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How many groups and semigroups and rings and the like - abstract algebra

lpetrich

lpetrich

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I'd started Abstract Algebra long ago, but I want to post on something more specific: how many abstract-algebra entities there are. I will consider only finite entities, those with finite sets of possible operation arguments. One can easily show that the number of each kind of entity must be finite, because the number of possible operation tables without constraints is an upper limit on it.

For a unary operation and order (set size) n, that upper limit is n^n: 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000, ...

For a binary operation and order n, that upper limit is n^(n^2): 1, 16, 19683, 4294967296, 298023223876953125, ...

But there is a problem. If one relabels the set members, one will get an equivalent operation, so one has to consider all the possible relabelings.

I will start with a unary function f to make it easy. The only constraint that I will impose is closure. For a set S, f(S) must be a subset of S. For the empty set, one gets the empty algebra. For a one-element set, one gets the trivial algebra: (f,{a}) has f(a) = a. For two elements, one has 4 possibilities:
f(a} = a, f(b) = a / f(a} = a, f(b) = b / f(a} = b, f(b) = a / f(a} = b, f(b) = b

Turning a and b into 1 and 2, I will use this shorthand: 11, 12, 21, 22

If we impose the constraint that f(S) = S, then the possible tables are 12, 21.

For finite sets, f(S) = S implies that f is a bijection, meaning that it is invertible. One can define an inverse function g such that f(g(x)) = g(f(x)) = x.
 
There is a problem, however, one can interchange 1 and 2. When we do that, f(S) in S gives us 3 distinct functions
{(12), (21), (11 22)}

while f(S) = S gives us 2 distinct functions
{(12), (21)}

Doing this analysis for order 3, I find 7 and 3 distinct functions

{(123), (231 312), (111 222 333), (132 213 321), (112 131 221 233 313 322), (113 121 122 133 223 323), (211 212 232 311 331 332)}

{(123), (231 312), (132 213 321)}

Bijections create permutations, and permutations are well-understood. A permutation is a combination of cyclic permutations, permutations like 1->2, 2->3, 3->1. The possible decompositions:

1
11 2
111 12 3
1111 112 13 22 4
11111 1112 113 122 14 23 5

How many permutations with each set of cycle lengths is n! / product of k^(m(k))*(m(k))! over cycle lengths k with number of cycles with that length m(k)

n is the total number, equal to sum of k*m(k) for cycle lengths k

The {m(k) over k} or else {individual k values} form an integer partition of n.

This has a generating function:
exp(sum of t(k)/k over cycle lengths k) = 1/n! * ( N({m}) * product of t(k)^(m(k)) over cycle lengths k )

For the number of partitions of each positive integer n, there is no simple formula, but there is a generating function:

sum of np(k)*t^k over k = product of 1/(1-t^k) over k

For more,  Partition (number theory),  Partition function (number theory)

The asymptotic formula for np(n) is 1/(4n*sqrt(3)) * exp(pi*sqrt(2n/3))
 
A000041 - OEIS - a(n) is the number of partitions of n (the partition numbers).
Mathematica built-in: PartitionsP

Starting from empty: 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173525

Turning to the general case of f(S) in S,
A001372 - OEIS - Number of mappings (or mapping patterns) from n points to themselves; number of endofunctions.

Starting from empty: 1, 1, 3, 7, 19, 47, 130, 343, 951, 2615, 7318, 20491, 57903, 163898, 466199, 1328993, 3799624, 10884049, 31241170, 89814958, 258604642, 745568756, 2152118306, 6218869389, 17988233052, 52078309200, 150899223268, 437571896993, 1269755237948, 3687025544605, 10712682919341, 31143566495273, 90587953109272, 263627037547365


There exist some formulas for the counts of full case of f(S) in S, but they are rather complicated. The asymptotic behavior is

c0 * c1^n / sqrt(n)
where
c0 = 0.442876769782206479836...
and
c1 = 2.9557652856519949747148...
(Otter's unrooted-tree constant)

This is much slower than the naive count, with its ignoring of relabelings, but it is still a rapid increase.
 
Going further with unary functions over finite sets, let us see what happens when we repeatedly apply it to the elements of S.

First, if the function repeats itself.
f(1) = 1
f(f(1)) = 1
f(f(f(1))) = 1
...
giving
1, 1, 1, 1, 1, ...

