#330 ⟨a, b | aa=1, abba=b⟩

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Properties

Element profile

Complete rewriting system

Format:
Word to reduce:
Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Reduction strategy:
Path to normal form: 1 
1
  1. b4 ⇒ b
  2. ba ⇒ ab2
  3. a2 ⇒ 1 
# ab:aa=1,abba=b b/a
bbbb=b
ba=abb
aa=1

Cayley table

Idempotents are shown in bold.

1ababb2ab2b3ab3
11ababb2ab2b3ab3
aa1abbab2b2ab3b3
bbab2b2ab3b3abbab2
ababb2ab2b3ab3babb2
b2b2abb3ab2bab3b2ab
ab2ab2bab3b2abb3ab2b
b3b3ab3babb2ab2b3ab3
ab3ab3b3abbab2b2ab3b3

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

33 unique, 1151 total

Σ#PresentationDescriptionRelated
7188⟨a, b | aab=1, bbbb=1⟩Isomorphic to ℤ8727 iso
8556⟨a, b | aba=b, aabb=1⟩Finite non-Abelian group with 8 elements28 iso
8898⟨a, b | aa=a, bbbb=a⟩Isomorphic to ℕ(8 = 4)48 iso
8918⟨a, b | aa=b, bbbb=a⟩Isomorphic to ℕ(8 = 1)34 iso
8919⟨a, b | aa=b, bbbb=b⟩Isomorphic to ℕ(8 = 2)46 iso
8961⟨a, b | aa=a, aba=bb⟩Finite non-commutative monoid with 8 elements5 iso
91605⟨a, b | aaa=bb, abb=b⟩Isomorphic to ℕ(8 = 3)55 iso
91606⟨a, b | aaa=bb, bab=a⟩Finite commutative monoid with 8 elements14 iso
91615⟨a, b | aab=aa, baa=b⟩Finite non-commutative monoid with 8 elements9 iso, 5 anti-iso
91650⟨a, b | aab=bb, baa=a⟩Finite non-commutative monoid with 8 elements4 iso, 11 anti-iso
92206⟨a, b | ab=aa, bbbb=a⟩Isomorphic to ℕ(8 = 5)32 iso
92220⟨a, b | bb=aa, aaab=a⟩Finite commutative monoid with 8 elements22 iso
92247⟨a, b | ab=aa, baa=bb⟩Finite non-commutative monoid with 8 elements1 iso
92256⟨a, b | ab=aa, bbb=aa⟩Finite non-commutative monoid with 8 elements1 iso
92258⟨a, b | ab=aa, bbb=ba⟩Finite non-commutative monoid with 8 elements
92883⟨a, b | aa=a, abbbb=b⟩Finite non-commutative monoid with 8 elements14 iso
93107⟨a, b | ab=a, baaa=bb⟩Finite non-commutative monoid with 8 elements9 iso
93123⟨a, b | ab=a, bbaa=bb⟩Finite non-commutative monoid with 8 elements10 iso
105191⟨a, b | aab=bb, aaaa=b⟩Isomorphic to ℕ(8 = 6)18 iso
106664⟨a, b | aab=b, aaaa=ba⟩Finite non-commutative monoid with 8 elements1 iso
107057⟨a, b | ab=aa, aaaa=bb⟩Finite non-commutative monoid with 8 elements7 iso
109380⟨a, b | ab=a, bbbb=baa⟩Finite non-commutative monoid with 8 elements2 iso
1112606⟨a, b | abbb=aa, bbbb=a⟩Isomorphic to ℕ(8 = 7)14 iso
1120047⟨a, b | aab=a, bbbb=bbb⟩Finite non-commutative monoid with 8 elements
1120051⟨a, b | aab=b, aaaa=abb⟩Finite commutative monoid with 8 elements1 iso
1124863⟨a, b | ab=a, aaaaaa=bb⟩Finite commutative monoid with 8 elements
1125112⟨a, b | ab=a, bbbbbb=aa⟩Finite commutative monoid with 8 elements
1125114⟨a, b | ab=a, bbbbbb=ba⟩Finite non-commutative monoid with 8 elements
1125411⟨a, b | ab=a, aaaaa=bbb⟩Finite commutative monoid with 8 elements
1125652⟨a, b | ab=a, bbbbb=aaa⟩Finite commutative monoid with 8 elements
1125658⟨a, b | ab=a, bbbbb=bba⟩Finite non-commutative monoid with 8 elements
1125897⟨a, b | ab=a, bbbb=aaaa⟩Finite commutative monoid with 8 elements
1125911⟨a, b | ab=a, bbbb=bbba⟩Finite non-commutative monoid with 8 elements

