Celebratio Mathematica — Neumann — Complete Bibliography

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W. D. Neumann: “On Malcev conditions,” J. Aust. Math. Soc. 17 : 3 (1974), pp. 376–384. Dedicated to my mother, in loving memory. MR 371781 Zbl 0294.08004 article

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 [1975] characterizing “Malcev definable classes of varieties” into a clearer perspective and is therefore worth mentioning.

We construct in a simple way a complete lattice \( L \) of equivalence classes of varieties containing a countable sublattice \( M \) (represented by the finitely presented varieties) such that:

equivalent varieties are indistinguishable by Malcev conditions, that is, they either all satisfy a given Malcev condition or all do not;
the classes of varieties defined by strong Malcev conditions are just (modulo the equivalence relation) the principal ideals in \( L \) generated by single elements of \( M \);
the classes of varieties defined by ordinary Malcev conditions (respectively, weak Malcev conditions) are the ideals in \( L \) generated by subsets of \( M \) (respectively, countable intersections of such ideals). Walter Taylor’s closure theorems follow easily.

Our lattice \( L \) is actually defined on a class, rather than a set. This can be avoided by a cardinality restriction on the number of defining operations that a variety may have; \( L \) will then be complete only up to this cardinality.