Then, if it doesn't.
f(1) = 2
Repeating f, we must evaluate f(2)
If f(2) = 1, then we get
1, 2, 1, 2, 1, 2, ...
If f(2) = 2, then we get
1, 2, 2, 2, 2, 2, ...

If f(2) = 3, then we must evaluate f(3)
If f(3) = 1, then we get
1, 2, 3, 1, 2, 3, ...
If f(3) = 2, then we get
1, 2, 3, 2, 3, 2, 3, ...
If f(3) = 3, then we get
1, 2, 3, 3, 3, 3, 3, ...

So we get a limit cycle with other elements approaching it from outward branches. The other elements need not be in the same branch, and they can merge with other elements as they approach the limit cycle. That's what makes the general formula so complicated.
 
Some functions are reducible: f(1) = 1, f(2) = 2 -- 12 -- can be divided into two parts, both of them f(a) = a. But f(1) = 2, f(2) = 1 -- 21 -- cannot be divided into two parts, so it is irreducible.

I've calculated the number of irreducible functions. It also requires a rather complicated formula, but I have implemented it, and I find A002861 - OEIS With the empty set added on,

1, 1, 2, 4, 9, 20, 51, 125, 329, 862, 2311, 6217, 16949, 46350, 127714, 353272, 981753, 2737539, 7659789, 21492286, 60466130, 170510030, 481867683, 1364424829, 3870373826, 10996890237, 31293083540, 89173833915, 254445242754, 726907585652, 2079012341822, 5952451249381, 17059518771406, 48937697963270

The fraction of the total slowly declines, and is approximately 1/(c0+c1*log(n)) for c0 ~ -8.5 and c1 ~ 3.8
 
I now go from unary to binary functions: binary algebras, groupoids, or magmas.

A naive count of groupoids of order n is n^(n^2) and that gets very big very fast:

1, 1, 16, 19683, 4294967296, 298023223876953125

Taking care of relabelings gives the number of different groupoids: A001329 - OEIS

1, 1, 10, 3330, 178981952, 2483527537094825, 14325590003318891522275680, 50976900301814584087291487087214170039, 155682086691137947272042502251643461917498835481022016

There is a rather complicated formula for this result, and it is asymptotic to n^(n^2)/n! = (naive number of order-n groupoids) / (number of permutations of n objects)
 
That is a very big field to work in, so let us consider various constraints on groupoids.

First, being commutative: a*b = b*a for all a, b in the groupoid.

For commutative ones, the count is A001425 - OEIS

1, 1, 4, 129, 43968, 254429900, 30468670170912, 91267244789189735259, 8048575431238519331999571800, 24051927835861852500932966021650993560, 2755731922430783367615449408031031255131879354330

This is also given by a formula, and this has an asymptotic value similar to the general case: n^(n*(n+1)/2) / n! = (naive count) / (permutations of n objects)


A groupoid has a left identity el if el*a = a for all a, and a right element er if a*er = a. If it has both a left and a right identity, then that two-sided identity is unique: e. There can be more than one left identity if there are no right identities, and likewise for right identities for no left identities.

For instance, if a*b = a, then every element is a right identity.

A groupoid has a left zero zl if zl*a = zl for all a, and a right zero zr if a*zr = zr. As with identities, there can be more than one left zero if there are no right zeros, and more than one right zero if there are no left identities. If there are both, then the zero is unique.

For instance, if a*b = a, then every element is a left zero.

Groupoids can have both identities and zeros.


For ones with an identity (or zero), the count is A090601 - OEIS

1, 2, 45, 43968, 6358196250, 236919104155855296, 3682959509036574988532481464, 35398008251644050232134479709365068115968, 292415292106611727928759157427747328169866020125762652311

Another formula, and asymptotic case n^((n-1)^2+1) / n! = (naive count) / (permutations)

For commutative ones with an identity (or zero), the count is A038017 - OEIS

1, 2, 15, 720, 409600, 3920030472, 775775333825891, 3837862827737186253664, 558740081065710564284870598075, 2755731923933734753149997221152548428020, 52099631413533260628548814884449469572205033391248

Another formula, and asymptotic case n^(n*(n-1)/2+1) / n! = (naive count) / (permutations)
 
I'll now turn to an analogue of division. For all elements a, b, there are unique elements x, y satisfying
a*x = b, y*a = b

A groupoid that satisfies this division property is a quasigroup. If it also has an identity, it is a loop.

