A category of quantum posets

Kornell, Andre; Lindenhovius, Bert; Mislove, Michael

doi:10.1016/j.indag.2022.07.001

Cited by 2 publications

(8 citation statements)

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“…In this section, we introduce the quantum down-set monad. Its construction shares similarities with the construction of the quantum power set in [14]. This construction yields a quantum poset, but the construction of this order seems to be a bit ad hoc.…”

Section: Quantum Posetsmentioning

confidence: 95%

“…In [14], the quantum power set monad P was introduced. We sketch how to derive this monad from the quantum down-set monad.…”

Section: The Quantum Power Setmentioning

confidence: 99%

“…We can now define the quantum power set P(X ) of a quantum poset X as the quantum set D(X , I X ), which can be made into a functor by composing D with the embedding of qSet into qPos. By playing with the composition of adjunctions, we can equip P with the structure of a monad such as in [14]. The ordered quantum power set qPow(X ) can be obtained as qDwn(X , I X ).…”

Section: The Quantum Power Setmentioning

confidence: 99%

“…Quantum posets were already defined by Weaver in [21]. The properties of the category of quantum posets in the framework of discrete quantum mathematics were investigated in [14] by Kornell, Mislove and the second author. The same authors proceeded in [13] by quantizing cpos by means of discrete quantization.…”

Section: Related Workmentioning

confidence: 99%

“…It was shown in [14] that the pointwise order of functions between quantum sets induces a Pos-enrichment of qPos. Similarly, in [13] it was shown that the category qCPO of quantum cpos is enriched over the category CPO of cpos, i.e., posets for which any monotonically increasing sequence has a supremum.…”

Section: Enrichmentmentioning

confidence: 99%

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Quantum Suplattices

Jenča,

Lindenhovius

2023

Electron. Proc. Theor. Comput. Sci.

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Building on the theory of quantum posets, we introduce a non-commutative version of suplattices, i.e., complete lattices whose morphisms are supremum-preserving maps, which form a step towards a new notion of quantum topological spaces. We show that the theory of these 'quantum suplattices' resembles the classical theory: the opposite quantum poset of a quantum suplattice is again a quantum suplattice, and quantum suplattices arise as algebras of a non-commutative version of the monad of downward-closed subsets of a poset. The existence of this monad is proved by introducing a noncommutative generalization of monotone relations between quantum posets, which form a compact closed category. Moreover, we introduce a non-commutative generalization of Galois connections and we prove that an upper Galois adjoint of a monotone map between quantum suplattices exists if and only if the map is a morphism of quantum suplattices. Finally, we prove a quantum version of the Knaster-Tarski fixpoint theorem: the quantum set of fixpoints of a monotone endomap on a quantum suplattice form a quantum suplattice.

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Section: Quantum Posetsmentioning

confidence: 95%

“…In [14], the quantum power set monad P was introduced. We sketch how to derive this monad from the quantum down-set monad.…”

Section: The Quantum Power Setmentioning

confidence: 99%

Section: The Quantum Power Setmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Enrichmentmentioning

confidence: 99%

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Quantum Suplattices

Jenča,

Lindenhovius

2023

Electron. Proc. Theor. Comput. Sci.

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Discrete quantum structures II: Examples

Kornell

2024

J. Noncommut. Geom.

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Part I of this paper established the basic properties of quantum predicate logic as the internal logic of discrete quantum structures. We now show that a majority of the established quantum generalizations of discrete structures are naturally axiomatizable within this internal logic. In particular, we axiomatize the quantum graphs of Duan, Severini, and Winter, the quantum metric spaces of Kuperberg and Weaver, the quantum isomorphisms of Atserias, Mančinska, Roberson, Šámal, Severini, and Varvitsiotis, and the quantum groups of Woronowicz. In each instance, we consider only those structures that are discrete in the sense that the underlying von Neumann algebra is hereditarily atomic.

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A category of quantum posets

Cited by 2 publications

References 51 publications

Quantum Suplattices

Quantum Suplattices

Discrete quantum structures II: Examples

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