#628 ⟨a, b | bb=aa, abab=1⟩

Up:Monoids with two generators and two relations
Prev:#618 ⟨a, b | ba=ab, aaaa=1⟩
Next:#639 ⟨a, b | ab=aa, baa=b⟩

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. a8 ⇒ 1
  2. ab ⇒ ba5
  3. b2 ⇒ a2 
# ab:bb=aa,abab=1 a/b
aaaaaaaa=1
ab=baaaaa
bb=aa

Cayley table

1aba2baa3ba2a4ba3a5ba4a6ba5a7ba6ba7
11aba2baa3ba2a4ba3a5ba4a6ba5a7ba6ba7
aaa2ba5a3ba6a4ba7a5ba6baa7ba21ba3ba4
bbbaa2ba2a3ba3a4ba4a5ba5a6ba6a7ba71a
a2a2a3ba2a4ba3a5ba4a6ba5a7ba61ba7abba
bababa2a7ba31ba4aba5a2ba6a3ba7a4ba5a6
a3a3a4ba7a5ba6baa7ba21ba3aba4a2ba5ba6
ba2ba2ba3a4ba4a5ba5a6ba6a7ba71babaa2a3
a4a4a5ba4a6ba5a7ba61ba7aba2baa3ba2ba3
ba3ba3ba4aba5a2ba6a3ba7a4ba5baa6ba2a71
a5a5a6baa7ba21ba3aba4a2ba5a3ba6a4ba7b
ba4ba4ba5a6ba6a7ba71babaa2ba2a3ba3a4a5
a6a6a7ba61ba7aba2baa3ba2a4ba3a5ba4ba5
ba5ba5ba6a3ba7a4ba5baa6ba2a7ba31ba4aa2
a7a71ba3aba4a2ba5a3ba6a4ba7a5ba6baba2
ba6ba6ba71babaa2ba2a3ba3a4ba4a5ba5a6a7
ba7ba7ba5baa6ba2a7ba31ba4aba5a2ba6a3a4

Right Cayley graph

Left Cayley graph

Others with same cardinality

20 unique, 117 total

Σ#PresentationDescriptionRelated
91331⟨a, b | aaaa=b, bbbb=1⟩Isomorphic to ℤ1667 iso
92051⟨a, b | aab=a, bbbb=b⟩Finite non-commutative monoid with 16 elements4 anti-iso
103808⟨a, b | aaab=ba, abab=1⟩Finite non-Abelian group with 16 elements7 iso
104630⟨a, b | aaaa=a, bbbb=a⟩Isomorphic to ℕ(16 = 4)1 iso
104648⟨a, b | aaaa=b, bbbb=a⟩Isomorphic to ℕ(16 = 1)5 iso
106205⟨a, b | aba=b, aaaabb=1⟩Finite non-Abelian group with 16 elements3 iso
1112164⟨a, b | aaaa=aa, bbbb=a⟩Isomorphic to ℕ(16 = 8)
1112194⟨a, b | aaaa=ab, bbbb=a⟩Isomorphic to ℕ(16 = 5)2 iso
1112212⟨a, b | aaaa=bb, bbbb=a⟩Isomorphic to ℕ(16 = 2)
1112306⟨a, b | aaab=bb, abba=a⟩Finite non-commutative monoid with 16 elements6 iso
1113251⟨a, b | bab=aab, bbb=aa⟩Finite non-commutative monoid with 16 elements
1113259⟨a, b | bab=aba, bbb=aa⟩Finite non-commutative monoid with 16 elements
1116028⟨a, b | aaa=ab, babb=bb⟩Finite non-commutative monoid with 16 elements
1116032⟨a, b | aaa=ab, bbaa=bb⟩Finite non-commutative monoid with 16 elements
1116060⟨a, b | aaa=bb, abab=aa⟩Finite non-commutative monoid with 16 elements
1116371⟨a, b | aba=aa, bbbb=ab⟩Finite non-commutative monoid with 16 elements
1118811⟨a, b | aaa=b, abbbbb=b⟩Isomorphic to ℕ(16 = 3)2 iso
1120251⟨a, b | aba=b, aaaa=abb⟩Finite non-commutative monoid with 16 elements
1121039⟨a, b | ab=aa, bbba=bbb⟩Finite non-commutative monoid with 16 elements
1124660⟨a, b | aa=a, ababab=bb⟩Finite non-commutative monoid with 16 elements

