Global representations of algebras with a near-unanimity term
Abstract
Global representations are subdirect representations satisfying a sheaf-like local-to-global patching principle. We develop a unified and simplified theory of such representations. For quasivarieties with a near-unanimity term, this framework recovers the main classical representation theorems through a common argument and yields a general representation by factors of bounded subdirect width. When the relatively subdirectly irreducible members form a universal class, we obtain an optimal result: every relatively congruence-distributive algebra admits a global representation by relatively globally indecomposable factors, which we characterize explicitly. We also provide a converse: global representations by factors of bounded relative subdirect width force the existence of a near-unanimity term. Adding semisimplicity to the hypotheses of the main theorem yields optimal representation results for filtral quasivarieties and dual discriminator varieties, together with a simple description of the globally indecomposable algebras. The technical engine behind these results is a new infinitary extension of the congruence-system component of the Baker--Pixley theorem.
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BibTeX
@misc{campercholi2026global,
title = {{Global representations of algebras with a near-unanimity term}},
author = {Miguel Campercholi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01445},
eprint = {2609.01445},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01445},
}
RIS
TY - UNPB
TI - Global representations of algebras with a near-unanimity term
AU - Miguel Campercholi
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01445
UR - https://arxiv.org/abs/2609.01445
AB - Global representations are subdirect representations satisfying a sheaf-like local-to-global patching principle. We develop a unified and simplified theory of such representations. For quasivarieties with a near-unanimity term, this framework recovers the main classical representation theorems through a common argument and yields a general representation by factors of bounded subdirect width. When the relatively subdirectly irreducible members form a universal class, we obtain an optimal result: every relatively congruence-distributive algebra admits a global representation by relatively globally indecomposable factors, which we characterize explicitly. We also provide a converse: global representations by factors of bounded relative subdirect width force the existence of a near-unanimity term. Adding semisimplicity to the hypotheses of the main theorem yields optimal representation results for filtral quasivarieties and dual discriminator varieties, together with a simple description of the globally indecomposable algebras. The technical engine behind these results is a new infinitary extension of the congruence-system component of the Baker--Pixley theorem.
ER -