Semigroups Varieties

Varieties of Semigroups

João Araújo (Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Center for Mathematics and Applications and CEMAT-CIÊNCIAS)

João Pedro Araújo (University of Stanford, California)

Peter J. Cameron (School of Mathematics and Statistics & CIRCA, University of St Andrews, Scotland; CEMAT-CIÊNCIAS, University of Lisbon)

Edmond W. H. Lee (Nova Southeastern University, Florida)

Jorge Raminhos (Universidade Aberta and CEMAT-CIÊNCIAS)

This project has been partially supported by the Fundação para a Ciência e a Tecnologia through the project CEMAT-CIÊNCIAS UID/Multi/04621/2013, and through project “Hilbert’s 24th problem” PTDC/MHC-FIL/2583/2014.

Any comments are welcome: jj.araujo@fct.unl.pt, raminhosj@gmail.com

Documentation

Introduction

Given a finite semigroup S as input (by specifiying the multiplication table, presentation, GAP smallsemi or smallgroup ID), this site returns:

The representative in the isomorphism class of S, whose vector S is lexicographically the least;
A basis for the variety var(S), if var(S) = var(B) for some variety whose identification system is available in the variety database.

The variety database (presently, the identification systems available include all 218 varieties generated by semigroups of order up to five) can also accessed by using the following options:

Input of the variety ID;
Search for a variety whose identity basis is equivalent to a given set of identities;
Return the list of all varieties in the database, implying or implied by a set of user identities, and draw a lattice of the resulting varieties.

The site will also display a list of known varieties generated by semigroups of order 6, and conjectures for the varieties not known.

Multiplication table

Enter a multiplication table of a semigroup S. Default elements are 1 to N, being N the order of the set, but base 0 instead of 1 or even specific elements can be defined at the option "Define set elements".
Elements can be separated by "," or spaces, and optionally include "[" and "]" to enclose each line and/or the full multiplication table. If the elements are all single-digit, all or part of the separators can be ommited. For instance, all below input strings can used to input the same multiplication table:

1 1 1 1 1 1 1 1 2	space separated
1,1,1,1,1,1,1,1,2	comma separated
1, 1, 1, 1, 1, 1, 1, 1, 2	mixed commas and spaces
[1, 1, 1, 1, 1, 1, 1, 1, 2]	"[" and "]" enclosed
[ [ 1, 1, 1 ], [ 1, 1, 1 ], [ 1, 1, 2 ] ]	GAP syntax
111 111 112	separators ommited (only for single digit elements)
111111112	separators ommited (only for single digit elements)

Using the GAP syntax option, it's possible to copy a multiplication table from GAP and paste it here. For example, to get the multiplication table of the group with ID=(5,1) at the SmallGroup GAP package:

gap> MultiplicationTable(SmallGroup(5,1));
[ [ 1, 2, 3, 4, 5 ], [ 2, 3, 4, 5, 1 ], [ 3, 4, 5, 1, 2 ], [ 4, 5, 1, 2, 3 ], [ 5, 1, 2, 3, 4 ] ]

The number of multiplication tables values must be exactly order², otherwise and error will be returned. Only semigroups will be accepted, so the associativity property is checked.

Semigroup Variety

After entering a multiplication table or selecting a GAP smallsemi ID, by pressing "Semigroup Variety", the site will give a basis for the variety var(S), if var(S) = var(B) for some variety whose identification system is available in the variety database. Semigroups up to order 100 are accepted, but the representative in the isomorphism class of S, whose vector S is lexicographically the least, will only be computed in case the order of S is less or equal to 10.

Group Variety

After selecting a GAP smallgroup ID, by pressing "Group Variety", the site will give known information about the group variety generated by the chosen group. Semigroups up to order 255 are accepted.

Presentation to Cayley

Identify a semigroup by entering its presentation in Prover9 syntax, or choose one of the examples provided.
The presentation is divided in two parts: the theory to follow and one more relations (identities) between initial elements of the set.

Use Prover9 syntax to specify the identities that define the theory and the relations. Finalize each identity with the character ".".

Examples:

Consider the following example presentations, and how to enter corresponding theory and relations into the site using Prover9 syntax:

Presentation	Theory	Relations
⟨ a,e \| ea²=a², e²=ae=e ⟩ = { a,e,a²,ea }	x(yz)=(xy)z.	(ea)a=aa. ee=ae. ae=e.
⟨ a \| a⁵ = 1 ⟩ = { a,a²,a³,a⁴,1 }	x(yz)=(xy)z. x1=x. 1x=x.	(((aa)a)a)a=1.
⟨ a,e \| ae=0, ea=a, e²=e ⟩ ∪ {1} = { 0,a,e,1 }	x(yz)=(xy)z. x0=0. 0x=0. x1=x. 1x=x.	ae=0. ea=a. e*e=e.

The site will create new elements of the set if necessary to close the multiplication table. If more than 20 elements are reached, an error will be returned.

Cayley to Presentation

After entering a multiplication table or selecting a GAP smallsemi ID, by pressing "Cayley to Presentation", the site will produce a presentation from the semigroup's multiplication table.

GAP smallsemi ID

Specify a semigroup by entering the ID defined at GAP package "smallsemi". Orders up to 6 are accepted, with the maximum sequence number corresponding to the number of different semigroups of that order (semigroups considered equivalent if isomorphic or anti-isomorphic):

Order	Sequence
2	1 - 4
3	1 - 18
4	1 - 126
5	1 - 1160
6	1 - 15973

GAP smallgroup ID

Specify a semigroup by entering the ID defined at GAP package "smallgroups". Orders up to 255 are accepted.

Define set of elements

Define the elements of S to use at "multiplication table" option. The elements specified will also be used on other outputs of the site.

Default elements are named 1 to N, with N equal to the order of S.
Another option is start elements with 0 up to N-1.
Choose "other" to enter other names for elements. Separate elements with commas or spaces. If an insufficient number of elements is provided, the site will use numbers to name the necessary additional elements.

Bands and other special cases

The site deals with special cases like bands, groups and order 6 semigroups thar generate non-finitely based varieties. Multiplication tables of some examples are:

Bands:

Band 16: [0000000,0100116,2222222,3333333,0103456,0133456,0100666]
Band 17: [0000000,0101156,2222222,0123356,0124456,0101656,0106656]
Band 18: [000000,010145,222222,012345,015445,555555]
Band 19: [000045,011145,012345,033345,004545,055545]
Band 20: [00000000,01011567,22222222,01033567,01044567,01055567,01715567,77777777]

Non-finitely based semigroups of order 6:

B12: [000000, 000112, 012020, 000335, 012345, 035050]
A12: [000000, 000112, 012122, 000335, 012345, 035355]
AG2: [000005, 000125, 012125, 000345, 034345, 555550]
L: [000000, 000001, 000002, 001031, 002042, 011335]

Variety by ID

Enter the variety ID (order of primitive generator and sequence number). With the current variety database, accepted sequence numbers are:

Order	Sequence
1	1 - 1
2	1 - 5
3	1 - 14
4	1 - 53
5	1 - 145

Identities

Enter one or more identities in Prover9 format, and the site will retrieve the variety whose identity basis is equivalent to the set of identities entered.

Use Prover9 syntax to specify the identities that define the theory and the relations. Finalize each identity with the character ".".

Example 1:

Enter the identities "y=y*z. x=x*x." (spaces are optional) and press "Go!".
Variety (2,3), whose identity basis is the identity xy=x, will be displayed, as these sets of identities are equivalent.

Example 2:

After acessing a variety, for instance with "Variety ID" option, it's possible to use the "Copy" button to copy the identity basis into the clipboard, in Prover9 format. Try to past these identities into the "Identities" option, and the site will find the corresponding variety.