Other isomorphic instances

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

58 total

Σ#PresentationMapping
81044⟨a, b | aa=1, ababbb=1⟩φ(a) = ba, φ(b) = a
81047⟨a, b | aa=1, abbbab=1⟩φ(a) = ba, φ(b) = a
81054⟨a, b | aa=1, bababb=1⟩φ(a) = ba, φ(b) = a
81130⟨a, b | aa=1, babbb=a⟩φ(a) = ba, φ(b) = a
104043⟨a, b | abb=aaa, abab=1⟩φ(a) = a, φ(b) = b
104048⟨a, b | abb=aaa, baba=1⟩φ(a) = a, φ(b) = b
104101⟨a, b | bab=aaa, aabb=1⟩φ(a) = a, φ(b) = baa
104103⟨a, b | bab=aaa, abba=1⟩φ(a) = a, φ(b) = baa
104105⟨a, b | bab=aaa, baab=1⟩φ(a) = a, φ(b) = baa
105867⟨a, b | aabb=1, aaaba=b⟩φ(a) = a, φ(b) = baa
105877⟨a, b | aabb=1, abaaa=b⟩φ(a) = a, φ(b) = baa
105899⟨a, b | abab=1, aaaba=b⟩φ(a) = a, φ(b) = b
105909⟨a, b | abab=1, abaaa=b⟩φ(a) = a, φ(b) = b
105931⟨a, b | abba=1, aaaba=b⟩φ(a) = a, φ(b) = baa
105960⟨a, b | abba=1, babbb=a⟩φ(a) = a, φ(b) = baa
105986⟨a, b | aaa=a, ababbb=1⟩φ(a) = ba, φ(b) = a
105989⟨a, b | aaa=a, abbbab=1⟩φ(a) = ba, φ(b) = a
105996⟨a, b | aaa=a, bababb=1⟩φ(a) = ba, φ(b) = a
106125⟨a, b | aab=b, ababbb=1⟩φ(a) = ba, φ(b) = a
106131⟨a, b | aab=b, abbbab=1⟩φ(a) = ba, φ(b) = a
106145⟨a, b | aab=b, bababb=1⟩φ(a) = ba, φ(b) = a
106148⟨a, b | aab=b, babbba=1⟩φ(a) = ba, φ(b) = a
106155⟨a, b | aab=b, bbabab=1⟩φ(a) = ba, φ(b) = a
106160⟨a, b | aab=b, bbbaba=1⟩φ(a) = ba, φ(b) = a
106216⟨a, b | aba=b, aabbbb=1⟩φ(a) = baaa, φ(b) = a
106224⟨a, b | aba=b, abbabb=1⟩φ(a) = baaa, φ(b) = a
106226⟨a, b | aba=b, abbbba=1⟩φ(a) = baaa, φ(b) = a
106231⟨a, b | aba=b, baabbb=1⟩φ(a) = baaa, φ(b) = a
106233⟨a, b | aba=b, babbab=1⟩φ(a) = baaa, φ(b) = a
106235⟨a, b | aba=b, bbaabb=1⟩φ(a) = baaa, φ(b) = a
109406⟨a, b | aa=1, aaababbb=1⟩φ(a) = ba, φ(b) = a
109412⟨a, b | aa=1, aaabbbab=1⟩φ(a) = ba, φ(b) = a
109424⟨a, b | aa=1, aabababb=1⟩φ(a) = ba, φ(b) = a
109427⟨a, b | aa=1, aababbba=1⟩φ(a) = ba, φ(b) = a
109432⟨a, b | aa=1, aabbabab=1⟩φ(a) = ba, φ(b) = a
109436⟨a, b | aa=1, aabbbaba=1⟩φ(a) = ba, φ(b) = a
109447⟨a, b | aa=1, abaaabbb=1⟩φ(a) = ba, φ(b) = a
109451⟨a, b | aa=1, abaabbab=1⟩φ(a) = ba, φ(b) = a
109455⟨a, b | aa=1, ababaabb=1⟩φ(a) = ba, φ(b) = a
109457⟨a, b | aa=1, abababba=1⟩φ(a) = ba, φ(b) = a
109459⟨a, b | aa=1, ababbaab=1⟩φ(a) = ba, φ(b) = a
109467⟨a, b | aa=1, abbaabab=1⟩φ(a) = ba, φ(b) = a
109475⟨a, b | aa=1, abbbaaab=1⟩φ(a) = ba, φ(b) = a
109489⟨a, b | aa=1, baaababb=1⟩φ(a) = ba, φ(b) = a
109493⟨a, b | aa=1, baababab=1⟩φ(a) = ba, φ(b) = a
109499⟨a, b | aa=1, babaaabb=1⟩φ(a) = ba, φ(b) = a
109700⟨a, b | aa=1, aababbb=a⟩φ(a) = ba, φ(b) = a
109710⟨a, b | aa=1, aabbbab=a⟩φ(a) = ba, φ(b) = a
109732⟨a, b | aa=1, abababb=a⟩φ(a) = ba, φ(b) = a
109736⟨a, b | aa=1, ababbba=a⟩φ(a) = ba, φ(b) = a
109744⟨a, b | aa=1, abbabab=a⟩φ(a) = ba, φ(b) = a
109766⟨a, b | aa=1, baaabbb=a⟩φ(a) = ba, φ(b) = a
109772⟨a, b | aa=1, baabbab=a⟩φ(a) = ba, φ(b) = a
109776⟨a, b | aa=1, babaabb=a⟩φ(a) = ba, φ(b) = a
1010005⟨a, b | aa=1, ababbb=aa⟩φ(a) = ba, φ(b) = a
1010017⟨a, b | aa=1, abbbab=aa⟩φ(a) = ba, φ(b) = a
1010043⟨a, b | aa=1, bababb=aa⟩φ(a) = ba, φ(b) = a
1010324⟨a, b | aa=1, babbb=aaa⟩φ(a) = ba, φ(b) = a