all its subvarieties are finitely generated as well. This applies in particular to all varieties of groups, loops, quasigroups and their expansions (e.g., modules, rings, Lie algebras,...).
Given the congruence lattice L of a finite algebra A with a Mal'cev term, we look for those sequences of operations on L that are sequences of higher commutator operations of expansions of A. The properties of higher commutators proved so far delimit the number of such sequences: the number is always at most countably infinite; if it is infinite, then L is the union of two proper subintervals with nonempty intersection.
We prove that every clone of operations on a finite set A, if it contains a Malcev operation, is finitely related -- i.e., identical with the clone of all operations respecting R for some finitary relation R over A. It follows that for a fixed finite set A, the set of all such Malcev clones is countable. This completes the solution of a problem that was first formulated in 1980, or earlier: how many Malcev clones can finite sets support? More generally, we prove that every finite algebra with few subpowers has a finitely related clone of term operations. Hence modulo term equivalence and a renaming of the elements, there are only countably many finite algebras with few subpowers, and thus only countably many finite algebras with a Malcev term.