Varieties Filter

Suppose we have some property and want to check which varieties in the database satisfy it. This can be done on the website, using these two distinct functionalities:

Returns the varieties whose identity basis is implied by the user's assumptions. To use it fill assumptions and press "Prover9" on the left side of the screen to search for proofs, or "Mace4" to search for countermodels.
Returns the varieties whose identity basis implies the user's goals. To use it fill goals and press "Prover9" on the right of the screen to search for proofs, or "Mace4" to search for countermodels.

Use Prover9 syntax to specify the identities defining assumptions and/or goals. Finalize each identity with the character ".".

It’s not necessary to specify the associativity property, it will be included by default.

Prover9, an automatic theorem prover, and its accompanying program, Mace 4, which look for countermodels, will run simultaneously to check the requested implications.

If the filtered set of varieties has less of 80 varieties, it’s possible to obtain a lattice of those varieties by pressing “Lattice”. The lattice graphic can be removed, getting back to the list, by pressing “List”.

Example:

Get only the varieties from the database whose semigroups are bands. These semigroups satisfy the identity x² ≈ x.

Fill the "Goals:" input field with the following string: "x*x=x.", and click "Prover9" in the right option "Varieties whose identity basis implies the user's goals:"

You should get the 15 varieties below:

Variety	Identity basis (apart from xx ≈ x)	Semigroups in this variety
(1,1)	x ≈ y	The trivial semigroup
(2,2)	xy ≈ yx	Semilattices
(2,3)	xy ≈ x	Left-zero semigroups
(2,4)	xy ≈ y	Right-zero semigroups
(3,7)	xyz ≈ xzy	Left normal bands
(3,8)	xyz ≈ yxz	Right normal bands
(3,10)	xyx ≈ xy	Semilattices of left-zero semigroups
(3,13)	xyx ≈ yx	Semilattices of right-zero semigroups
(4,32)	xyzx ≈ xzyx	Normal bands
(4,34)	xyxz ≈ xyz	Left quasi-normal bands
(4,44)	xyzy ≈ xzy	Right quasi-normal bands
(4,48)	xyx ≈ x	Rectangular bands
(5,101)	xyxzx ≈ xyzx	Regular bands
(5,102)	xyzxz ≈ xyz	Left semi-normal bands
(5,142)	xyz ≈ xzxyz	Right semi-normal bands

Press “Lattice” to get the lattice of these varieties.

Add “x*y=y*x.” to the "Goals" field to get “x*x=x. x*y=y*x.”. Press “Prover9” in the right side of the screen. This time only varieties whose semigroups are commutative bands will be obtained (only 3 varieties in the current database).

Variety

Primitive
Generator

Identity
Basis

Common Generator

# max.
subvar.

V(1,1)

x ≈ y

none

None

V(2,1)

1	1
1	1

x² ≈ xy
xy ≈ yx

\(N_2 \) = ⟨ a | a² = 0 ⟩ = { 0,a }

V(2,2)

1	1
1	2

x² ≈ x
xy ≈ yx

\(S\ell_2 \) = { 0,1 }

V(2,3)

1	1
2	2

ax ≈ a

\(LZ_2 \) = ⟨ a,b | a² = ab = a, b² = ba = b ⟩ = { a,b }

V(2,4)

1	2
1	2

xa ≈ a

\(RZ_2 \) = ⟨ a,b | a² = ba = a, b² = ab = b ⟩ = { a,b }

V(2,5)

1	2
2	1

x²a ≈ a
xy ≈ yx

\(\mathbb{Z}_2 \) = ⟨ a | a² = 1 ⟩ = { a,1 }

V(3,1)

1	1	1
1	1	1
1	1	2

x³ ≈ xyz
xy ≈ yx

\(N_3 \) = ⟨ a | a³ = 0 ⟩ = { 0,a,a² }

V(3,2)

1	1	1
1	1	1
1	1	3

x²a ≈ xa
xy ≈ yx

none

V(3,3)

1	1	1
1	1	1
1	2	3

x²a ≈ xa
xy² ≈ yx²

\(J \) = ⟨ a,e | ae=0, ea=a, e²=e ⟩ = { 0,a,e }

V(3,4)

1	1	1
1	1	1
3	3	3

x² ≈ xy

none

V(3,5)

1	1	1
1	1	2
1	1	3

ax² ≈ ax
x²y ≈ y²x

\(\overleftarrow{J} \) = ⟨ a,e | ae=a, ea=0, e²=e ⟩ = { 0,a,e }

V(3,6)

1	1	1
1	1	2
1	2	3

x³ ≈ x²
xy ≈ yx

\(N_2^1 \) = ⟨ a | a² = 0 ⟩ ∪ { 1 } = {0,a,1}

V(3,7)

1	1	1
1	2	1
3	3	3

x² ≈ x
axy ≈ ayx

none

V(3,8)

1	1	1
1	2	3
1	2	3

x² ≈ x
xya ≈ yxa

none

V(3,9)

1	1	1
1	2	3
1	3	2

x³ ≈ x
xy ≈ yx

none

V(3,10)

1	1	1
1	2	3
3	3	3

x² ≈ x
xyx ≈ xy

\(LZ_2^1 \) =
⟨ a,b | a² = ab = a, b² = ba = b ⟩ ∪ { 1 }
= {a,b,1}

V(3,11)

1	1	3
1	1	3
1	1	3

x² ≈ yx

none

V(3,12)

1	1	3
1	1	3
3	3	1

x²ab ≈ ab
xy ≈ yx

none

V(3,13)

1	1	3
1	2	3
1	3	3

x² ≈ x
xyx ≈ yx

\(RZ_2^1 \) =
⟨ a,b | a² = ba = a, b² = ab = b ⟩ ∪ { 1 }
= {a,b,1}

V(3,14)

1	2	3
2	3	1
3	1	2

x³a ≈ a
xy ≈ yx

\(\mathbb{Z}_3 \) = ⟨ a | a³ = 1 ⟩ = { a,a²,1}

V(4,1)

1	1	1	1
1	1	1	1
1	1	1	1
1	1	2	1

x² ≈ yzt

\(F_4 \) = ⟨ a,b | a²=b²=ab=0⟩ = { 0,a,b,ba}

V(4,2)

1	1	1	1
1	1	1	1
1	1	1	1
1	1	2	2

x³ ≈ yzt

none

V(4,3)

1	1	1	1
1	1	1	1
1	1	1	2
1	1	2	1

x² ≈ xyz
xy ≈ yx

\(G_4 \) = ⟨ a,b | a²=b²=0,ab=ba⟩ = { 0,a,b,ab}

V(4,4)

1	1	1	1
1	1	1	1
1	1	1	2
1	1	2	3

x⁴ ≈ xyzt
x²y ≈ xy²
xy ≈ yx

\(N_4 \) = ⟨ a | a⁴ = 0 ⟩ = { 0,a,a²,a³}

V(4,5)

1	1	1	1
1	1	1	1
1	1	1	3
1	2	1	4

x³ ≈ x²
xyx ≈ x²y²
xyx ≈ y²x²
ax²b ≈ axb

none

V(4,6)

1	1	1	1
1	1	1	1
1	1	1	3
1	2	3	4

x³ ≈ x²
x²y² ≈ y²x²
xya ≈ yxa

none

V(4,7)

1	1	1	1
1	1	1	1
1	1	2	1
1	1	1	4

x²ab ≈ xab
xy ≈ yx

none

V(4,8)

1	1	1	1
1	1	1	1
1	1	2	1
4	4	4	4

x³ ≈ xyz

none

V(4,9)

1	1	1	1
1	1	1	1
1	1	3	1
4	4	4	4

x²a ≈ xa
ax² ≈ ax
axy ≈ ayx

none

V(4,10)

1	1	1	1
1	1	1	1
1	1	3	4
1	1	3	4

x²a ≈ xa
ax² ≈ ax
xya ≈ yxa

none

V(4,11)

1	1	1	1
1	1	1	1
1	1	3	4
1	1	4	3

x³a ≈ xa
xy ≈ yx

none

V(4,12)

