A Survey of Tensor Products and Related Constructions in Two Lectures

We survey tensor products of lattices with zero and related constructions focused on two topics: amenable lattices and box products. PART I. FIRST LECTURE: AMENABLE LATTICES Abstract. Let A be a finite lattice. Then A is amenable (A⊗B is a lattice, for every lattice B with zero) iff A (as a join-semilattice) is sharply transferable (whenever A has an embedding φ into IdL, the ideal lattice of a lattice L, then A has an embedding ψ into L satisfying ψ(x) ∈ φ(x) and ψ(x) / ∈ φ(y), if y < x). In Section 1, we survey tensor products. In Section 2, we introduce transferability. These two topics are brought together in Section 3 in the characterization theorem of amenable lattices. Let A be a finite lattice. Then A is amenable (A⊗B is a lattice, for every lattice B with zero) iff A (as a join-semilattice) is sharply transferable (whenever A has an embedding φ into IdL, the ideal lattice of a lattice L, then A has an embedding ψ into L satisfying ψ(x) ∈ φ(x) and ψ(x) / ∈ φ(y), if y < x). In Section 1, we survey tensor products. In Section 2, we introduce transferability. These two topics are brought together in Section 3 in the characterization theorem of amenable lattices.