A modular inherently nonfinitely based lattice

Proof. As observed in McNulty [7], a locally finite variety V of finite type is inherently nonfinitely based if and only if for infinitely many natural numbers N , there is a non-locally-finite algebra each of whose N -generated subalgebras belongs to V. We prove the theorem by establishing these facts. We assume the reader is familiar with the basic facts of modular lattices; see [1], [2], [6]. Let B (for bottom) be the sublattice of L∞ consisting of all elements of finite height and let T consist of all elements of finite depth. Of course L∞ is the ordinal (or linear) sum B + T of these sublattices.