1	1	1	1
1	1	1	1
1	1	3	4
4	4	4	4

x²a ≈ xa
ax² ≈ ax
xyx ≈ xy

none

V(4,13)

1	1	1	1
1	1	1	1
1	2	3	1
4	4	4	4

x²a ≈ xa
axy² ≈ ayx²

none

V(4,14)

1	1	1	1
1	1	1	1
1	2	3	4
1	2	3	4

x²a ≈ xa
xya ≈ yxa

none

V(4,15)

1	1	1	1
1	1	1	1
1	2	3	4
1	2	4	3

x³a ≈ xa
x²y² ≈ y²x²
xya ≈ yxa

none

V(4,16)

1	1	1	1
1	1	1	1
1	2	3	4
4	4	4	4

x²a ≈ xa
xyx ≈ xy²

none

V(4,17)

1	1	1	1
1	1	1	2
1	1	1	3
1	1	3	4

x³ ≈ x²
x²y² ≈ y²x²
axy ≈ ayx

none

V(4,18)

1	1	1	1
1	1	1	2
1	1	2	3
1	2	3	4

x⁴ ≈ x³
xy ≈ yx

\(N_3^1 \) = ⟨ a | a³ = 0 ⟩ ∪ { 1 } = {0,a,a²,1}

V(4,19)

1	1	1	1
1	1	1	2
1	2	3	1
1	1	1	4

x³ ≈ x²
x²yx² ≈ yxy
x²y² ≈ y²x²

\(B_0 \) =
⟨ a,e,f | e²=e, f²=f, ef=fe=0, ea = af = a ⟩
= { 0,a,e,f }

V(4,20)

1	1	1	1
1	1	1	2
1	2	3	2
1	1	1	4

x³ ≈ x²
x²yx² ≈ yxy

\(A_0 \) = ⟨ e,f | e²=e, f²=f, ef = 0 ⟩ = { 0,e,f,fe }

V(4,21)

1	1	1	1
1	1	1	2
1	2	3	3
1	2	3	4

x³ ≈ x²
x²y² ≈ y²x²
xyx ≈ yx²

\(J^1 \) =
⟨ a,e | ae=0, ea=a, e²=e ⟩ ∪ {1}
= { 0,a,e,1 }

V(4,22)

1	1	1	1
1	1	1	2
3	3	3	3
1	1	1	4

ax² ≈ ax
axy ≈ ayx

none

V(4,23)

1	1	1	1
1	1	1	2
3	3	3	3
1	1	3	4

ax² ≈ ax
xyx ≈ x²y

none

V(4,24)

1	1	1	1
1	1	1	2
3	3	3	3
1	2	1	4

x³ ≈ x²
axy ≈ ayx

none

V(4,25)

1	1	1	1
1	1	1	2
3	3	3	3
1	2	3	4

x³ ≈ x²
xyx ≈ x²y

none

V(4,26)

1	1	1	1
1	1	1	3
3	3	3	3
4	4	4	4

abx ≈ ab

\(P_2 \) = ⟨ a,e | a²e=a², e²=ea=e ⟩ = { a,e,a²,ae }

V(4,27)

1	1	1	1
1	1	2	2
1	1	3	3
1	2	3	4

x³ ≈ x²
x²y² ≈ y²x²
xyx ≈ x²y

\(\overleftarrow{J^1} \) = ⟨ a,e | ae=a, ea=0, e²=e ⟩ ∪ {1} = { 0,a,e,1 }

V(4,28)

1	1	1	1
1	1	2	2
1	1	3	4
1	1	3	4

ax² ≈ ax
x²ya ≈ y²xa

none

V(4,29)

1	1	1	1
1	1	2	2
1	1	3	4
1	1	4	3

ax³ ≈ ax
x²y² ≈ y²x²
axy ≈ ayx

none

V(4,30)

1	1	1	1
1	1	2	2
1	2	3	4
1	2	3	4

x³ ≈ x²
xya ≈ yxa

none

V(4,31)

1	1	1	1
1	1	2	2
1	2	3	4
1	2	4	3

x⁴ ≈ x²
xy ≈ yx

none

V(4,32)

1	1	1	1
1	2	1	4
3	3	3	3
1	2	1	4

x² ≈ x
axya ≈ ayxa

none

V(4,33)

1	1	1	1
1	2	1	4
3	3	3	3
1	4	1	2

x³ ≈ x
axy ≈ ayx

none

V(4,34)

1	1	1	1
1	2	3	4
1	2	3	4
4	4	4	4

x² ≈ x
xyxa ≈ xya

none

V(4,35)

1	1	1	1
1	2	3	4
1	3	2	4
4	4	4	4

x³ ≈ x
xyx ≈ x²y

none

V(4,36)

1	1	1	1
1	2	3	4
1	3	4	2
1	4	2	3

x⁴ ≈ x
xy ≈ yx

none

V(4,37)

1	1	1	1
1	2	3	4
3	3	3	3
3	4	1	2

x³ ≈ x
xyxy ≈ xy²x

\(O_2 \) = ⟨ a,e | e²=ea=e, a²=1⟩ = { a,e,ae,1}

V(4,38)

1	1	1	4
1	1	1	4
1	1	2	4
1	1	1	4

x³ ≈ yzx

none

V(4,39)

1	1	1	4
1	1	1	4
1	1	2	4
4	4	4	1

x²abc ≈ abc
xy ≈ yx

none

V(4,40)

1	1	1	4
1	1	1	4
1	1	3	4
1	1	4	4

x²a ≈ xa
ax² ≈ ax
xyx ≈ yx

none

V(4,41)

1	1	1	4
1	1	1	4
1	2	3	4
1	1	4	4

x²a ≈ xa
xyx ≈ yx²

none

V(4,42)

1	1	1	4
1	1	2	4
1	1	3	4
1	1	4	4

ax² ≈ ax
xyx ≈ y²x

none

V(4,43)

1	1	1	4
1	1	2	4
1	2	3	4
1	1	4	4

x³ ≈ x²
xyx ≈ yx²

none

V(4,44)

1	1	1	4
1	2	2	4
1	3	3	4
1	4	4	4

x² ≈ x
axyx ≈ ayx

none

V(4,45)

1	1	1	4
1	2	3	4
1	2	3	4
4	4	4	1

x³ ≈ x
xya ≈ yxa

none

V(4,46)

1	1	1	4
1	2	3	4
1	3	2	4
1	4	4	4

x³ ≈ x
xyx ≈ yx²

none

V(4,47)

1	1	3	3
1	2	3	4
1	3	3	1
1	4	3	2

x³ ≈ x
xyxy ≈ yx²y

\(\overleftarrow{O_2} \) = ⟨ a,e | e²=ae=e, a²=1⟩ = { a,e,ea,1}

V(4,48)

1	1	3	3
2	2	4	4
1	1	3	3
2	2	4	4

xyx ≈ x

none

V(4,49)

1	1	3	3
2	2	4	4
3	3	1	1
4	4	2	2

ax² ≈ a
axy ≈ ayx

none

V(4,50)

1	1	3	4
1	1	3	4
1	1	3	4
1	3	3	4

xab ≈ ab

\(\overleftarrow{P_2} \) = ⟨ a,e | ea²=a², e²=ae=e ⟩ = { a,e,a²,ea }

V(4,51)

1	1	3	4
1	1	3	4
3	3	4	1
4	4	1	3

x³ab ≈ ab
xy ≈ yx

none

V(4,52)

1	2	3	4
1	2	3	4
3	4	1	2
3	4	1	2

x²a ≈ a
xya ≈ yxa

none

V(4,53)

