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THE SEMILATTICE, □, AND ⊠−TENSOR PRODUCTS IN QUANTUM LOGIC
, 2004
"... Abstract. Given two complete atomistic lattices L1 and L2, we define a set S = S(L1, L2) 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 ..."
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Cited by 3 (3 self)
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Abstract. Given two complete atomistic lattices L1 and L2, we define a set S = S(L1, L2) 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 L1 ∧○L2 is the separated product of Aerts [1]. For atomistic lattices with 1 (not complete), L1 ∧○L2 ∼ = L1□L2 the box product of Grätzer and Wehrung [7], and, in case L1 and L2 are moreover coatomistic, L1 ∧○L2 ∼ = L1 ⊠ L2 the lattice tensor product (Th. 2.2). The top element L1 ∨○L2 is the (complete) joinsemilattice 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 L1 and L2 (true if L1 and L2 are moreover orthomodular with the covering property), we prove that S is a singleton if and only if L1 or L2 is distributive (Th. 4.3, Cor. 4.7), if and only if L1 ∨○L2 has the covering property (Th. 7.4). Our main result reads: L ∈ S is orthocomplemented if and only if L = L1 ∧○L2 (Th. 6.8). For L1
THE SEMILATTICE, BOX, AND LATTICETENSOR PRODUCTS IN QUANTUM LOGIC
, 2005
"... Abstract. Given two complete atomistic lattices L1 and L2, we define a set S = S(L1, L2) of complete atomistic lattices by means of three axioms (natural regarding the description of separated quantum compound systems), or in terms of a universal property with respect to a given class of bimorphisms ..."
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Cited by 2 (2 self)
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Abstract. Given two complete atomistic lattices L1 and L2, we define a set S = S(L1, L2) of complete atomistic lattices by means of three axioms (natural regarding the description of separated quantum compound systems), or in terms of a universal property with respect to a given class of bimorphisms. We prove that S is a complete lattice. The bottom element L1 ∧○L2 is the separated product of Aerts. For atomistic lattices with 1 (not complete), L1 ∧○L2 ∼ = L1□L2 the box product of Grätzer and Wehrung, and, in case L1 and L2 are moreover coatomistic, L1 ∧○L2 ∼ = L1⊠L2 the lattice tensor product. The top element L1 ∨○L2 is the (complete) joinsemilattice tensor product of Fraser, which is isomorphic to the tensor products of Chu and Shmuely. With some additional hypotheses on L1 and L2 (true if L1 and L2 are moreover orthomodular with the covering property), we prove that S is a singleton if and only if L1 or L2 is distributive, if and only if L1 ∨○L2 has the covering property. Our main result reads: L ∈ S admits an orthocomplementation if and only if L = L1 ∧○L2. For L1 and L2 moreover irreducible, we characterize the automorphisms of each L ∈ S in terms of those of L1 and L2. At the end, we construct an example L1 ⇓○L2 in S which has the covering property. 1.
Orthocomplementation and compound systems
"... Abstract. In their 1936 founding paper on quantum logic, Birkhoff and von Neumann postulated that the lattice describing the experimental propositions concerning a quantum system is orthocomplemented. We prove that this postulate fails for the lattice Lsep describing a compound system consisting of ..."
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Cited by 1 (1 self)
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Abstract. In their 1936 founding paper on quantum logic, Birkhoff and von Neumann postulated that the lattice describing the experimental propositions concerning a quantum system is orthocomplemented. We prove that this postulate fails for the lattice Lsep describing a compound system consisting of so called separated quantum systems. By separated we mean two systems prepared in different “rooms ” of the lab, and before any interaction takes place. In that case the state of the compound system is necessarily a product state. As a consequence, Dirac’s superposition principle fails, and therefore Lsep cannot satisfy all Piron’s axioms. In previous works, assuming that Lsep is orthocomplemented, it was argued that Lsep is not orthomodular and fails to have the covering property. Here we prove that Lsep cannot admit and orthocomplementation. Moreover, we propose a natural model for Lsep which has the covering property. PACS numbers: 03.65.Ta, 03.65.Ca 1.
THE CHU CONSTRUCTION IN QUANTUM LOGIC
, 2008
"... Abstract. The Chu construction is used to define a ∗−autonomous structure on a category of complete atomistic coatomistic lattices, denoted by ..."
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Abstract. The Chu construction is used to define a ∗−autonomous structure on a category of complete atomistic coatomistic lattices, denoted by
Contents
"... Abstract. We review recent results on congruence lattices of (infinite) lattices. We discuss results obtained with box products, as well as categorical, ringtheoretical, and topological results. ..."
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Abstract. We review recent results on congruence lattices of (infinite) lattices. We discuss results obtained with box products, as well as categorical, ringtheoretical, and topological results.