Weak Congruences in Universal Algebra
by
Andreja Tepavcevic
Institute of Mathematics, University of Novi Sad, Yugoslavia
Coauthors: Branimir Seselja

A book "Weak Congruences in Universal Algebra" by B. Seselja and A. Tepavcevic, Institute of Mathematics, Symbol, Novi Sad, published in December 2001, will be presented. The book contains most of the results concerning weak congruences (symmetric, transitive and compatible relations on an algebra) from the appearance of the first papers on the topic fifteen years ago. New results about some identities valid on a weak congruence lattice will be given. Open problems will be discussed, like abstract representation problem of lattices by weak congruences and a problem on the Congruence Intersection Property (CIP).

In structural investigations of algebras and varieties, the best known ordered structures are lattices of subalgebras and those of congruences. These two lattices are usually studied separately, since they consist of different kinds of object connected with an algebra: subsets of the universe and subsets of its square. Weak congruences appear to be a tool for common investigations of both, congruences and subalgebras of the same algebra. Being symmetric and transitive subalgebras of the square, these relations can be understood as congruences on all subalgebras, among which diagonal relations represent subalgebras themselves. Thus, the collection Cw A of weak congruences on an algebra A is a lattice under inclusion, and its sublattices are Con  A, Sub A, and Con B, for every subalgebra B.

It turns out that Cw A provides information about properties of an algebra which could not be deduced from the lattices of its subalgebras and congruences (e.g., regularity, Hamiltonian property, extension of congruences etc.). In addition, all algebras representing a given lattice by weak congruences must fulfill these algebraic conditions.

Weak congruences were introduced in universal algebra together with other compatible relations generalizing the concept of a congruence (implicitly by F. Sik and his Ph.D. student T.D. Mai in 1974, and under the present name by Vojvodi\'c and Seselja in 1988).

The aim of this text is to present basic properties of weak congruences, particularly of lattices of these, and to highlight their applications in universal algebra. The book is appearing after systematic investigations conducted by the authors over several years, and contains the complete bibliography on these and related topics until 2001.

Chapter 1 is an overview of distributive and other special elements of lattices, which serve as a tool for describing properties of weak congruence lattices. Proofs are given only for results which could not be found in the literature.

Main definitions and properties of notions concerning weak congruences and the corresponding lattices are presented in Chapter 2. This part contains also some historical remarks about weak equivalences and compatible relations.

Weak congruences in known classes of algebras and varieties like groups, rings, Abelian and Hamiltonian algebras are discussed in Chapter 3.

Chapter 4 is devoted to different aspects of representations of lattices by weak congruences. As a concrete application of the results in this part, Appendix contains representations by weak congruences of all lattices with at most six elements.

Finally, a generalization of weak congruences to other compatible relations is presented in Chapter 5.

Complete proofs are given mostly for theorems appearing here for the first time. In addition, formulations of some theorems and proofs differ from those in original papers. However, most of the theorems are commented by a sketch of the proof, or by a suitable example.

Open problems connected with weak congruences:

1. Which Abelian or Hamiltonian varieties possess the CIP (Congruence Intersection Property)?

Congruence Intersection Property is equivalent with the distributivity of the diagonal element in the lattice of weak congruences).

2. Weak congruence lattice representation problem. Let L be an algebraic lattice and a in L. Is there an algebra such that its weak congruence lattice is isomorphic with L, the diagonal relation being the image of a under the isomorphism.

Date received: April 30, 2002