Stone profiniteness, reticulation spectra, and profinite-completion obstructions for residuated lattices
Abstract
We develop a profinite theory for bounded commutative integral residuated lattices around three related questions. First, we prove a Stone--profinite theorem: a topological residuated lattice is profinite exactly when its underlying space is Stone. The proof refines clopen congruences of the semiring reduct into clopen congruences in the full residuated-lattice signature via the 1-class filter. Second, we solve the classwise Grätzer-type spectral representation problem: for every class K of residuated lattices, the K-representable prime-filter spectra are precisely the Priestley spectra of the bounded distributive lattices in the reticulation image Ret(K). This reduction is sharp, and we compute Ret(K) for several natural finite and logical classes, including finite MTL-, BL-, MV-, and n-potent/EDPC varieties. Third, we study profinite completions. We give an intrinsic criterion in terms of dense subalgebras and clopen congruences, prove a positive completion theorem under finite generation of finite-index congruences in n-potent varieties, and establish a finite-tag obstruction theorem. In particular, using the Heyting non-completion examples of Bezhanishvili--Bezhanishvili--Moraschini--Stronkowski and the three-element Łukasiewicz chain L3, we obtain genuinely non-idempotent profinite residuated lattices which are not profinite completions.
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BibTeX
@misc{yang2026stone,
title = {{Stone profiniteness, reticulation spectra, and profinite-completion obstructions for residuated lattices}},
author = {Jiang Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00016},
eprint = {2610.00016},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00016},
}
RIS
TY - UNPB
TI - Stone profiniteness, reticulation spectra, and profinite-completion obstructions for residuated lattices
AU - Jiang Yang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00016
UR - https://arxiv.org/abs/2610.00016
AB - We develop a profinite theory for bounded commutative integral residuated lattices around three related questions. First, we prove a Stone--profinite theorem: a topological residuated lattice is profinite exactly when its underlying space is Stone. The proof refines clopen congruences of the semiring reduct into clopen congruences in the full residuated-lattice signature via the 1-class filter. Second, we solve the classwise Grätzer-type spectral representation problem: for every class K of residuated lattices, the K-representable prime-filter spectra are precisely the Priestley spectra of the bounded distributive lattices in the reticulation image Ret(K). This reduction is sharp, and we compute Ret(K) for several natural finite and logical classes, including finite MTL-, BL-, MV-, and n-potent/EDPC varieties. Third, we study profinite completions. We give an intrinsic criterion in terms of dense subalgebras and clopen congruences, prove a positive completion theorem under finite generation of finite-index congruences in n-potent varieties, and establish a finite-tag obstruction theorem. In particular, using the Heyting non-completion examples of Bezhanishvili--Bezhanishvili--Moraschini--Stronkowski and the three-element Łukasiewicz chain L3, we obtain genuinely non-idempotent profinite residuated lattices which are not profinite completions.
ER -