Equational Theories of Interval Semirings of Posets
Abstract
We study the equational theory and subvariety structure of the ai-semiring variety V∞ generated by all flat semirings S(a1⋯ ak), where the letters ai are pairwise distinct. Using interval semirings of posets, we characterize its subdirectly irreducible members and describe variety membership in terms of jointly separating families of strict order-preserving maps. We obtain explicit finite identity bases for V∞ and each Vk generated by S(a1⋯ ak). Consequently, every flat semiring S(W) associated with a nonempty set W of linear words is finitely based. This yields finitely based ai-semirings with exactly k-nilpotent multiplicative reduct for each k≥ 1. For each k≥ 1, let Bk be the subvariety of V∞ defined by the (k+1)-nilpotent identity. We prove that each Bk is generated by a finite interval semiring and that every proper subvariety of V∞ is contained in some Bk. The variety B3 is a Cross variety with exactly 11 subvarieties, whereas [Vk,Bk], [Vk,Vk+1], and [Bk−1,Bk] each contain continuum many subvarieties for every k≥ 4. In particular, this provides infinitely many finitely based finite semirings S(a1⋯ ak) whose generated variety has continuum many subvarieties for every k≥ 5.
Related
BibTeX
@misc{gao2026equational,
title = {{Equational Theories of Interval Semirings of Posets}},
author = {Zidong Gao and Yilin Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06351},
eprint = {2610.06351},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06351},
}
RIS
TY - UNPB
TI - Equational Theories of Interval Semirings of Posets
AU - Zidong Gao
AU - Yilin Zhou
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06351
UR - https://arxiv.org/abs/2610.06351
AB - We study the equational theory and subvariety structure of the ai-semiring variety V_∞ generated by all flat semirings S(a_1⋯ a_k), where the letters a_i are pairwise distinct. Using interval semirings of posets, we characterize its subdirectly irreducible members and describe variety membership in terms of jointly separating families of strict order-preserving maps. We obtain explicit finite identity bases for V_∞ and each V_k generated by S(a_1⋯ a_k). Consequently, every flat semiring S(W) associated with a nonempty set W of linear words is finitely based. This yields finitely based ai-semirings with exactly k-nilpotent multiplicative reduct for each k≥ 1. For each k≥ 1, let B_k be the subvariety of V_∞ defined by the (k+1)-nilpotent identity. We prove that each B_k is generated by a finite interval semiring and that every proper subvariety of V_∞ is contained in some B_k. The variety B_3 is a Cross variety with exactly 11 subvarieties, whereas [V_k,B_k], [V_k,V_k+1], and [B_k−1,B_k] each contain continuum many subvarieties for every k≥ 4. In particular, this provides infinitely many finitely based finite semirings S(a_1⋯ a_k) whose generated variety has continuum many subvarieties for every k≥ 5.
ER -