Identify all varieties generated by a semigroup of order 6
Statement
A variety of semigroups is a class closed under sub-semigroups, homomorphic images, and direct products — equivalently, the semigroups satisfying some set of identities. For a finite semigroup $S$, let $\mathbf{V}(S)=HSP(S)$ be the variety it generates. Every variety generated by a semigroup of order $\le 5$ has been completely identified (equational basis, subvariety lattice, generators). Identify all varieties generated by a semigroup of order $6$: for each of the $15{,}973$ semigroups of order $6$, determine a finite equational basis for $\mathbf{V}(S)$ (or show it is non-finitely based) and its position in the lattice of subvarieties.
Acceptance. ADVANCES: for a batch of order-$6$ semigroups not yet resolved on the companion website, a finite equational basis for each $\mathbf{V}(S)$ (or a proof it is non-finitely based) with its subvariety-lattice placement, each basis machine-verified. FULLY RESOLVES: the complete identified list for all order-$6$ semigroups. Provide the bases plus a verification script (e.g. GAP/Prover9 checking the identities and their completeness).
Background
Araújo, J. P. Araújo, Bentz, Cameron, Kinyon et al., 'A Survey on Varieties Generated by Small Semigroups and a Companion Website' (arXiv:1911.05817; J. Algebra 2023), Problem 7.1. Order $\le 5$ is fully done in the survey (bases and subvariety lattices tabulated) via a computational pipeline (GAP `Smallsemi`, identity-checking, automated reasoning, SAT solvers, finite model builders). Order $6$ is the stated open frontier: the companion website lists the varieties determined so far and only conjectures for those not yet resolved. Prior related result: Lee & Zhang (LMS J. Comput. Math. 18 (2015)) settled the finite-basis STATUS of every order-$6$ semigroup (exactly four minimal non-finitely-based semigroups of order $6$ among the $15{,}973$); the variety-IDENTIFICATION problem here is strictly finer and open. An attacker must bring: the survey's exact GAP pipeline, applied class-by-class; the work is embarrassingly decomposable (per generator, per variety), each claimed basis machine-checkable, each sub-classification independently publishable.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | A Survey on Varieties Generated by Small Semigroups and a Companion Website — Problem 7.1 | arxiv |
Investigations · 0
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