#332 ⟨a, b | aa=1, abbb=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. b5 ⇒ b
  2. ab ⇒ b3
  3. a2 ⇒ 1 
# ab:aa=1,abbb=b b/a
bbbbb=b
ab=bbb
aa=1

Cayley table

Idempotents are shown in bold.

1abbab2b2ab3b3ab4b4a
11abbab2b2ab3b3ab4b4a
aa1b3b3ab4b4abbab2b2a
bbbab2b2ab3b3ab4b4abba
bababb4b4abbab2b2ab3b3a
b2b2b2ab3b3ab4b4abbab2b2a
b2ab2ab2bbab2b2ab3b3ab4b4a
b3b3b3ab4b4abbab2b2ab3b3a
b3ab3ab3b2b2ab3b3ab4b4abba
b4b4b4abbab2b2ab3b3ab4b4a
b4ab4ab4b3b3ab4b4abbab2b2a

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

31 unique, 834 total

Σ#PresentationDescriptionRelated
8507⟨a, b | aaa=b, abbb=1⟩Isomorphic to ℤ10536 iso
8656⟨a, b | bb=aa, aba=a⟩Finite non-commutative monoid with 10 elements1 iso
8657⟨a, b | bb=aa, aba=b⟩Finite non-commutative monoid with 10 elements1 iso
8970⟨a, b | aa=a, bbb=ab⟩Finite non-commutative monoid with 10 elements3 iso
91654⟨a, b | aab=bb, bba=a⟩Finite non-commutative monoid with 10 elements6 iso
92012⟨a, b | aaa=b, abbb=a⟩Isomorphic to ℕ(10 = 1)59 iso
92013⟨a, b | aaa=b, abbb=b⟩Isomorphic to ℕ(10 = 3)35 iso
92277⟨a, b | bb=aa, aba=aa⟩Finite non-commutative monoid with 10 elements1 iso
92894⟨a, b | aa=a, bbbbb=a⟩Isomorphic to ℕ(10 = 5)17 iso
92935⟨a, b | aa=b, bbbbb=b⟩Isomorphic to ℕ(10 = 2)57 iso
104637⟨a, b | aaaa=b, aabb=b⟩Isomorphic to ℕ(10 = 4)20 iso
105346⟨a, b | aaa=bb, abb=ab⟩Finite non-commutative monoid with 10 elements2 iso, 1 anti-iso
105356⟨a, b | aab=aa, aba=bb⟩Finite non-commutative monoid with 10 elements
106587⟨a, b | aaa=b, abbb=bb⟩Isomorphic to ℕ(10 = 6)3 iso
106633⟨a, b | aab=a, baaa=bb⟩Finite non-commutative monoid with 10 elements20 iso, 1 anti-iso
107089⟨a, b | ab=aa, baaa=bb⟩Finite non-commutative monoid with 10 elements3 iso
108619⟨a, b | aa=a, abbbbb=b⟩Finite non-commutative monoid with 10 elements5 iso
109051⟨a, b | ab=a, baaaa=bb⟩Finite non-commutative monoid with 10 elements4 iso
1110729⟨a, b | aaab=baa, abab=1⟩Finite non-Abelian group with 10 elements3 iso
1112157⟨a, b | aaaa=aa, abba=b⟩Finite non-commutative monoid with 10 elements
1112181⟨a, b | aaaa=ab, baaa=b⟩Finite non-commutative monoid with 10 elements1 iso
1112196⟨a, b | aaaa=bb, aaab=a⟩Finite commutative monoid with 10 elements7 iso
1112197⟨a, b | aaaa=bb, aaab=b⟩Finite commutative monoid with 10 elements1 iso
1112452⟨a, b | aabb=aa, bbbb=a⟩Isomorphic to ℕ(10 = 8)3 iso
1115738⟨a, b | aab=bb, aaaaa=b⟩Isomorphic to ℕ(10 = 7)7 iso
1119564⟨a, b | aab=b, abbbb=aa⟩Finite commutative monoid with 10 elements5 iso
1119899⟨a, b | aaa=b, abbb=bbb⟩Isomorphic to ℕ(10 = 9)1 iso
1120052⟨a, b | aab=b, aaaa=baa⟩Finite non-commutative monoid with 10 elements
1120253⟨a, b | aba=b, aaaa=bbb⟩Finite commutative monoid with 10 elements
1125905⟨a, b | ab=a, bbbb=baaa⟩Finite non-commutative monoid with 10 elements
1125909⟨a, b | ab=a, bbbb=bbaa⟩Finite non-commutative monoid with 10 elements

