Congruence-preserving subdirect products

An algebra A is said to be a congruence-preserving extension of a subalgebra B if the mapping from the congruence lattice of B to that of A, assigning to each congruence relation β on B the minimal congruence relation on A containing β, is an isomorphism. We give a necessary and sufficient condition on the congruence lattice of a subdirect product B of finitely many algebras in a congruence-distributive variety that the full direct product be a congruence-preserving extension of B. We give several applications to congruence lattices of lattices. Received May 25, 2000; accepted in final form January 22, 2001.     

Serial rings and subdirect products

A basic Artinian serial ring can be realized as the subdirect product of factor rings of (S, M)-upper triangular matrix rings with S a local Artinian ring and M the maximal ideal of S. As an application the serial subdirect product of (S, M)-rings is shown to have self-duality.     

Quotients of subdirect products of groups

We find the sharp bound for the nilpotency class of the quotients of subdirect products of groups.     

Joins of projective varieties and multisecant spaces

Let , , be integral varieties. For any integers 0$">, , and set and . Let be the set of all linear -spaces contained in a linear -space spanned by points of , points of , ..., points of . Here we study some cases where has the expected dimension. The case was recently considered by Chiantini and Coppens and we follow their ideas. The two main results of the paper consider cases where each is a surface, more particularly:

or

     

Joins and secant varieties of subvarieties of a Grassmannian

Here we give the geometric foundations of the theory of joins and higher secant varieties.     

Notes on sums of algebras, subdirect products and subdirectly irreducible algebras

The paper shows that certain sums of algebras admit a representation as subdirect products. As a corollary, one obtains a characterization of subdirectly irreducible algebras. Examples of modes show that the characterization gives a good basis for a detailed investigation of the subdirectly irreducible algebras. Received September 10, 1999; accepted in final form January 3, 2001.