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PreprintCC BY-NC-ND 4.0arXiv

A topological Mundici representation theorem

  • Yifan Meng
  • Lili Shen
  • Dexue Zhang
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Abstract

We lift Mundici's equivalence between MV-algebras and abelian ℓ-groups with a strong unit to the topological setting. To this end, we prove that, for every topological MV-algebra, the associated Mundici group can be endowed with a topology that extends the topology of the MV-algebra and makes it into an abelian topological ℓ-group. Among such topologies there is a finest one; we identify this topology as the topological-group reflection of the bounded-stage final topology, and construct an explicit zero-neighbourhood base in terms of the original MV-algebra topology. Using this topology, we establish a topological Mundici adjunction that restricts to an equivalence between topological MV-algebras and Mundici-fine abelian topological ℓ-groups with a strong unit. Here, Mundici-fine means that the group topology is the finest ℓ-group extension of its unit-interval topology. We also identify extremal ℓ-group extensions under additional hypotheses, and establish a quasitopological correspondence clarifying the distinction between separate and joint continuity.

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Cite

BibTeXRIS
BibTeX
@misc{meng2026topological,
  title = {{A topological Mundici representation theorem}},
  author = {Yifan Meng and Lili Shen and Dexue Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05786},
  eprint = {2611.05786},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05786},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - A topological Mundici representation theorem
AU  - Yifan Meng
AU  - Lili Shen
AU  - Dexue Zhang
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.05786
UR  - https://arxiv.org/abs/2611.05786
AB  - We lift Mundici's equivalence between MV-algebras and abelian ℓ-groups with a strong unit to the topological setting. To this end, we prove that, for every topological MV-algebra, the associated Mundici group can be endowed with a topology that extends the topology of the MV-algebra and makes it into an abelian topological ℓ-group. Among such topologies there is a finest one; we identify this topology as the topological-group reflection of the bounded-stage final topology, and construct an explicit zero-neighbourhood base in terms of the original MV-algebra topology. Using this topology, we establish a topological Mundici adjunction that restricts to an equivalence between topological MV-algebras and Mundici-fine abelian topological ℓ-groups with a strong unit. Here, Mundici-fine means that the group topology is the finest ℓ-group extension of its unit-interval topology. We also identify extremal ℓ-group extensions under additional hypotheses, and establish a quasitopological correspondence clarifying the distinction between separate and joint continuity.
ER  -