#556 ⟨a, b | aba=b, aabb=1⟩

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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. a4 ⇒ 1
  2. ab ⇒ ba3
  3. b2 ⇒ a2 
# ab:aba=b,aabb=1 a/b
aaaa=1
ab=baaa
bb=aa

Cayley table

1aba2baa3ba2ba3
11aba2baa3ba2ba3
aaa2ba3a3b1baba2
bbbaa2ba2a3ba31a
a2a2a3ba21ba3abba
bababa2aba3a2ba31
a3a31baaba2a2ba3b
ba2ba2ba31babaa2a3
ba3ba3ba3ba1ba2aa2

Right Cayley graph

Left Cayley graph

Others with same cardinality

33 unique, 1183 total

Σ#PresentationDescriptionRelated
7188⟨a, b | aab=1, bbbb=1⟩Isomorphic to ℤ8727 iso
7330⟨a, b | aa=1, abba=b⟩Finite non-commutative monoid with 8 elements60 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.

28 total

Σ#PresentationMapping
8558⟨a, b | aba=b, abba=1⟩φ(a) = a, φ(b) = b
8560⟨a, b | aba=b, baab=1⟩φ(a) = a, φ(b) = b
103806⟨a, b | aaab=ba, aabb=1⟩φ(a) = a, φ(b) = b
103809⟨a, b | aaab=ba, abba=1⟩φ(a) = a, φ(b) = b
103812⟨a, b | aaab=ba, baab=1⟩φ(a) = a, φ(b) = b
103815⟨a, b | aaab=ba, bbaa=1⟩φ(a) = a, φ(b) = b
103846⟨a, b | aaba=ab, aabb=1⟩φ(a) = a, φ(b) = b
103849⟨a, b | aaba=ab, abba=1⟩φ(a) = a, φ(b) = b
103852⟨a, b | aaba=ab, baab=1⟩φ(a) = a, φ(b) = b
103855⟨a, b | aaba=ab, bbaa=1⟩φ(a) = a, φ(b) = b
103922⟨a, b | abab=aa, abba=1⟩φ(a) = a, φ(b) = b
103925⟨a, b | abab=aa, baab=1⟩φ(a) = a, φ(b) = b
103928⟨a, b | abab=aa, bbaa=1⟩φ(a) = a, φ(b) = b
103989⟨a, b | baba=aa, bbaa=1⟩φ(a) = a, φ(b) = b
105596⟨a, b | aabb=1, aababa=1⟩φ(a) = a, φ(b) = b
105602⟨a, b | aabb=1, abaaab=1⟩φ(a) = a, φ(b) = b
105604⟨a, b | aabb=1, ababaa=1⟩φ(a) = a, φ(b) = b
105613⟨a, b | aabb=1, baaaba=1⟩φ(a) = a, φ(b) = b
105616⟨a, b | aabb=1, babaaa=1⟩φ(a) = a, φ(b) = b
105663⟨a, b | abba=1, aaabab=1⟩φ(a) = a, φ(b) = b
105667⟨a, b | abba=1, aababa=1⟩φ(a) = a, φ(b) = b
105673⟨a, b | abba=1, abaaab=1⟩φ(a) = a, φ(b) = b
105678⟨a, b | abba=1, ababbb=1⟩φ(a) = a, φ(b) = b
105681⟨a, b | abba=1, abbbab=1⟩φ(a) = a, φ(b) = b
105688⟨a, b | abba=1, bababb=1⟩φ(a) = a, φ(b) = b
106207⟨a, b | aba=b, aaabab=1⟩φ(a) = a, φ(b) = b
106211⟨a, b | aba=b, aababa=1⟩φ(a) = a, φ(b) = b
106217⟨a, b | aba=b, abaaab=1⟩φ(a) = a, φ(b) = b