The operation table of a quasigroup is a Latin square, a square where every row and column is a permutation of its symbols. For a loop, one row and one column have some canonical order.

Number of quasigroups: A057991 - OEIS

1, 1, 1, 5, 35, 1411, 1130531, 12198455835, 2697818331680661, 15224734061438247321497, 2750892211809150446995735533513, 19464657391668924966791023043937578299025

Number of loops: A057771 - OEIS

0, 1, 1, 1, 2, 6, 109, 23746, 106228849, 9365022303540, 20890436195945769617, 1478157455158044452849321016

Number of commutative quasigroups: A057992 - OEIS

1, 1, 1, 3, 7, 11, 491, 6381

Number of commutative loops: A089925 - OEIS

0, 1, 1, 1, 2, 1, 8, 17, 2265, 30583

Number of Latin squares: A002860 - OEIS

1, 2, 12, 576, 161280, 812851200, 61479419904000, 108776032459082956800, 5524751496156892842531225600, 9982437658213039871725064756920320000, 776966836171770144107444346734230682311065600000

Number of reduced (loop-like) Latin squares: A000315 - OEIS

1, 1, 1, 4, 56, 9408, 16942080, 535281401856, 377597570964258816, 7580721483160132811489280, 5363937773277371298119673540771840

(full number) = (reduced number) * n! * (n-1)! -- general Latin squares are row and column permutations of reduced ones

OEIS has no formulas for the counts of any of these entities.
 
Quasigroups and loops are not associative in general, so I turn to adding associativity to groupoids:

(a*b)*c = a*(b*c)

This gives semigroups. Associativity has the nice property that semigroups can be implemented as matrices, since matrix multiplication is associative. Semigroups are also related to finite state machines, since an input is a function that makes a new internal state from an old one. That is easy to implement in matrix form for a finite set of internal states.

Number of semigroups: A027851 - OEIS

1, 1, 5, 24, 188, 1915, 28634, 1627672, 3684030417, 105978177936292

Number of semigroups where equivalence includes reversing the operator: A001423 - OEIS

1, 1, 4, 18, 126, 1160, 15973, 836021, 1843120128, 52989400714478, 12418001077381302684

Number of commutative semigroups: A001426 - OEIS

1, 1, 3, 12, 58, 325, 2143, 17291, 221805, 11545843, 3518930337

-

An inverse semigroup has the property that for all x, there is a unique y that satisfies x = x*y*x. Alternately, x and y are related by x = x*y*x and y = y*x*y

Number of inverse semigroups: A118099 - OEIS

1, 3, 8, 24, 76, 284, 1195

Number of inverse semigroups with reversing the operator: A001428 - OEIS

1, 2, 5, 16, 52, 208, 911, 4637, 26422, 169163, 1198651, 9324047, 78860687, 719606005, 7035514642

Number of commutative inverse semigroups: A234843 - OEIS

1, 2, 5, 16, 51, 201, 877, 4443, 25284, 161698, 1145508, 8910291, 75373563, 687950735, 6727985390
 
A monoid is a semigroup with an identity element.

Number of monoids: A058129 - OEIS

0, 1, 2, 7, 35, 228, 2237, 31559, 1668997

Number of operator-reversal monoids: A058133 - OEIS

0, 1, 2, 6, 27, 156, 1373, 17730, 858977, 1844075697, 52991253973742

Number of commutative monoids: A058131 - OEIS

0, 1, 2, 5, 19, 78, 421, 2637

An inverse monoid satisfies what an inverse semigroup satisfies.

Number of inverse monoids: A234844 - OEIS

1, 2, 4, 11, 27, 89, 310, 1311, 6253, 34325, 212247, 1466180, 11167987, 92889294, 836021796

Number of commutative inverse monoids: A234845 - OEIS

1, 2, 4, 11, 27, 87, 300, 1259, 5988, 32812, 202784, 1400541, 10669344, 88761928, 799112310
 
It's sometimes not very clear from OEIS where a sequence starts, at size parameter 0 (empty set) or size parameter 1 (one-member set).

That aside, a less restrictive version of the inverse condition is the regularity condition: for every element a, there is at least one element x such that a = a*x*a.

