Compatible additions on a six-element commutative semigroup: equational bases and subvariety lattices
Abstract
Let M be the six-element commutative semigroup occurring as the common multiplicative reduct of the semirings SR6 and TR6. The closing paragraph of Shao, Ren, and Gao~\cite{ShaoRenGao2026} asks for the finite-basis and subvariety questions for the four remaining compatible additions on M. We answer these questions for the four isomorphism types R01,R02,R11,R12. First, we classify all compatible additions on M: there are nine labelled additions and six isomorphism types, parametrized by Rij with 0≤ i≤ j≤ 2. For each of the four new types we give a graph-theoretic criterion for every identity, an explicit infinite basis, and a proof of nonfinite basability. The generated varieties V(R01) and V(R02) have eleven subvarieties each, while V(R11) has sixty-six. The lattice Sub(V(R12)) is countably infinite. Every identity in this variety reduces to a subset of twenty-five fixed identities together with two monotone path families γn and gammaDn. This yields a canonical signature (H,p,q), complete normal forms, explicit meet and join operations, and a formula for all covers. There are 153 fixed nodes, 43 one-parameter families, and 9 two-parameter families; exactly eighteen subvarieties are finitely based, and the unique limit subvariety is V(SR6). The strong nonfinite-basis status of the four finite semirings remains open.
Related
- The regionally proximal relation for commutative semigroup actions
- Subvariety intervals and limit varieties of zero extensions of mirror-word semirings
- The unique limit subvariety of the max-plus variety and finite basis properties
- Matrix varieties over S7: dimension-three stability and continuum-sized subvariety intervals
- The finite basis problem and subvariety intervals for max-plus matrix semirings
BibTeX
@misc{wang2026compatible,
title = {{Compatible additions on a six-element commutative semigroup: equational bases and subvariety lattices}},
author = {Lili Wang and Jinjing Wu and Aifa Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22667},
eprint = {2609.22667},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22667},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Compatible additions on a six-element commutative semigroup: equational bases and subvariety lattices
AU - Lili Wang
AU - Jinjing Wu
AU - Aifa Wang
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22667
UR - https://arxiv.org/abs/2609.22667
AB - Let M be the six-element commutative semigroup occurring as the common multiplicative reduct of the semirings SR_6 and TR_6. The closing paragraph of Shao, Ren, and Gao~\cite{ShaoRenGao2026} asks for the finite-basis and subvariety questions for the four remaining compatible additions on M. We answer these questions for the four isomorphism types R_01,R_02,R_11,R_12. First, we classify all compatible additions on M: there are nine labelled additions and six isomorphism types, parametrized by R_ij with 0≤ i≤ j≤ 2. For each of the four new types we give a graph-theoretic criterion for every identity, an explicit infinite basis, and a proof of nonfinite basability. The generated varieties V(R_01) and V(R_02) have eleven subvarieties each, while V(R_11) has sixty-six. The lattice Sub(V(R_12)) is countably infinite. Every identity in this variety reduces to a subset of twenty-five fixed identities together with two monotone path families γ_n and gammaD_n. This yields a canonical signature (H,p,q), complete normal forms, explicit meet and join operations, and a formula for all covers. There are 153 fixed nodes, 43 one-parameter families, and 9 two-parameter families; exactly eighteen subvarieties are finitely based, and the unique limit subvariety is V(SR_6). The strong nonfinite-basis status of the four finite semirings remains open.
ER -