# Finite generation of congruence preserving functions

@article{Aichinger2015FiniteGO, title={Finite generation of congruence preserving functions}, author={Erhard Aichinger and Marijana Lazic and Nebojsa Mudrinski}, journal={Monatshefte f{\"u}r Mathematik}, year={2015}, volume={181}, pages={35-62} }

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.

## 6 Citations

Congruence preserving expansions of nilpotent algebras

- MathematicsInt. J. Algebra Comput.
- 2020

Those nilpotent algebras of prime power order and finite type in congruence modular varieties that have infinitely many polynomially inequivalent congruent expansions are characterized.

Rings of congruence preserving functions, II

- Mathematics
- 2020

Abstract For several classes of special p-groups G, of exponent p, p > 2, we show that the near-ring, of congruence preserving functions on G is a ring if and only if G is a 1-affine complete group.

Rings of congruence preserving functions

- MathematicsMonatshefte fur Mathematik
- 2018

This work investigates the question “When is C_0 (G)$$C0(G) a ring?” and obtains information externally via the lattice structure of the normal subgroups of G and internally via structural properties of the group G.

Congruence lattices forcing nilpotency

- Mathematics
- 2016

Given a lattice $\mathbb{L}$ and a class $K$ of algebraic structures, we say that $\mathbb{L}$ \emph{forces nilpotency} in $K$ if every algebra $\mathbf{A} \in K$ whose congruence lattice…

Projectivity and linkage for completely join irreducible ideals of an expanded group

- MathematicsAlgebra universalis
- 2019

Specialized from lattice theory to the lattice of ideals of an expanded group, the equivalence relation of projectivity between two intervals of ideals I[A, B] and I[C, D] of an expanded group V…

The Weak Descending Chain Condition on Right Ideals for Nearrings

- Mathematics
- 2019

The purpose of this paper is to introduce a weaker form of the descending chain condition on right ideals than the standard one. We shall see that fundamental results about socle and Frattini series…

## References

SHOWING 1-10 OF 23 REFERENCES

Types of polynomial completeness of expanded groups

- Mathematics
- 2009

Abstract.The polynomial functions of an algebra preserve all congruence relations. In addition, if the algebra is finite, they preserve the labelling of the congruence lattice in the sense of Tame…

Congruence modular varieties with small free spectra

- Mathematics
- 1999

Abstract. Let A be a finite algebra that generates a congruence modular variety. We show that the free spectrum of
${\cal V}({\bf A})$ fails to have a doubly exponentially lower bound if and only if…

Some applications of higher commutators in Mal’cev algebras

- Mathematics
- 2010

We establish several properties of Bulatov’s higher commutator operations in congruence permutable varieties. We use higher commutators to prove that for a finite nilpotent algebra of finite type…

A Course in the Theory of Groups

- Mathematics
- 1982

This is a detailed introduction to the theory of groups: finite and infinite; commutative and non-commutative. Presupposing only a basic knowledge of modern algebra, it introduces the reader to the…

Polynomial Completeness in Algebraic Systems

- Mathematics
- 2000

ALGEBRAS, LATTICES, AND VARIETIES Algebras, Languages, Clones, Varieties Congruence Properties CHARACTERIZATIONS OF EQUIVALENCE LATTICES Introduction Arithmeticity Compatible Function Lifting…

On Hagemann's and Herrmann's characterization of strictly affine complete algebras

- Mathematics
- 2000

Abstract. In [4], J.Hagemann and C. Herrmann have characterized those algebras that are strictly k-affine complete for all
$ k \in {\Bbb N} $. In the present note, we prove a very similar…

Commutator theory for congruence modular varieties

- Mathematics
- 1989

Introduction In the theory of groups, the important concepts of Abelian group, solvable group, nilpotent group, the center of a group and centraliz-ers, are all defined from the binary operation [x,…

Algebras, Lattices, Varieties

- Mathematics
- 1987

Introduction. Preliminaries. Basic concepts. Lattices. Unary and binary operations. Fundamental algebraic results. Unique factorization. Bibliography. Table of notation. Index.

Sequences of Commutator Operations

- MathematicsOrder
- 2013

Given the congruence lattice of a finite algebra A with a Mal’cev term, it is shown that if it is infinite, then ${{\mathbb{L}}}$ is the union of two proper subintervals with nonempty intersection.