Two operations, one semigroup: rectangular doppelsemigroups as pairs of inflations of a band, and abelian dimonoids as abelian g-dimonoids

Abstract

A doppelsemigroup is a set carrying two associative operations ⊣ and ⊢ that satisfy the two interassociativity laws (x ⊣ y) ⊢ z=x ⊣ (y ⊢ z) and (x ⊢ y) ⊣ z=x ⊢ (y ⊣ z); it is rectangular when both operations obey x · y · z=x · z, in which idempotency is not required. Gavrylkiv, correcting the published count of three-element doppelsemigroups from 75 to 77 because "the nontrivial rectangular doppelsemigroups were overlooked", tabulates the number rds(n) of rectangular doppelsemigroups of order n up to isomorphism for n ≤ 6 and asks, as his Problem 1, for the values at n ≥ 7. We prove a structure theorem: a pair of operations on a set is a rectangular doppelsemigroup precisely when both operations are inflations of one and the same associative rectangular operation, along two arbitrary and mutually unrelated maps fixing its products. The interassociativity laws are thus free on top of a shared band, and the band is intrinsic — the two operations have the same idempotents, the same image, and agree there. Four consequences follow: the rectangular interassociates of a rectangular semigroup are exactly the inflations of its own band; a rectangular doppelsemigroup whose two operations differ has at least three elements, one such exists at every n ≥ 3, and the smallest member of that family is on the nose the structure the 2019 classification overlooked, so the published erratum is explained rather than exhibited; a second dichotomy sits beside the published "strong ⇔ trivial"; and, using the structure theorem to generate candidates and a compiled certificate checker to verify them, rds(7) ≥ 571, rds(8) ≥ 2193 and rds(9) ≥ 8594 — the first three values Problem 1 asks for, each as that many pairwise non-isomorphic algebras. The same machinery reproduces every published value of rds from n=3 on, and the rectangular-semigroup counts of OEIS A390162 at n=3,…,9. A second freeness theorem concerns Loday's dimonoids and the g-dimonoids, or generalized dimonoids, which drop Loday's axiom (L₂). Two tables of Gavrylkiv print one and the same column of six integers, 1,4,17,103,791,10870, for the abelian dimonoids and for the abelian g-dimonoids, and neither paper remarks on it. We show that the agreement is forced at every order: under the abelian law x ⊣ y=y ⊢ x each of (L₁), (L₂), (L₃) unwinds, modulo associativity alone, to the single identity (xy)z=(xz)y, so the abelian dimonoids and the abelian g-dimonoids are literally the same algebras and adm(n)=agdm(n) for every n — two of the eight columns the two papers leave open are one column. The dimonoid half of that criterion is a 2013 lemma of A. V. Zhuchok and is credited, not claimed; what is new is that it survives the deletion of (L₂). This is an identity between counting problems and not a new value: the first unpublished order n=7 is priced below at roughly twenty CPU-days for the enumeration used here and is not attempted. The published g-dimonoid tables at orders 2 and 3 and six published order-4 values are also re-derived from the axioms, the abelian one of them by a single computation serving both varieties. Every theorem below is machine-checked in Lean 4.

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