The Semilattice, □, and ⊠−tensor Products in Quantum Logic

Given two complete atomistic lattices L 1 and L 2 , we define a set S = S(L 1 , L 2) of complete atomistic lattices by means of four axioms (natural regarding the logic of compound quantum systems), or in terms of a universal property with respect to a given class of bimorphisms (Th. 3.3). S is a complete lattice (Th. 0.10). The bottom element L 1 ∧ L 2 is the separated product of Aerts [1]. For atomistic lattices with 1 (not complete), L 1 ∧ L 2 ∼ = L 1 L 2 the box product of Grätzer and Wehrung [7], and, in case L 1 and L 2 are moreover coatomistic, L 1 ∧ L 2 ∼ = L 1 ⊠ L 2 the lattice tensor product (Th. 2.2). The top element L 1 ∨ L 2 is the (complete) join-semilattice tensor product of Fraser [5] (Th. 2.12), which is isomorphic to the tensor of Chu [2] and Shmuely [17] (Th. 2.5). With some additional hypotheses on L 1 and L 2 (true if L 1 and L 2 are moreover orthomodular with the covering property), we prove that S is a singleton if and only if L 1 or L 2 is distributive (Th. 4.3, Cor. 4.7), if and only if L 1 ∨ L 2 has the covering property (Th. 7.4). Our main result reads: L ∈ S is orthocomplemented if and only if L = L 1 ∧ L 2 (Th. 6.8). For L 1 and L 2 moreover irreducible, we classify the automorphisms of each L ∈ S in terms of those of L 1 and L 2 (Th. 5.6). At the end, we construct an example L 1 ⇓ L 2 in S which has the covering property (Th. 8.4). 0. T-tensor products, ∧ and ∨ Let us start with our main definitions and postpone the introduction. Most of the time, we adopt the notations and terminology used in the book of Maeda and Maeda [11]. Notation 0.1. A set L i ⊆ 2 Σi , closed under arbitrary set-intersections, containing Σ i , ∅, and all singletons, ordered by set-inclusion, is called a simple closure space on Σ i. The bottom and top elements are denoted by 0 and 1 respectively. A simple closure space is a complete atomistic lattice, and if L is a complete atomistic lattice, then {Σ[a] …