1	2	3	4
2	1	4	3
3	4	2	1
4	3	1	2

x⁴a ≈ a
xy ≈ yx

\(\mathbb{Z}_4 \) = ⟨ a | a⁴ = 1 ⟩ = { a,a²,a³,1}

V(5,1)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	1
1	1	1	1	2
1	1	2	2	4

x⁴ ≈ xyzt
x²y ≈ xy²
xya ≈ yxa
axy ≈ ayx

none

V(5,2)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	1
1	1	1	2	1
1	1	3	1	5

x²ab ≈ xab
xy² ≈ yx²
xya ≈ yxa

none

V(5,3)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	1
1	1	1	4	1
1	1	2	1	1

x³ ≈ x²
x²ab ≈ xab
xya ≈ yxa
axy ≈ ayx

none

V(5,4)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	1
1	1	1	4	1
1	1	2	1	2

x²ab ≈ xab
xya ≈ yxa
axy ≈ ayx

none

V(5,5)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	1
1	1	2	1	1
1	2	1	4	5

x³ ≈ x²
x²ab ≈ xab
xyx ≈ yx²
xy² ≈ yx²

none

V(5,6)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	1
1	1	2	1	1
5	5	5	5	5

x² ≈ xyz

none

V(5,7)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	1
1	1	3	4	5
1	3	3	4	5

x²a ≈ xa
xyx ≈ yx²
xya² ≈ yxa²

none

V(5,8)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	2
1	1	1	2	3
1	1	2	3	4

x⁵ ≈ xyztu
x²ya ≈ xy²a
xy ≈ yx

\(N_5 \) = ⟨ a | a⁵ = 0 ⟩ = { 0,a,a²,a³,a⁴ }

V(5,9)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	2
1	1	1	4	1
1	1	2	1	1

x³ ≈ x²
x²ab ≈ xab
xy ≈ yx

none

V(5,10)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	2
1	1	1	4	1
1	1	2	1	3

x²abc ≈ xabc
x²y ≈ xy²
xy ≈ yx

none

V(5,11)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	2
1	2	3	4	3
1	1	1	1	2

x²ab ≈ xab
xyx ≈ xy³
xy³ ≈ yx³

none

V(5,12)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	2
1	2	3	4	4
1	2	3	4	4

x³ ≈ x²
x²ab ≈ xab
xy² ≈ yx²

none

V(5,13)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	2
4	4	4	4	4
1	1	2	1	3

x⁴ ≈ xyzt
x²y ≈ xy²
axy ≈ ayx

none

V(5,14)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	3
1	1	1	1	4
1	2	1	4	5

x³ ≈ x²
x²yx ≈ xyx
x²y² ≈ y²x²
axyb ≈ ayxb

none

V(5,15)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	3
1	1	1	2	1
1	1	1	1	5

x²y ≈ y²x
abx² ≈ abx
axy ≈ ayx

none

V(5,16)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	3
1	1	1	2	1
1	1	3	1	5

x³a ≈ x²a
xy ≈ yx

none

V(5,17)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	3
1	1	1	3	4
1	2	3	4	5

x⁴ ≈ x³
x²y² ≈ y²x²
xya ≈ yxa

none

V(5,18)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	3
4	4	4	4	4
1	2	1	1	5

x³ ≈ x²
xyx ≈ x²y²
ax²b ≈ axb
axy² ≈ ayx²

none

V(5,19)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	3
4	4	4	4	4
1	2	3	1	5

x³ ≈ x²
x²yx ≈ xyx
xy²x ≈ x²y²
axyb ≈ ayxb

none

V(5,20)

1	1	1	1	1
1	1	1	1	1
1	1	1	1	3
4	4	4	4	4
1	2	3	4	5

x³ ≈ x²
x²yx ≈ xyx
xyxa ≈ x²ya
xyz²x ≈ x²yz²

none

V(5,21)

1	1	1	1	1
1	1	1	1	1
1	1	1	2	3
1	1	1	2	3
1	2	3	4	5

x³a ≈ x²a
x²y³ ≈ y²x³
xya ≈ yxa

none

V(5,22)

1	1	1	1	1
1	1	1	1	1
1	1	1	3	3
1	1	1	4	4
1	2	3	4	5

x³ ≈ x²
x²yx ≈ xyx
x²y² ≈ y²x²
xyxa ≈ x²ya

none

V(5,23)

1	1	1	1	1
1	1	1	1	1
1	1	1	3	3
1	2	1	4	5
1	2	1	4	5

x³ ≈ x²
xyx ≈ y²x²
ax²b ≈ axb

none

V(5,24)

1	1	1	1	1
1	1	1	1	1
1	1	1	3	3
1	2	1	4	5
1	2	1	5	4

x⁴ ≈ x²
x²yx² ≈ xyx
x²yx ≈ xyx²
x²y² ≈ y²x²
ax³b ≈ axb

none

V(5,25)

1	1	1	1	1
1	1	1	1	1
1	1	1	3	3
1	2	3	4	5
1	2	3	5	4

x⁴ ≈ x²
x²y² ≈ y²x²
xya ≈ yxa

none

V(5,26)

1	1	1	1	1
1	1	1	1	1
1	1	2	1	1
1	1	1	4	1
5	5	5	5	5

x²ab ≈ xab
x²y ≈ xy²
axy ≈ ayx

none

V(5,27)

1	1	1	1	1
1	1	1	1	1
1	1	2	1	1
1	1	1	4	5
1	1	1	4	5

x²ab ≈ xab
x²y ≈ xy²
xya ≈ yxa

none

V(5,28)

1	1	1	1	1
1	1	1	1	1
1	1	2	1	1
1	1	1	4	5
1	1	1	5	4

x³ab ≈ xab
x³y ≈ xy³
xy ≈ yx

none

V(5,29)

1	1	1	1	1
1	1	1	1	1
1	1	2	1	1
1	1	1	4	5
5	5	5	5	5

x²ab ≈ xab
x²y ≈ xy²
xyx ≈ x²y

none

V(5,30)

1	1	1	1	1
1	1	1	1	1
1	1	2	1	4
4	4	4	4	4
5	5	5	5	5

xyx ≈ xyz

none

V(5,31)

1	1	1	1	1
1	1	1	1	1
1	1	3	1	5
4	4	4	4	4
1	1	3	1	5

x²a ≈ xa
ax² ≈ ax
axyb ≈ ayxb

none

V(5,32)

1	1	1	1	1
1	1	1	1	1
1	1	3	1	5
4	4	4	4	4
1	1	5	1	3

xyxyxy ≈ xy
axy ≈ ayx

none

V(5,33)

1	1	1	1	1
1	1	1	1	1
1	1	3	4	5
1	1	3	4	5
5	5	5	5	5

x²a ≈ xa
ax² ≈ ax
xyxa ≈ xya

none

V(5,34)

1	1	1	1	1
1	1	1	1	1
1	1	3	4	5
1	1	4	3	5
5	5	5	5	5

x³a ≈ xa
ax³ ≈ ax
xyx ≈ x²y

none

V(5,35)

1	1	1	1	1
1	1	1	1	1
1	1	3	4	5
1	1	4	5	3
1	1	5	3	4

x⁴a ≈ xa
xy ≈ yx

none

V(5,36)

1	1	1	1	1
1	1	1	1	1
1	1	3	4	5
4	4	4	4	4
4	4	5	1	3

x³a ≈ xa
ax³ ≈ ax
xyxy ≈ xy²x

none

V(5,37)

1	1	1	1	1
1	1	1	1	1
1	2	3	1	5
4	4	4	4	4
1	2	3	1	5

x²a ≈ xa
axyb ≈ ayxb

none

V(5,38)

1	1	1	1	1
1	1	1	1	1
1	2	3	1	5
4	4	4	4	4
1	2	5	1	3

x³a ≈ xa
xyx ≈ x²y³
axy³ ≈ ayx³

none

V(5,39)

1	1	1	1	1
1	1	1	1	1
1	2	3	4	5
1	2	3	4	5
5	5	5	5	5

x²a ≈ xa
xyxa ≈ xya

none

V(5,40)

1	1	1	1	1
1	1	1	1	1
1	2	3	4	5
1	2	4	3	5
5	5	5	5	5

x³a ≈ xa
xyx ≈ x²y³

none

V(5,41)

