A finite basis theorem for residually finite, congruence meet-semidistributive varieties

同余交半分配簇类不是强 Maltsev 的

The class of congruence meet semidistributive varieties is not strong Maltsev

本文证明同余交半分配簇类无法由单一有限恒等式集刻画，从而说明其不是强 Maltsev 类。

一般同余交半分配代数的 Maltsev 条件

Maltsev conditions for general congruence meet-semidistributive algebras

本文提出一个新的相对简单的 Maltsev 条件来刻画交半分配簇，并给出强 Maltsev 条件的候选，以解决该簇是否可由强 Maltsev 条件刻画的未知问题。

Absolute Retracts and Essential Extensions in Congruence Modular Varieties

This paper studies absolute retracts in congruence modular varieties of universal algebras. It is shown that every absolute retract with finite dimensional congruence lattice is a product of subdirectly irreducible algebras. Further, every absolute retract in a residually small variety is the product of an abelian algebra and a centerless algebra.

Congruence lifting of diagrams of finite Boolean semilattices requires large congruence varieties

We construct a diagram D, indexed by a finite partially ordered set, of finite Boolean semilattices and (v,0,1)-embeddings, with top semilattice $2^4$, such that for any variety V of algebras, if D has a lifting, with respect to the congruence lattice functor, by algebras and homomorphisms in V, then there exists an algebra $U$ in V such that the congruence lattice of $U$ contains, as a 0,1-sublattice, the five-element modular nondistributive lattice $M_3$. In particular, V has an algebra whose congruence lattice is neither join- nor meet-semidistributive. Using earlier work of K.A. Kearnes an…

Supernilpotent Taylor algebras are nilpotent

We develop the theory of the higher commutator for Taylor varieties. A new higher commutator operation called the hypercommutator is defined using a type of invariant relation called a higher dimensional congruence. The hypercommutator is shown to be symmetric and satisfy an inequality relating nested terms. For a Taylor algebra the term condition higher commutator and the hypercommutator are equal when evaluated at a constant tuple, and it follows that every supernilpotent Taylor algebra is nilpotent. We end with a characterization of congruence meet-semidistributive varieties in terms of the…