A variety is primitive if every subquasivariety is equational, i.e. a subvariety. In this paper, we explore the connection between primitive lattice varieties and Whitman’s condition (W). For example, if every finite subdirectly irreducible lattice in a locally finite variety V satisfies Whitman’s condition (W), then V is primitive. This allows us to construct infinitely many sequences of primitive lattice varieties, and to show that there are 2ℵ0 such varieties. Some lattices that fail (W) also generate primitive varieties. But if I is a (W)-failure interval in a finite subdirectly irreducible lattice L, and L[I] denotes the lattice with I doubled, then V(L[I]) is never primitive.
P. Jipsen and J. B. Nation, Primitive lattice varieties, International Journal of Algebra and Computation 32 (2022), 717-752. https://doi.org/10.1142/S021819672250031X
World Scientific Publishing
Available for download on Tuesday, March 28, 2023