Other isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

60 total

Σ#PresentationMapping
7364⟨a, b | aa=1, aba=bb⟩φ(a) = a, φ(b) = b
7367⟨a, b | aa=1, abb=ba⟩φ(a) = a, φ(b) = b
81115⟨a, b | aa=1, ababa=b⟩φ(a) = a, φ(b) = ab
81185⟨a, b | aa=1, abab=ba⟩φ(a) = a, φ(b) = ab
81250⟨a, b | aa=1, bab=aba⟩φ(a) = a, φ(b) = ab
93349⟨a, b | aa=1, aaabba=b⟩φ(a) = a, φ(b) = b
93369⟨a, b | aa=1, abaaba=b⟩φ(a) = a, φ(b) = b
93482⟨a, b | aa=1, aaaba=bb⟩φ(a) = a, φ(b) = b
93485⟨a, b | aa=1, aaabb=ba⟩φ(a) = a, φ(b) = b
93495⟨a, b | aa=1, aabba=ab⟩φ(a) = a, φ(b) = b
93504⟨a, b | aa=1, abaab=ba⟩φ(a) = a, φ(b) = b
93620⟨a, b | aa=1, aaab=bba⟩φ(a) = a, φ(b) = b
93625⟨a, b | aa=1, aaba=abb⟩φ(a) = a, φ(b) = b
93632⟨a, b | aa=1, aabb=aba⟩φ(a) = a, φ(b) = b
93647⟨a, b | aa=1, abba=aab⟩φ(a) = a, φ(b) = b
93662⟨a, b | aa=1, baab=aba⟩φ(a) = a, φ(b) = b
109677⟨a, b | aa=1, aaababa=b⟩φ(a) = a, φ(b) = ab
109719⟨a, b | aa=1, abaaaba=b⟩φ(a) = a, φ(b) = ab
109949⟨a, b | aa=1, aaabab=ba⟩φ(a) = a, φ(b) = ab
109964⟨a, b | aa=1, aababa=ab⟩φ(a) = a, φ(b) = ab
109988⟨a, b | aa=1, abaaab=ba⟩φ(a) = a, φ(b) = ab
1010219⟨a, b | aa=1, aaaba=bab⟩φ(a) = a, φ(b) = ab
1010238⟨a, b | aa=1, aabab=aba⟩φ(a) = a, φ(b) = ab
1010269⟨a, b | aa=1, ababa=aab⟩φ(a) = a, φ(b) = ab
1010306⟨a, b | aa=1, baaab=aba⟩φ(a) = a, φ(b) = ab
1010483⟨a, b | aa=1, abab=aaba⟩φ(a) = a, φ(b) = ab
1010500⟨a, b | aa=1, baaa=abab⟩φ(a) = a, φ(b) = ab
1126453⟨a, b | aa=1, aaaaabba=b⟩φ(a) = a, φ(b) = b
1126475⟨a, b | aa=1, aaabaaba=b⟩φ(a) = a, φ(b) = b
1126487⟨a, b | aa=1, aaabbaaa=b⟩φ(a) = a, φ(b) = b
1126559⟨a, b | aa=1, abaaaaba=b⟩φ(a) = a, φ(b) = b
1126978⟨a, b | aa=1, aaaaaba=bb⟩φ(a) = a, φ(b) = b
1126981⟨a, b | aa=1, aaaaabb=ba⟩φ(a) = a, φ(b) = b
1126992⟨a, b | aa=1, aaaabba=ab⟩φ(a) = a, φ(b) = b
1127001⟨a, b | aa=1, aaabaaa=bb⟩φ(a) = a, φ(b) = b
1127004⟨a, b | aa=1, aaabaab=ba⟩φ(a) = a, φ(b) = b
1127016⟨a, b | aa=1, aaabbaa=ba⟩φ(a) = a, φ(b) = b
1127035⟨a, b | aa=1, aabaaba=ab⟩φ(a) = a, φ(b) = b
1127086⟨a, b | aa=1, abaaaab=ba⟩φ(a) = a, φ(b) = b
1127508⟨a, b | aa=1, aaaaab=bba⟩φ(a) = a, φ(b) = b
1127513⟨a, b | aa=1, aaaaba=abb⟩φ(a) = a, φ(b) = b
1127520⟨a, b | aa=1, aaaabb=aba⟩φ(a) = a, φ(b) = b
1127532⟨a, b | aa=1, aaabaa=bba⟩φ(a) = a, φ(b) = b
1127543⟨a, b | aa=1, aaabba=aab⟩φ(a) = a, φ(b) = b
1127546⟨a, b | aa=1, aaabba=baa⟩φ(a) = a, φ(b) = b
1127560⟨a, b | aa=1, aabaab=aba⟩φ(a) = a, φ(b) = b
1127584⟨a, b | aa=1, aabbaa=aba⟩φ(a) = a, φ(b) = b
1127621⟨a, b | aa=1, abaaba=aab⟩φ(a) = a, φ(b) = b
1127698⟨a, b | aa=1, baaaab=aba⟩φ(a) = a, φ(b) = b
1128040⟨a, b | aa=1, aaaab=abba⟩φ(a) = a, φ(b) = b
1128053⟨a, b | aa=1, aaaba=aabb⟩φ(a) = a, φ(b) = b
1128059⟨a, b | aa=1, aaaba=baab⟩φ(a) = a, φ(b) = b
1128062⟨a, b | aa=1, aaaba=bbaa⟩φ(a) = a, φ(b) = b
1128068⟨a, b | aa=1, aaabb=aaba⟩φ(a) = a, φ(b) = b
1128074⟨a, b | aa=1, aaabb=baaa⟩φ(a) = a, φ(b) = b
1128087⟨a, b | aa=1, aabaa=abba⟩φ(a) = a, φ(b) = b
1128109⟨a, b | aa=1, aabba=aaab⟩φ(a) = a, φ(b) = b
1128112⟨a, b | aa=1, aabba=abaa⟩φ(a) = a, φ(b) = b
1128142⟨a, b | aa=1, abaab=aaba⟩φ(a) = a, φ(b) = b
1128148⟨a, b | aa=1, abaab=baaa⟩φ(a) = a, φ(b) = b