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

1,4

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

Table of n, a(n) for n=1..15.

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

Formula

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

Crossrefs

Cf. A023814, A001423.

Cf. A383885, A383871.

Sequence in context: A350264 A113376 A205311 * A258338 A245197 A281340

Adjacent sequences: A383883 A383884 A383885 * A383887 A383888 A383889

Keyword

nonn

Author

Elijah Beregovsky, May 13 2025

Status

approved