Another interesting property is the cancellative property. A semigroup is left cancellative if a*b = a*c implies b = c, right cancellative if b*a = c*a implies b = c, and plain cancellative if both left and right cancellative.

A finite cancellative semigroup is a group.

A monogenic semigroup has one generator: all the elements are powers of it. A finite one will repeat itself. I write them as (initial elements) (repeating elements in parens). Thus 1(23) is 1^2 = 2, 1*2 = 3, 1*3 = 2, ... If the period is one, then it's called aperiodic.

 Semigroup with two elements

1. Null semigroup: 11.11 and 22.22
Commut, zero

2a. Left-zero semigroup: 11.22
2b. Right-zero semigroup: 12.12

3. Boolean semigroup: 11.12 and 12.22
Commut, identity, inverse

4. Z2 group: 12.21 and 21.12
Commut, identity, inverse

Of these, (3) and (4) are monoids and (4) is a group.

 Semigroup with three elements

1. Z3 group: 123.231.312
Commut, identity, inverse

2. Monogenic: 232.323.232
Commut

3. Aperiodic: 233.333.333
Commut

4. {-1,0,1} under mult: 321.222.123
Commut, identity, zero, inverse

5. 311.123.133
Commut, identity, inverse

6. 311.133.133
Commut

7. Null: 333.333.333

8. 333.323.333
Commut

9. 323.222.323
Commut

10. 313.123.333
Commut, identity

11a. 333.222.333
11b. 323.323.323

12a. 333.123.333
12b. 313.323.333

13. Semilattice: 123.223.333
Commut, identity, inverse

14. Semilattice: 133.323.333
Commut, inverse

15a. Idempotent: 111.222.113
13b. Idempotent: 121.121.123
Regular

16a. Idempotent: 113.223.333
16b. Idempotent: 123.123.333
Regular

17a. Left zero: 111.222.333
17b. Right zero: 123.123.123
Regular

18a. Idempotent (left flip-flop): 111.222.123
18b. Idempotent (right flip-flop): 121.122.123
Regular, identity

Not very sure about what qualifies as an inverse semigroup - my attempt to find what's inverse and what's regular are in diagreement with OEIS's counts. But my evaluations of the commutative and identity properties agree with OEIS's counts.
 
I now consider groups: monoids with inverses: a*inv(a) = inv(a)*a = (identity)

Number of finite groups with each order: A000001 - OEIS

1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 13, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, 1, 2, 3, 4, 1, 6, 1, 52, 15, 2, 1, 15, 1, 2, 1, 12, 1, 10, 1, 4, 2

Commutative groups are often called Abelian ones.

Number of finite abelian groups with each order: A000688 - OEIS

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1

Mathematica has built-in functions, FiniteGroupCount and FiniteAbelianGroupCount, which can go much farther than what OEIS lists.

For finite abelian groups, there is a formula.

Every abelian group is known: it is a product of cyclic groups of prime-power order. If a cyclic group's order is the product of powers of more than one prime, then it can be broken apart into prime-power cyclic groups. This is because Z(n1*n2) = Z(n1)*Z(n2) if n1 and n2 are relatively prime (coprime). For instance, Z(6) = Z(2)*Z(3) and Z(12) = Z(4)*Z(3). But Z(4) does not reduce to Z(2)*Z(2).

Thus, for order 4, there are Z(2)*Z(2) and Z(4), and for order 8, Z(2)*Z(2)*Z(2), Z(2)*Z(4), and Z(8).

The powers of 2 are for 4 = 2^2, 1 1 and 2, and for 8 = 2^3, 1 1 1 and 1 2 and 3.

Thus, we find all integer partitions of a power, Mathematica's IntegerPartitions with count PartitionsP.

So for order n factorizing into (product over primes p of p^(m(p))), the order is (product over p of (number of partitions of m(p)))

-

There isn't an analogous formula for groups in general.
 
There is a simple way of making groupoids from other groupoids. Composition. Take groupoids (G1,*1) and (G2,*2). Compose a product groupoid, (G12,*12) where the elements of G12 are (a1,a2) with a1 in G1 and a2 in G2. The operation is

a*b = (a1*b1,a2*b2)

So from two order-2 groupoids, one can make an order-4 one. One can calculate the number of irreducible groupoids, one that cannot be decomposed in this fashion, from the total number of groupoids.
m(1) = n(1)
m(2) = n(2)
m(3) = n(3)
m(4) = n(4) - m(2)*(m(2)+1)/2
m(5) = n(5)
m(6) = n(6) - m(2)*m(3)
m(7) = n(7)
m(8) = n(8) - m(2)*(m(2)+1)*(m(2)+2)/6 - m(2)*m(4)

One has to be careful with repeated irreducible ones, to avoid duplicates.