1	1	1	1	1
1	1	1	1	1
1	2	3	4	5
1	2	4	5	3
1	2	5	3	4

x⁴a ≈ xa
x²y² ≈ y²x²

none

V(5,42)

1	1	1	1	1
1	1	1	1	2
1	1	1	1	3
1	1	1	2	4
1	2	1	4	5

x⁴ ≈ x³
x²y² ≈ y²x²
axy ≈ ayx

none

V(5,43)

1	1	1	1	1
1	1	1	1	2
1	1	1	1	3
1	1	2	1	1
1	1	1	1	5

x³ ≈ x²
x²y ≈ y²x
xyx ≈ x²y
abx² ≈ abx

none

V(5,44)

1	1	1	1	1
1	1	1	1	2
1	1	1	1	3
1	1	2	1	1
1	2	1	4	5

x³ ≈ x²
x²yx² ≈ xyx
x²ay² ≈ y²ax²
axaya ≈ ayaxa

\(Q \) =
⟨ a,b,e | e²=e, ea=a, be=b, ae=eb=ba=0 ⟩
= { 0,a,b,ae,ab}

V(5,45)

1	1	1	1	1
1	1	1	1	2
1	1	1	1	3
1	1	2	1	4
1	2	3	4	5

x³ ≈ x²
xyx ≈ x²y
xyx ≈ yx²

\(F_4^1 \) = ⟨ a,b | a²=b²=ab=0⟩ ∪ {1} = { 0,a,b,ba,1}

V(5,46)

1	1	1	1	1
1	1	1	1	2
1	1	1	1	3
1	1	2	2	3
1	1	1	1	5

x³y ≈ xyx
x³y ≈ y³x
abx² ≈ abx

none

V(5,47)

1	1	1	1	1
1	1	1	1	2
1	1	1	1	3
1	1	2	2	4
1	1	3	3	5

x³y² ≈ y³x²
ax³ ≈ ax²
axy ≈ ayx

none

V(5,48)

1	1	1	1	1
1	1	1	1	2
1	1	1	1	3
1	1	2	2	4
1	2	3	4	5

x⁴ ≈ x³
xyx ≈ x²y
xyx ≈ yx²

none

V(5,49)

1	1	1	1	1
1	1	1	1	2
1	1	1	1	3
1	1	3	4	1
1	2	1	1	5

x³ ≈ x²
x²yx ≈ xyx
xy²x ≈ yx²y
x²y² ≈ y²x²
axya ≈ ayxa

none

V(5,50)

1	1	1	1	1
1	1	1	1	2
1	1	1	1	3
1	1	3	4	3
1	2	1	1	5

x³ ≈ x²
x²yx ≈ xyx
xy²x ≈ yx²y
axya ≈ ayxa

none

V(5,51)

1	1	1	1	1
1	1	1	1	2
1	1	1	1	3
1	1	3	4	4
1	1	3	4	5

x³ ≈ x²
x²yx ≈ xyx
x²y² ≈ y²x²
axyx ≈ ayx²

none

V(5,52)

1	1	1	1	1
1	1	1	1	2
1	1	1	2	3
1	1	2	3	4
1	2	3	4	5

x⁵ ≈ x⁴
xy ≈ yx

\(N_4^1 \) = ⟨ a | a⁴ = 0 ⟩ ∪ { 1 } = {0,a,a²,a³,1}

V(5,53)

1	1	1	1	1
1	1	1	1	2
1	1	1	3	3
1	2	1	4	4
1	2	3	4	5

x³ ≈ x²
x²yx² ≈ xyx
xyxy ≈ yxyx
x²y² ≈ y²x²
xyxzx ≈ xyzx

\(B_0^1 \) =
⟨ a,e,f | e²=e, f²=f, ef=fe=0, ea = af = a ⟩ ∪ {1}
= { 0,a,e,f,1 }

V(5,54)

1	1	1	1	1
1	1	1	1	2
1	1	2	1	3
4	4	4	4	4
1	2	3	1	5

x⁴ ≈ x³
axy ≈ ayx

none

V(5,55)

1	1	1	1	1
1	1	1	1	2
1	1	2	1	3
4	4	4	4	4
1	2	3	4	5

x⁴ ≈ x³
xyx ≈ x²y

none

V(5,56)

1	1	1	1	1
1	1	1	1	2
1	2	3	1	1
4	4	4	4	4
1	1	1	1	5

x³ ≈ x²
xyx ≈ x²y²
ax²y² ≈ ay²x²

none

V(5,57)

1	1	1	1	1
1	1	1	1	2
1	2	3	1	1
4	4	4	4	4
1	1	1	4	5

x³ ≈ x²
x²yx² ≈ xyx
xy²x ≈ xyx
xyx ≈ x²y²
xyxzx ≈ xyzx

none

V(5,58)

1	1	1	1	1
1	1	1	1	2
1	2	3	1	2
4	4	4	4	4
1	1	1	1	5

x³ ≈ x²
x²yx² ≈ xyx
axyx ≈ ayxy

none

V(5,59)

1	1	1	1	1
1	1	1	1	2
1	2	3	1	2
4	4	4	4	4
1	1	1	4	5

x³ ≈ x²
x²yx² ≈ xyx
xy²x ≈ xyx
xyxy ≈ xyx
xyxzx ≈ xyzx

none

V(5,60)

1	1	1	1	1
1	1	1	1	2
1	2	3	1	3
4	4	4	4	4
1	2	3	1	5

x³ ≈ x²
x²yx ≈ xyx
axyx ≈ ayx²
ax²y² ≈ ay²x²

none

V(5,61)

1	1	1	1	1
1	1	1	1	2
1	2	3	1	3
4	4	4	4	4
1	2	3	4	5

x³ ≈ x²
x²yx² ≈ xyx
xy²x ≈ x²y²
xyxzx ≈ xyzx

none

V(5,62)

1	1	1	1	1
1	1	1	1	2
1	2	3	4	1
1	2	3	4	1
1	1	1	1	5

x³ ≈ x²
xyx ≈ y²x²
x²y²a ≈ y²x²a

none

V(5,63)

1	1	1	1	1
1	1	1	1	2
1	2	3	4	1
1	2	3	4	2
1	1	1	1	5

x³ ≈ x²
xyxyx ≈ xyx
axaya ≈ ayaxa
xyxy ≈ yxy

none

V(5,64)

1	1	1	1	1
1	1	1	1	2
1	2	3	4	1
1	2	4	3	1
1	1	1	1	5

x⁴ ≈ x²
x²yx² ≈ xyx
xy³x ≈ xyx
xy²x ≈ yx²y
x²y² ≈ y²x²
axya ≈ ayxa

none

V(5,65)

1	1	1	1	1
1	1	1	1	2
1	2	3	4	2
1	2	3	4	2
1	1	1	1	5

x³ ≈ x²
x²yx² ≈ xyx
xyxa ≈ yxya

none

V(5,66)

1	1	1	1	1
1	1	1	1	2
1	2	3	4	2
1	2	4	3	2
1	1	1	1	5

x⁴ ≈ x²
x²yx² ≈ xyx
xy³x ≈ xyx
xy²x ≈ yx²y
axya ≈ ayxa

none

V(5,67)

1	1	1	1	1
1	1	1	1	2
1	2	3	4	3
1	2	3	4	3
1	2	3	4	5

x³ ≈ x²
xyx ≈ yx²
x²y²a ≈ y²x²a

none

V(5,68)

1	1	1	1	1
1	1	1	1	2
1	2	3	4	3
1	2	4	3	4
1	2	3	4	5

x⁴ ≈ x²
x²y² ≈ y²x²
xyx ≈ yx²

none

V(5,69)

1	1	1	1	1
1	1	1	1	3
1	1	1	3	3
1	1	1	4	4
1	1	1	5	5

ax² ≈ ax
xyx ≈ x²y
a²xy ≈ a²yx

none

V(5,70)

