Direct-product rigidity and factor reconstruction for monoids with zero
Abstract
We study direct products of monoids with zero and give a criterion under which their factors can be recovered from the multiplicative structure alone. While classical decomposition theory encodes direct products through factor congruences, central elements, and refinement properties, we give a concrete multiplicative reconstruction mechanism. If the factors have no nontrivial complemented central idempotents, then the coordinate idempotents are precisely the atoms and coatoms of the complemented-central-idempotent poset, and their multiplicative stabilizers are precisely the coordinate factors and cofactors. It follows that every isomorphism between such products is monomial, yielding the corresponding wreath-product description of automorphism groups. More generally, every product decomposition is obtained by grouping the original factors; in particular, strict refinement follows. We apply these results to multiplicative monoids of directly indecomposable unital rings, including connected commutative rings, and to arithmetic examples arising from residue-class rings.
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BibTeX
@misc{atalaye2026directproduct,
title = {{Direct-product rigidity and factor reconstruction for monoids with zero}},
author = {Joseph Atalaye and Liam Baker and Sophie Marques},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08815},
eprint = {2609.08815},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08815},
}
RIS
TY - UNPB
TI - Direct-product rigidity and factor reconstruction for monoids with zero
AU - Joseph Atalaye
AU - Liam Baker
AU - Sophie Marques
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.08815
UR - https://arxiv.org/abs/2609.08815
AB - We study direct products of monoids with zero and give a criterion under which their factors can be recovered from the multiplicative structure alone. While classical decomposition theory encodes direct products through factor congruences, central elements, and refinement properties, we give a concrete multiplicative reconstruction mechanism. If the factors have no nontrivial complemented central idempotents, then the coordinate idempotents are precisely the atoms and coatoms of the complemented-central-idempotent poset, and their multiplicative stabilizers are precisely the coordinate factors and cofactors. It follows that every isomorphism between such products is monomial, yielding the corresponding wreath-product description of automorphism groups. More generally, every product decomposition is obtained by grouping the original factors; in particular, strict refinement follows. We apply these results to multiplicative monoids of directly indecomposable unital rings, including connected commutative rings, and to arithmetic examples arising from residue-class rings.
ER -