Abstract

Quantum analogues of sets are defined by two simple assumptions, allowing enumeration, reminiscent of the Gram–Schmidt orthogonalization process. It is shown that any symmetric quantum set is a classical set of irreducible components, and that each irreducible component of size $>3$ is representable by an orthomodular space over a skew field with involution. For finite or sufficiently large irreducible components, invariance of quantum cardinality is proved. Topological quantum sets are introduced as quantum analogues of topological spaces; irreducible ones of size $>3$ are shown to be representable by Hilbert spaces over ${\mathbb{R}}$, ${\mathbb{C}}$, or ${\mathbb{H}}$. Symmetric quantum sets are characterized as a class of $L$-algebras with an intrinsic geometry, and they are shown to be equivalent to Piron’s quantum formalism. Equivalences between symmetric quantum sets and several other structures are established. To any symmetric quantum set, a group with a right invariant lattice structure is associated as a complete invariant. A simple and self-contained proof of Solèr’s theorem is included, which is used to prove that sufficiently large irreducible symmetric quantum sets come from a classical Hilbert space.

中文翻译：

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对称量子集和 L-代数

摘要

集合的量子类似物由两个简单的假设定义，允许枚举，让人想起 Gram-Schmidt 正交化过程。结果表明，任何对称量子集都是经典的不可约分量集，并且每个大小为 $>3$ 的不可约分量都可以由具有对合的斜场上的正交模空间表示。对于有限或足够大的不可约分量，证明了量子基数的不变性。引入拓扑量子集作为拓扑空间的量子类似物；大小为 $>3$ 的不可约空间可以用 ${\mathbb{R}}$、${\mathbb{C}}$ 或 ${\mathbb{H}}$ 上的希尔伯特空间表示。对称量子集的特征是一类具有内在几何的$L$-代数，并且它们被证明等同于Piron的量子形式。建立了对称量子集和其他几个结构之间的等价性。对于任何对称量子集，具有右不变晶格结构的群都关联为完全不变量。包括一个简单且独立的 Solèr 定理证明，用于证明足够大的不可约对称量子集来自经典希尔伯特空间。