1	1	1	1	1
1	1	1	1	3
1	1	1	3	3
1	2	3	4	4
1	2	3	5	5

x³ ≈ x²
x²yx² ≈ xyx
xyz²x ≈ x²yz²
axaya ≈ ayaxa

none

V(5,71)

1	1	1	1	1
1	1	1	1	3
3	3	3	3	3
1	1	1	4	1
5	5	5	5	5

xyxy ≈ xy
abxy ≈ abyx

none

V(5,72)

1	1	1	1	1
1	1	1	1	3
3	3	3	3	3
1	2	3	4	1
5	5	5	5	5

x³ ≈ x²
xyx ≈ xy²

none

V(5,73)

1	1	1	1	1
1	1	1	2	2
1	1	1	3	3
1	1	1	4	4
1	1	2	4	4

x³ ≈ x²
x²y ≈ y²x
abx² ≈ abx

none

V(5,74)

1	1	1	1	1
1	1	1	2	2
1	1	1	3	3
1	1	3	4	5
1	1	3	4	5

x³ ≈ x²
x²yx ≈ xyx
xy²x ≈ y²x²
axyb ≈ ayxb

none

V(5,75)

1	1	1	1	1
1	1	1	2	2
1	1	1	3	3
1	1	3	4	5
1	1	3	5	4

x⁴ ≈ x²
x²y² ≈ y²x²
axy ≈ ayx

none

V(5,76)

1	1	1	1	1
1	1	1	2	2
1	1	1	3	3
1	1	3	4	5
1	3	3	4	5

x³ ≈ x²
x²yx² ≈ xyx
xy²zx ≈ y²zx²
axaya ≈ ayaxa

none

V(5,77)

1	1	1	1	1
1	1	1	2	2
1	1	1	3	3
1	2	3	4	5
1	3	2	5	4

x⁴ ≈ x²
x³yx ≈ xyx
x²y ≈ yx²
xyxy ≈ x²y²

none

V(5,78)

1	1	1	1	1
1	1	1	2	2
1	1	2	3	3
1	2	3	4	5
1	2	3	4	5

x⁴ ≈ x³
xya ≈ yxa

none

V(5,79)

1	1	1	1	1
1	1	1	2	2
1	1	2	3	3
1	2	3	4	5
1	2	3	5	4

x⁵ ≈ x³
xy ≈ yx

none

V(5,80)

1	1	1	1	1
1	1	1	2	2
1	2	3	1	2
1	1	1	4	4
1	1	1	5	5

x³ ≈ x²
xyxyx ≈ xyx
axaya ≈ ayaxa
xyxy ≈ xyx

none

V(5,81)

1	1	1	1	1
1	1	1	2	2
1	2	3	2	3
1	1	1	4	4
1	2	3	4	5

x³ ≈ x²
x²yx² ≈ xyx
xyxy ≈ yxyx
xyxzx ≈ xyzx

\(A_0^1 \) =
⟨ e,f | e²=e, f²=f, ef = 0 ⟩ ∪ {1}
= { 0,e,f,fe,1 }

V(5,82)

1	1	1	1	1
1	1	1	2	2
3	3	3	3	3
1	1	1	4	4
1	2	1	4	5

x³ ≈ x²
xyx ≈ x²y
ax²y² ≈ ay²x²

none

V(5,83)

1	1	1	1	1
1	1	1	2	2
3	3	3	3	3
1	1	1	4	5
1	1	1	4	5

ax² ≈ ax
axyb ≈ ayxb

none

V(5,84)

1	1	1	1	1
1	1	1	2	2
3	3	3	3	3
1	1	1	4	5
1	1	1	5	4

ax³ ≈ ax
axy ≈ ayx

none

V(5,85)

1	1	1	1	1
1	1	1	2	2
3	3	3	3	3
1	1	3	4	5
1	1	3	4	5

ax² ≈ ax
xyxa ≈ x²ya

none

V(5,86)

1	1	1	1	1
1	1	1	2	2
3	3	3	3	3
1	1	3	4	5
1	1	3	5	4

ax³ ≈ ax
xyx ≈ x²y

none

V(5,87)

1	1	1	1	1
1	1	1	2	2
3	3	3	3	3
1	1	3	4	5
3	3	1	5	4

ax³ ≈ ax
xyx² ≈ x³y
xyxy ≈ xy²x

none

V(5,88)

1	1	1	1	1
1	1	1	2	2
3	3	3	3	3
1	2	1	4	5
1	2	1	4	5

x³ ≈ x²
x²yx ≈ xyx
axyb ≈ ayxb

none

V(5,89)

1	1	1	1	1
1	1	1	2	2
3	3	3	3	3
1	2	1	4	5
1	2	1	5	4

x⁴ ≈ x²
axy ≈ ayx

none

V(5,90)

1	1	1	1	1
1	1	1	2	2
3	3	3	3	3
1	2	3	4	5
1	2	3	4	5

x³ ≈ x²
x²yx ≈ xyx
xyxa ≈ x²ya

none

V(5,91)

1	1	1	1	1
1	1	1	2	2
3	3	3	3	3
1	2	3	4	5
1	2	3	5	4

x⁴ ≈ x²
xyx ≈ x²y

none

V(5,92)

1	1	1	1	1
1	1	1	2	3
1	2	3	1	1
1	1	1	4	5
1	4	5	1	1

x³ ≈ x²
xyxyx ≈ xyx
x²y² ≈ y²x²

\(B_2 \) =
⟨ a,b | a²=b²=0, aba=a, bab=b ⟩
= { 0,a,b,ab,ba }

V(5,93)

1	1	1	1	1
1	1	1	2	3
1	2	3	2	3
1	1	1	4	5
1	4	5	4	5

x³ ≈ x²
xyxyx ≈ xyx
axaya ≈ ayaxa

\(A_2 \) =
⟨ a,e | e²=eae=e, aea=a, a²=0 ⟩
= { 0,a,e,ae,ea }

V(5,94)

1	1	1	1	1
1	1	1	2	3
3	3	3	3	3
1	1	1	4	5
5	5	5	5	5

ax² ≈ ax
axyx ≈ axy

none

V(5,95)

1	1	1	1	1
1	1	1	2	3
3	3	3	3	3
1	2	3	4	5
5	5	5	5	5

x³ ≈ x²
x²yx ≈ x²y
xyx² ≈ xyx
xyxy ≈ xy²x
xyaxy ≈ xyayx
xayxy ≈ xay²x
xaybxy ≈ xaybyx

\(P_2^1 \) =
⟨ a,e | a²e=a², e²=ea=e ⟩ ∪ {1}
= { a,e,a²,ae,1 }

V(5,96)

1	1	1	1	1
1	1	2	2	2
1	1	3	3	3
1	2	3	4	5
1	2	3	4	5

x³ ≈ x²
x²yx ≈ xyx
xyxa ≈ x²ya
x²y²a ≈ y²x²a

none

V(5,97)

1	1	1	1	1
1	1	2	2	2
1	1	3	3	3
1	2	3	4	5
1	2	3	5	4

x⁴ ≈ x²
x²y² ≈ y²x²
xyx ≈ x²y

none

V(5,98)

1	1	1	1	1
1	1	2	2	2
1	1	3	3	5
1	2	3	4	5
1	1	3	5	5

x³ ≈ x²
x²yx² ≈ xyx
xy²x ≈ y²x²
xyxzx ≈ xyzx

none

V(5,99)

1	1	1	1	1
1	1	2	2	2
1	1	3	4	5
1	1	4	5	3
1	1	5	3	4

ax⁴ ≈ ax
x²y² ≈ y²x²

none

V(5,100)

1	1	1	1	1
1	1	2	2	2
1	2	3	4	5
1	2	4	5	3
1	2	5	3	4

x⁵ ≈ x²
xy ≈ yx

none

V(5,101)