With the number of abelian groups, I indeed find that the number of irreducible ones is 1 for a prime power, 0 otherwise.
 
I will now create some comparison tables.

Algebra12345678
Groupoid 11033301.79e82.48e151,43e255.10e371.56e53
Irreducible gpd11033301.79e82.48e152.43e255.10e371.56353
Commutative gpd14129439682.54e83.05e139.13e198.05e27
Irreducible cmt gpd
14129439582.54e83.05e139.13e198.05e27
Gpd w/ identity
1
2
45
43968
6.36e9
2.37e17
3.68e27
3.54e40
Irreducible gpd id
1
2
45
43965
6.36e9
2.37e17
3.68e27
3.54e60
Commutative gpd id
1
2
15
720
409600
3.92e9
7.76e14
3.84e21
Irreducible cmt gpd id
1
2
15
717
409600
3.92e9
7.76e14
3.84e21

The reducible ones barely make a dent in the totals.
 
I will now add the division property.

Algebra
1
2
3
4
5
6
7
8
Quasigroup
1
1
5
35
1411
1.13e6
1.212e10
2.70e15
Irreducible qsg
1
1
5
34
1411
1.13e6
1.22e10
2.70e15
Commutative qsg
1
1
3
7
11
491
6381
Irreducible cmt qsg
1
1
3
6
11
488
6381
Loop - qsg with identity
1
1
1
2
6
109
23746
1.06e6
Irreducible loop
1
1
1
1
6
108
23746
1.06e6
Commutative loop
1
1
1
2
1
8
17
2265
Irreducible cmt loop
1
1
1
1
1
7
17
2263

They are becoming more manageable.
 
I will add the associative property.


Algebra123456789
Semigroup15241881915286341.67e63.68e91.06e14
Irreducible smg15241731915285141.67e63.68e91.06e14
Commutative smg1312583252143172912218051.15e7
Irreducible cmt smg1312523252107172912216391.15e7
Monoid - smg w/ identity127352282237315591.67e6
Irreducible mnd127322282223315591.67e6
Commutative monoid12519784212637
Irreducible cmt mnd12516784112637
Group - mnd w/ inverses111212152
Irreducible group111111131
Commutative group111211132
Irreducible cmt group111110111

So the number gets very small when one adds a lot of constraints.
 
The On-Line Encyclopedia of Integer Sequences® (OEIS®) was a valuable resource as I collected these numbers of algebraic entities. It is good for guessing what sequence one might have. Enter the sequence values that one already knows and its software will find which sequences match it.

For instance, 1,1,2,3,5,8,13 gives the Fibonacci sequence as its first hit. Its second hit is the number of transitive rooted trees, a sequence that deviates from the Fibonacci sequence in its 12th member: 88 instead of 89.

 On-Line Encyclopedia of Integer Sequences
AMS :: The On-line Encyclopedia of Integer Sequences, or, Confessions of a Sequence Addict: An MAA Invited Address by Neil J. A. Sloane
The On-Line Encyclopedia of Integer Sequences by Neil J.A. Sloane
Its creator:  Neil Sloane

"Founded in 1964 by N.J.A. Sloane"

That was long before the Internet, of course, and back then  Packet switching, how the Internet works, was being researched.

Neil Sloane was a graduate student back then, and he started collecting integer sequences to support his work on combinatorics. He first stored them on punched cards, and only later on more accessible digital media. He wrote two books of collections of sequences, "A Handbook of Integer Sequences" in 1973 with 2,372 sequences and "The Encyclopedia of Integer Sequences" in 1995 with 5,488 sequences.

Publishing sequences in book form was awkward, and NS decided to go online, first with an e-mail service in 1994, then with a website in 1996. The site was first hosted by AT&T, and later on outside hosting. The site is now managed by The OEIS Foundation Inc founded in 2009. It now has more than 300,000 entries.
 
The OEIS hosts several sequences that are not strictly speaking one-dimensional sequences of integers.

