Matrix varieties over S7: dimension-three stability and continuum-sized subvariety intervals
Abstract
We study matrix semirings over the three-element flat additively idempotent semiring with elements one, a, and infinity, where the square of a is infinity. We determine the associated matrix-variety chain completely. The scalar, two-by-two, and three-by-three cases generate three distinct varieties, while every matrix dimension at least three generates the same variety. The proof shows that every failure of an identity in arbitrary dimension is already witnessed on three indices, and it yields a coordinate criterion for all identities in the stable variety. We also realize every graph semiring arising from a directed graph of in-degree and out-degree at most one as a divisor of a direct power of the two-by-two matrix semiring. Consequently, the variety generated by all three-nilpotent flat semirings is contained in the two-by-two matrix variety. Directed cycles and independent reversal identities then embed the power-set lattice of the odd primes into each interval between the base variety and a nontrivial matrix variety. Thus every such interval has continuum cardinality and contains continuum-sized chains and antichains. Finally, we determine the last three powers of the multiplicative subsemiring obtained by deleting the constant all-one matrix, and we develop general matrix operators on the lattice of additively idempotent semiring varieties, including stable closures, stable cores, and propagation of equality along matrix-dimension chains.
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BibTeX
@misc{jiao2026matrix,
title = {{Matrix varieties over $S_7$: dimension-three stability and continuum-sized subvariety intervals}},
author = {Jun Jiao and Xiaolei Shao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32359},
eprint = {2609.32359},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32359},
}
RIS
TY - UNPB
TI - Matrix varieties over S_7: dimension-three stability and continuum-sized subvariety intervals
AU - Jun Jiao
AU - Xiaolei Shao
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.32359
UR - https://arxiv.org/abs/2609.32359
AB - We study matrix semirings over the three-element flat additively idempotent semiring with elements one, a, and infinity, where the square of a is infinity. We determine the associated matrix-variety chain completely. The scalar, two-by-two, and three-by-three cases generate three distinct varieties, while every matrix dimension at least three generates the same variety. The proof shows that every failure of an identity in arbitrary dimension is already witnessed on three indices, and it yields a coordinate criterion for all identities in the stable variety. We also realize every graph semiring arising from a directed graph of in-degree and out-degree at most one as a divisor of a direct power of the two-by-two matrix semiring. Consequently, the variety generated by all three-nilpotent flat semirings is contained in the two-by-two matrix variety. Directed cycles and independent reversal identities then embed the power-set lattice of the odd primes into each interval between the base variety and a nontrivial matrix variety. Thus every such interval has continuum cardinality and contains continuum-sized chains and antichains. Finally, we determine the last three powers of the multiplicative subsemiring obtained by deleting the constant all-one matrix, and we develop general matrix operators on the lattice of additively idempotent semiring varieties, including stable closures, stable cores, and propagation of equality along matrix-dimension chains.
ER -