1	1	1	1	1
1	2	1	2	5
3	3	3	3	3
1	2	3	4	5
1	2	1	5	5

x² ≈ x
xyxzx ≈ xyzx

none

V(5,102)

1	1	1	1	1
1	2	1	4	5
3	3	3	3	3
1	2	5	4	5
5	5	5	5	5

x² ≈ x
xay ≈ xayxy

none

V(5,103)

1	1	1	1	1
1	2	1	4	5
3	3	3	3	3
1	4	1	5	2
1	5	1	2	4

x⁴ ≈ x
axy ≈ ayx

none

V(5,104)

1	1	1	1	1
1	2	2	4	5
1	2	2	4	5
1	2	2	4	5
1	2	4	4	5

xyxy ≈ xy
xyab ≈ yxab

none

V(5,105)

1	1	1	1	1
1	2	3	4	5
1	3	2	5	4
1	4	5	3	2
1	5	4	2	3

x⁵ ≈ x
xy ≈ yx

none

V(5,106)

1	1	1	1	1
1	2	3	4	5
1	3	4	2	5
1	4	2	3	5
5	5	5	5	5

x⁴ ≈ x
xyx ≈ x²y

none

V(5,107)

1	1	1	1	5
1	1	1	1	5
1	1	1	1	5
1	1	2	1	5
1	1	1	1	5

x² ≈ yzx

none

V(5,108)

1	1	1	1	5
1	1	1	1	5
1	1	1	1	5
1	1	2	1	5
5	5	5	5	1

x² ≈ y²
x³ab ≈ xab
xya ≈ yxa
axy ≈ ayx

none

V(5,109)

1	1	1	1	5
1	1	1	1	5
1	1	1	1	5
1	1	2	2	5
5	5	5	5	1

x³ ≈ xy²
x³ab ≈ xab
xya ≈ yxa
axy ≈ ayx

none

V(5,110)

1	1	1	1	5
1	1	1	1	5
1	1	1	2	5
1	1	2	1	5
5	5	5	5	1

x² ≈ y²
x³ab ≈ xab
xy ≈ yx

none

V(5,111)

1	1	1	1	5
1	1	1	1	5
1	1	1	2	5
1	1	2	3	5
1	1	1	1	5

x⁴ ≈ yztx
x²y ≈ xy²
xya ≈ yxa

none

V(5,112)

1	1	1	1	5
1	1	1	1	5
1	1	1	2	5
1	1	2	3	5
5	5	5	5	1

x³a ≈ xy²a
x³abc ≈ xabc
xy ≈ yx

none

V(5,113)

1	1	1	1	5
1	1	1	1	5
1	1	2	1	5
1	1	1	4	5
1	1	1	5	5

x²ab ≈ xab
x²y ≈ xy²
xyx ≈ yx²

none

V(5,114)

1	1	1	1	5
1	1	1	1	5
1	1	3	3	5
1	1	4	4	5
1	1	5	5	5

x²a ≈ xa
ax² ≈ ax
axyx ≈ ayx

none

V(5,115)

1	1	1	1	5
1	1	1	1	5
1	1	3	4	5
1	1	3	4	5
5	5	5	5	1

xyxyxy ≈ xy
xya ≈ yxa

none

V(5,116)

1	1	1	1	5
1	1	1	1	5
1	1	3	4	5
1	1	4	3	5
1	1	5	5	5

x³a ≈ xa
ax³ ≈ ax
xyx ≈ yx²

none

V(5,117)

1	1	1	1	5
1	1	1	1	5
1	2	3	3	5
1	2	4	4	5
1	1	5	5	5

x²a ≈ xa
axyx ≈ ayx²

none

V(5,118)

1	1	1	1	5
1	1	1	1	5
1	2	3	4	5
1	2	3	4	5
5	5	5	5	1

x³a ≈ xa
xya ≈ yxa

none

V(5,119)

1	1	1	1	5
1	1	1	1	5
1	2	3	4	5
1	2	4	3	5
1	1	5	5	5

x³a ≈ xa
xyx ≈ yx²

none

V(5,120)

1	1	1	1	5
1	1	1	2	5
1	1	1	3	5
1	1	3	4	5
1	1	1	5	5

x³ ≈ x²
xyx² ≈ xyx
axyx ≈ ayx²
xy²zx ≈ y²zx²

none

V(5,121)

1	1	1	1	5
1	1	1	2	5
1	1	2	3	5
1	2	3	4	5
1	1	1	5	5

x⁴ ≈ x³
xyx ≈ yx²

none

V(5,122)

1	1	1	1	5
1	1	1	2	5
1	2	3	1	5
1	1	1	4	5
1	1	1	5	5

x³ ≈ x²
x²yx² ≈ xyx
xy²x ≈ xyx
xyx ≈ y²x²
xyxzx ≈ xyzx

none

V(5,123)

1	1	1	1	5
1	1	1	2	5
1	2	3	2	5
1	1	1	4	5
1	1	1	5	5

x³ ≈ x²
x²yx² ≈ xyx
xy²x ≈ xyx
xyxy ≈ yxy
xyxzx ≈ xyzx

none

V(5,124)

1	1	1	1	5
1	1	2	2	5
1	1	3	3	5
1	1	4	4	5
1	1	5	5	5

ax² ≈ ax
axyx ≈ ayx

none

V(5,125)

1	1	1	1	5
1	1	2	2	5
1	1	3	4	5
1	1	3	4	5
5	5	5	5	1

ax³ ≈ ax
xyx ≈ y³x²
x³ya ≈ y³xa

none

V(5,126)

1	1	1	1	5
1	1	2	2	5
1	1	3	4	5
1	1	4	3	5
1	1	5	5	5

ax³ ≈ ax
xyx ≈ y³x²

none

V(5,127)

1	1	1	1	5
1	1	2	2	5
1	2	3	3	5
1	2	4	4	5
1	1	5	5	5

x³ ≈ x²
x²yx ≈ xyx
axyx ≈ ayx²

none

V(5,128)

1	1	1	1	5
1	1	2	2	5
1	2	3	4	5
1	2	3	4	5
5	5	5	5	1

x⁴ ≈ x²
xya ≈ yxa

none

V(5,129)

1	1	1	1	5
1	1	2	2	5
1	2	3	4	5
1	2	4	3	5
1	1	5	5	5

x⁴ ≈ x²
xyx ≈ yx²

none

V(5,130)

1	1	1	1	5
1	2	3	4	5
1	3	4	2	5
1	4	2	3	5
1	1	1	1	5

x⁴ ≈ x
xya ≈ yxa

none

V(5,131)

1	1	1	1	5
1	2	3	4	5
1	3	4	2	5
1	4	2	3	5
1	5	5	5	5

x⁴ ≈ x
xyx ≈ yx²

none

V(5,132)

1	1	1	1	5
1	2	3	4	5
1	3	4	2	5
1	4	2	3	5
5	5	5	5	1

x⁷ ≈ x
xy ≈ yx

none

V(5,133)

1	1	1	4	4
1	1	1	4	4
1	1	3	4	5
1	1	4	4	1
1	1	5	4	3

x³a ≈ xa
ax³ ≈ ax
xyxy ≈ yx²y

none

V(5,134)

1	1	1	4	4
1	1	1	4	4
1	2	3	4	5
1	1	4	4	1
1	2	5	4	3

x³a ≈ xa
x²yx ≈ yx³
xyxy ≈ yx²y

none

V(5,135)

1	1	1	4	4
1	1	1	4	4
3	3	3	5	5
1	1	1	4	4
3	3	3	5	5

xyz ≈ xz

none

V(5,136)

1	1	1	4	4
1	1	1	4	4
3	3	3	5	5
4	4	4	1	1
5	5	5	3	3

x³a ≈ xa
a²x² ≈ a²
axy ≈ ayx

none

V(5,137)

1	1	1	4	5
1	1	1	4	5
1	1	2	4	5
1	1	1	4	5
1	1	4	4	5

xyx ≈ zyx

none

V(5,138)

