On Malcev conditions

Abstract

This note gives a way of looking at Malcev conditions for varieties as ideals in a certain lattice. Though this viewpoint (so far) yields no new results, we feel it puts Walter Taylor's results [3] characterizing “Malcev definable classes of varieties” into a clearer perspective and is therefore worth mentioning. The author is grateful to R. Wille for pointing out W. Taylor's paper to him.

Near-unanimity is decomposable

“…The lattice L of interpretability types was introduced in [12] and thoroughly studied in [8]. Maltsev conditions, which are associated with many important properties of varieties -such as permutability or distributivity of congruence lattices -correspond very nicely to filters of L. We assume the reader is familiar with the basic notions of interpretability of varieties and Maltsev conditions.…”

Some universal properties of the category of clones

“…Identifying equally interpretable varieties one gets a partially ordered class L which is a complete lattice in the sense that each subset has an infimum and a supremum, the lattice of interpretability types of varieties (see e. g. [14], [6]; further references may be found in the latter).…”

“…The lattice L of interpretability types of varieties of (finitary monosorted) universal algebras was introduced and started to be investigated in [14]. Then an issue [6] of Memoirs of the AMS by O. C. García and W. Taylor was devoted to the study of L. Many open problems are formulated at the end of [6].…”

Interpretations into 𝑝-algebras

“…The concept of interpretability of one variety into another was introduced by Neumann [4] in the study of Malt'sev conditions; it was later developed by Garcia and Taylor [3]. Roughly speaking, a variety V is interpretable in a variety W if the operations of V can be defined in terms of the operations of W in such a way that the (underlying sets of the) algebras in W with these new operations are algebras in V .…”