For instance, 1,1,1,1,2,1,1,3,3,1,1,4,6,4,1 gives Pascal's triangle, stored as a sequence of rows: 1 then 1,1 then 1,2,1 then 1,3,3,1 then 1,4,6,4,1 then ...

It also stores sequences of digits of irrational numbers, both in decimal and in binary, and it stores some continued-fraction expansions. It stores some sequences of fractions as separate sequences of numerators and denominators.


I tried putting into OEIS my counts of irreducible groupoids and related algebraic entities, but I had no success. Even the count of irreducible abelian groups failed. For the record, that count is 1 for a prime power and 0 otherwise.
 
I mentioned "idempotent" in a previous post. An idempotent element is one whose square equals itself: for x, x*x = x.

This is also true of any power of x in the system, even if it is not associative.

The associative property: for every a,b,c: (a*b)*c = a*(b*c)

A weaker version is the alternating property, a two-variable version of associativity. For every a,b:
(a*a)*b = a*(a*b)
(a*b)*a = a*(b*a)
(b*a)*a = b*(a*a)

Even weaker is power-associativity. When calculating some power x^n, it does not matter how one groups the x's in it.
(x*x)*x = x*(x*x)
((x*x)*x)*x = (x*(x*x))*x = (x*x)*(x*x) = x*((x*x)*x) = x*(x*(x*x))
etc.

Power-associativity < alternativity < associativity


In ordinary arithmetic, 0 is idempotent in addition, 1 is idempotent in multiplication. A group has only one idempotent element, its identity.

In Boolean algebra, "and" and "or" have both "true" and "false" idempotent.

Fuzzy logic violates idempotence, except for the "Zadeh norm": a and b = min(a,b), a or b = max(a,b)
 
A semigroup with every element idempotent is called a band.
If it is commutative, then it is called a semilattice.

Number of each kind of algebra with all-idempotent elements. The second list of numbers is for irreducible ones.

Idempotent groupoids - A038018 - OEIS - general numbers of idempotents: A038021 - OEIS

1, 1, 3, 138, 700688, 794734575200, 307047114275109035760, 61899500454067972015948863454485, 9279375475116928325576506574232168143663715776
1, 1, 3, 138, 700682, ...

Asymptotic to n^(n*(n-1)) / n!

Idempotent commutative groupoids - A030257 - OEIS - general numbers of idempotents: A038021 - OEIS

1, 1, 1, 7, 192, 82355, 653502972, 110826042515867, 479732982053513924168, 62082231641825701423422054735, 275573192431752191557427399293883120600, 47363301285150007842253190185182901101879369430257
1, 1, 1, 7, 191, ...

Asymptotic to n^((n-1)*(n-2)/2) / n!

For order 2, these groupoids are idempotent:

11.12 (equvalent to 12.22), 11.22 12.12

All of them are semigroups and the first one is a commutative monoid.

For order 3, these commutative groupoids are idempotent:

111.121.113 _ 111.122.123 _ 112.121.213 _ 112.122.223 _ 112.123.233 _ 113.122.323 _ 132.321.213

Two of them are semigroups: 111.121.113 _ 111.122.123 and the second one is a monoid.

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Idempotent semigroups - general numbers of idempotents: A058108 - OEIS

1, 3, 10, 46, 251, 1682, 13213, 119826, 1228712
1, 3, 10, 40, 251, 1652, ...

Idempotent commutative semigroups - A006966 - OEIS - general numbers of idempotents: A058116 - OEIS

1, 1, 2, 5, 15, 53, 222, 1078, 5994, 37622, 262776, 2018305, 16873364, 152233518, 1471613387, 15150569446, 165269824761, 1901910625578, 23003059864006
1, 1, 2, 4, 15, 51, 222, 1073, ...

Idempotent monoids - general numbers of idempotents: A058137 - OEIS

1, 1, 3, 10, 46, 251, 1682
1, 1, 3, 9, 46, 248, 1682

Idempotent commutative monoids - A006966 - OEIS - general numbers of idempotents: A058142 - OEIS

1, 1, 1, 2, 5, 15, 53, 222, 1078, 5994, 37622, 262776, 2018305, 16873364, 152233518, 1471613387, 15150569446, 165269824761, 1901910625578, 23003059864006
1, 1, 1, 1, 5, 14, 53, 220, 1077, ...
 
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