1	1	1	4	5
1	1	1	4	5
1	1	2	4	5
4	4	4	5	1
5	5	5	1	4

x⁴ab ≈ xab
x³ ≈ y³
xy ≈ yx

none

V(5,139)

1	1	1	4	5
1	1	1	4	5
1	2	3	4	5
1	1	1	4	5
1	4	5	4	5

x²a ≈ xa
xyxa ≈ yxa

none

V(5,140)

1	1	1	4	5
1	1	2	4	5
1	1	3	4	5
1	1	4	4	5
1	4	1	4	5

x³ ≈ x²
xyx ≈ y²x

none

V(5,141)

1	1	1	4	5
1	1	2	4	5
1	2	3	4	5
1	1	4	4	5
1	4	5	4	5

x³ ≈ x²
x²yx ≈ xyx
xyx² ≈ yx²
xyxy ≈ yx²y
xyaxy ≈ yxaxy
xyxay ≈ yx²ay
xyaxby ≈ yxaxby

\(\overleftarrow{P_2^1} \) = ⟨ a,e | ea²=a², e²=ae=e ⟩ ∪ {1} = { a,e,a²,ea,1 }

V(5,142)

1	1	1	4	5
1	2	2	4	5
1	3	3	4	5
1	1	5	4	5
1	5	5	4	5

x² ≈ x
xay ≈ xyxay

none

V(5,143)

1	1	3	4	5
1	1	3	4	5
1	1	3	4	5
4	4	5	1	3
4	4	5	1	3

ax³ ≈ ax
x²a² ≈ a²
xya ≈ yxa

none

V(5,144)

1	1	3	4	5
1	1	3	4	5
3	3	1	5	4
4	4	5	3	1
5	5	4	1	3

x⁴ab ≈ ab
xy ≈ yx

none

V(5,145)

1	2	3	4	5
2	3	4	5	1
3	4	5	1	2
4	5	1	2	3
5	1	2	3	4

x⁵a ≈ a
xy ≈ yx

\(\mathbb{Z}_5 \) = ⟨ a | a⁵ = 1 ⟩ = { a,a²,a³,a⁴,1 }

W(6,1)

1	1	1	1	1	1
1	1	1	1	1	1
1	1	1	1	1	1
1	1	1	1	1	1
1	1	1	2	1	1
1	1	3	1	1	6

x³ ≈ x²
xy² ≈ yx²
ax²b ≈ axb

none

W(6,2)

1	1	1	1	1	1
1	1	1	1	1	1
1	1	1	1	1	1
1	1	1	1	1	4
1	1	2	1	1	1
1	1	1	1	1	6

x³ ≈ x²
x²y ≈ y²x
ax²b ≈ axb

none

W(6,3)

1	1	1	1	1	1
1	1	1	1	1	1
1	1	1	1	1	1
1	1	2	1	1	5
5	5	5	5	5	5
6	6	6	6	6	6

x³ ≈ x²
xyx ≈ xyz

none

W(6,4)

1	1	1	1	5	6
1	1	1	1	5	6
1	1	1	1	5	6
1	1	2	1	5	6
1	1	1	1	5	6
1	1	1	5	5	6

x³ ≈ x²
xyx ≈ zyx

none

W(6,5)

1	2	3	4	5	6
2	1	4	3	6	5
3	4	5	6	1	2
4	3	6	5	2	1
5	6	1	2	3	4
6	5	2	1	4	3

x⁶a ≈ a
xy ≈ yx

\(\mathbb{Z}_6 \) = ⟨ a | a⁶ = 1 ⟩ = { a,a²,a³,a⁴,a⁵,1 }

W(7,1)

1	1	1	1	1	1	1
1	1	1	1	1	1	1
1	1	1	1	1	1	2
1	1	1	1	1	2	1
1	1	1	1	1	2	2
1	1	1	2	2	3	5
1	1	2	1	2	5	4

x³ ≈ xyzt
x²y ≈ xy²
xy ≈ yx

none

W(7,2)

1	2	3	4	5	6	7
2	3	4	5	6	7	1
3	4	5	6	7	1	2
4	5	6	7	1	2	3
5	6	7	1	2	3	4
6	7	1	2	3	4	5
7	1	2	3	4	5	6

x⁷a ≈ a
xy ≈ yx

\(\mathbb{Z}_7 \) = ⟨ a | a⁷ = 1 ⟩ = { a,a²,a³,a⁴,a⁵,a⁶,1 }

W(8,1)

1	1	1	1	1	1	1	1
1	1	1	1	1	1	1	1
1	1	1	1	1	1	1	2
1	1	1	1	1	1	2	1
1	1	1	1	1	2	1	1
1	1	1	1	2	1	3	4
1	1	1	2	1	3	1	5
1	1	2	1	1	4	5	1

x² ≈ xyzt
xy ≈ yx

none

W(8,2)

1	1	1	1	1	1	1	1
1	1	1	1	1	1	1	1
1	1	1	1	1	1	1	2
1	1	1	1	1	1	2	1
1	1	1	1	1	2	1	1
1	1	1	1	2	1	3	4
1	1	1	2	1	3	1	5
1	1	2	1	1	4	5	2

x²y ≈ xyzt
xy ≈ yx

none

W(8,3)

1	2	3	4	5	6	7	8
2	1	4	3	6	5	8	7
3	4	2	1	7	8	6	5
4	3	1	2	8	7	5	6
5	6	7	8	3	4	2	1
6	5	8	7	4	3	1	2
7	8	6	5	2	1	4	3
8	7	5	6	1	2	3	4

x⁸a ≈ a
xy ≈ yx

\(\mathbb{Z}_8 \) = ⟨ a | a⁸ = 1 ⟩ = { a,a²,a³,a⁴,a⁵,a⁶,a⁷,1}

W(9,1)

1	2	3	4	5	6	7	8	9
2	3	1	5	6	4	8	9	7
3	1	2	6	4	5	9	7	8
4	5	6	7	8	9	2	3	1
5	6	4	8	9	7	3	1	2
6	4	5	9	7	8	1	2	3
7	8	9	2	3	1	5	6	4
8	9	7	3	1	2	6	4	5
9	7	8	1	2	3	4	5	6

x⁹a ≈ a
xy ≈ yx

\(\mathbb{Z}_9 \) = ⟨ a | a⁹ = 1 ⟩ = { a,a²,a³,a⁴,a⁵,a⁶,a⁷,a⁸,1}

Introduction
Semigroups Multiplication table Semigroup Variety Group Variety Cayley to Presentation Presentation to Cayley GAP smallsemi ID GAP smallgroup ID Define set of elements Bands and other special cases	Varieties Variety by ID Identities Varieties Filter Conjectures

Varieties of Semigroups

J.Araújo, J.P.Araújo, P.J.Cameron, E.W.H.Lee, J.Raminhos

João Araújo (Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Center for Mathematics and Applications and CEMAT-CIÊNCIAS)

João Pedro Araújo (University of Stanford, California)

Peter J. Cameron (School of Mathematics and Statistics & CIRCA, University of St Andrews, Scotland; CEMAT-CIÊNCIAS, University of Lisbon)

Edmond W. H. Lee (Nova Southeastern University, Florida)

Jorge Raminhos (Universidade Aberta and CEMAT-CIÊNCIAS)

This project has been partially supported by the Fundação para a Ciência e a Tecnologia through the project CEMAT-CIÊNCIAS UID/Multi/04621/2013, and through project “Hilbert’s 24th problem” PTDC/MHC-FIL/2583/2014.

Any comments are welcome: jj.araujo@fct.unl.pt, raminhosj@gmail.com

Documentation

Introduction

Multiplication table

Semigroup Variety

Group Variety

Presentation to Cayley

Cayley to Presentation

GAP smallsemi ID

GAP smallgroup ID

Define set of elements

Bands and other special cases

Variety by ID

Identities

Varieties Filter