Quantum Suplattices

TL;DR

This paper develops a non-commutative generalization of suplattices, extending classical lattice theory into the quantum realm and establishing foundational properties and theorems for quantum topological spaces.

Contribution

It introduces quantum suplattices, explores their properties, and proves key theorems like the quantum Galois connection and a quantum fixpoint theorem, advancing quantum topology.

Findings

Quantum suplattices are closed under opposite operations.

Existence of a non-commutative monad of downward-closed subsets.

Quantum fixpoints form quantum suplattices.

Abstract

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 non-commutative 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.