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Proper varieties of distributive l-pregroups are periodic

  • Nikolaos Galatos
  • Simon Santschi
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Abstract

We show that every proper variety of distributive ℓ-pregroups is periodic; as a result, the subvariety lattice of the variety of distributive ℓ-pregroups decomposes into a disjoint union of bounded intervals each consisting of all properly n-periodic varieties, for different n ∈ ℤ+. On the way, we study non-periodic (one-generated) subalgebras of F(ℤ) and show that they either contain a non-periodic idempotent or they contain idempotents of arbitrary big periodicity. The same results holds for non-periodic one-generated subalgebras of F(J overrightarrow× ℤ), where J is a chain, generated by a composite element and the resulting idempotents can be taken to be local in the associated wreath product.

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Cite

BibTeXRIS
BibTeX
@misc{galatos2026proper,
  title = {{Proper varieties of distributive l-pregroups are periodic}},
  author = {Nikolaos Galatos and Simon Santschi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16775},
  eprint = {2610.16775},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16775},
}
RIS
TY  - UNPB
TI  - Proper varieties of distributive l-pregroups are periodic
AU  - Nikolaos Galatos
AU  - Simon Santschi
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16775
UR  - https://arxiv.org/abs/2610.16775
AB  - We show that every proper variety of distributive ℓ-pregroups is periodic; as a result, the subvariety lattice of the variety of distributive ℓ-pregroups decomposes into a disjoint union of bounded intervals each consisting of all properly n-periodic varieties, for different n ∈ ℤ^+. On the way, we study non-periodic (one-generated) subalgebras of F(ℤ) and show that they either contain a non-periodic idempotent or they contain idempotents of arbitrary big periodicity. The same results holds for non-periodic one-generated subalgebras of F(J overrightarrow× ℤ), where J is a chain, generated by a composite element and the resulting idempotents can be taken to be local in the associated wreath product.
ER  -