A383886 Number of 3-nilpotent semigroups, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

0, 0, 1, 8, 84, 2660, 609797, 1831687022, 52966239062973, 12417282095522918811, 26530703289252298687053072, 1008860098093547692911901804990610, 1378288413994605341053354105969660808031163, 36959929418354255758713676933402538920157765946956889, 14799968982226242179794503639146983952853044950740907666303436922

Offset: 1

Author

Elijah Beregovsky, May 13 2025

Keywords

• nonn

Comments

A semigroup S is nilpotent if there exists a natural number r such that the set S^r of all products of r elements of S has size 1.

If r is the smallest such number, then S is said to have nilpotency degree r.

This sequence counts semigroups S that have an element e such that for all x,y,z in S x*y*z = e.

In 1976 Kleitman, Rothschild and Spencer gave an argument asserting that the proportion of 3-nilpotent semigroups, amongst all semigroups of order n, is asymptotically 1. Later opinion regards their argument as incomplete, and no satisfactory proof has been found.

References

• H. Jürgensen, F. Migliorini, and J. Szép, Semigroups. Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest, 1991.

Links

• Andreas Distler and James D. Mitchell, The number of nilpotent semigroups of degree 3, arXiv:1201.3529 [math.CO], 2012.
• Igor Dolinka, D. G. FitzGerald, and James D. Mitchell, Semirigidity and the enumeration of nilpotent semigroups of index three, arXiv:2411.00466 [math.CO], 2024.
• Pierre A. Grillet, Counting Semigroups, Communications in Algebra, 43(2), 574-596, (2014).
• D. J. Kleitman, B. R. Rothschild, and J. H. Spencer, The number of semigroups of order n, Proc. Amer. Math. Soc. 55 (1976), 227-232.
• Index entries for sequences related to semigroups

Crossrefs

Cf. A023814, A001423.

Cf. A383885, A383871.

Formula

a(n) = A383871(n)/2n! * (1+o(1)). See Grillet paper in Links.