Other isomorphic instances

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

51 total

Σ#PresentationMapping
7373⟨a, b | aa=1, bbb=ab⟩φ(a) = a, φ(b) = b
93351⟨a, b | aa=1, aaabbb=b⟩φ(a) = a, φ(b) = b
93371⟨a, b | aa=1, abaabb=b⟩φ(a) = a, φ(b) = b
93373⟨a, b | aa=1, ababab=b⟩φ(a) = a, φ(b) = b
93379⟨a, b | aa=1, abbaab=b⟩φ(a) = a, φ(b) = b
93499⟨a, b | aa=1, aabbb=ab⟩φ(a) = a, φ(b) = b
93528⟨a, b | aa=1, baabb=ab⟩φ(a) = a, φ(b) = b
93532⟨a, b | aa=1, babab=ab⟩φ(a) = a, φ(b) = b
93621⟨a, b | aa=1, aaab=bbb⟩φ(a) = a, φ(b) = b
93653⟨a, b | aa=1, abbb=aab⟩φ(a) = a, φ(b) = b
93656⟨a, b | aa=1, abbb=baa⟩φ(a) = a, φ(b) = b
1126455⟨a, b | aa=1, aaaaabbb=b⟩φ(a) = a, φ(b) = b
1126477⟨a, b | aa=1, aaabaabb=b⟩φ(a) = a, φ(b) = b
1126481⟨a, b | aa=1, aaababab=b⟩φ(a) = a, φ(b) = b
1126489⟨a, b | aa=1, aaabbaab=b⟩φ(a) = a, φ(b) = b
1126495⟨a, b | aa=1, aaabbbaa=b⟩φ(a) = a, φ(b) = b
1126561⟨a, b | aa=1, abaaaabb=b⟩φ(a) = a, φ(b) = b
1126563⟨a, b | aa=1, abaaabab=b⟩φ(a) = a, φ(b) = b
1126569⟨a, b | aa=1, abaabaab=b⟩φ(a) = a, φ(b) = b
1126581⟨a, b | aa=1, ababaaab=b⟩φ(a) = a, φ(b) = b
1126603⟨a, b | aa=1, abbaaaab=b⟩φ(a) = a, φ(b) = b
1126996⟨a, b | aa=1, aaaabbb=ab⟩φ(a) = a, φ(b) = b
1127024⟨a, b | aa=1, aaabbba=ba⟩φ(a) = a, φ(b) = b
1127039⟨a, b | aa=1, aabaabb=ab⟩φ(a) = a, φ(b) = b
1127046⟨a, b | aa=1, aababab=ab⟩φ(a) = a, φ(b) = b
1127058⟨a, b | aa=1, aabbaab=ab⟩φ(a) = a, φ(b) = b
1127070⟨a, b | aa=1, aabbbaa=ab⟩φ(a) = a, φ(b) = b
1127101⟨a, b | aa=1, abaabba=ba⟩φ(a) = a, φ(b) = b
1127172⟨a, b | aa=1, baaaabb=ab⟩φ(a) = a, φ(b) = b
1127176⟨a, b | aa=1, baaabab=ab⟩φ(a) = a, φ(b) = b
1127184⟨a, b | aa=1, baabaab=ab⟩φ(a) = a, φ(b) = b
1127509⟨a, b | aa=1, aaaaab=bbb⟩φ(a) = a, φ(b) = b
1127533⟨a, b | aa=1, aaabaa=bbb⟩φ(a) = a, φ(b) = b
1127551⟨a, b | aa=1, aaabbb=aab⟩φ(a) = a, φ(b) = b
1127554⟨a, b | aa=1, aaabbb=baa⟩φ(a) = a, φ(b) = b
1127598⟨a, b | aa=1, aabbba=aba⟩φ(a) = a, φ(b) = b
1127627⟨a, b | aa=1, abaabb=aab⟩φ(a) = a, φ(b) = b
1127630⟨a, b | aa=1, abaabb=baa⟩φ(a) = a, φ(b) = b
1127635⟨a, b | aa=1, ababab=aab⟩φ(a) = a, φ(b) = b
1127638⟨a, b | aa=1, ababab=baa⟩φ(a) = a, φ(b) = b
1127659⟨a, b | aa=1, abbaab=aab⟩φ(a) = a, φ(b) = b
1127662⟨a, b | aa=1, abbaab=baa⟩φ(a) = a, φ(b) = b
1128041⟨a, b | aa=1, aaaab=abbb⟩φ(a) = a, φ(b) = b
1128064⟨a, b | aa=1, aaaba=bbba⟩φ(a) = a, φ(b) = b
1128088⟨a, b | aa=1, aabaa=abbb⟩φ(a) = a, φ(b) = b
1128125⟨a, b | aa=1, aabbb=aaab⟩φ(a) = a, φ(b) = b
1128128⟨a, b | aa=1, aabbb=abaa⟩φ(a) = a, φ(b) = b
1128200⟨a, b | aa=1, abbba=aaba⟩φ(a) = a, φ(b) = b
1128235⟨a, b | aa=1, baabb=aaab⟩φ(a) = a, φ(b) = b
1128238⟨a, b | aa=1, baabb=abaa⟩φ(a) = a, φ(b) = b
1128251⟨a, b | aa=1, babab=aaab⟩φ(a) = a, φ(b) = b

Other anti-isomorphic instances

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

34 total

Σ#PresentationMapping
93363⟨a, b | aa=1, aabbba=b⟩φ(a) = a, φ(b) = b
93500⟨a, b | aa=1, aabbb=ba⟩φ(a) = a, φ(b) = b
93518⟨a, b | aa=1, abbba=ab⟩φ(a) = a, φ(b) = b
93529⟨a, b | aa=1, baabb=ba⟩φ(a) = a, φ(b) = b
93629⟨a, b | aa=1, aaba=bbb⟩φ(a) = a, φ(b) = b
93654⟨a, b | aa=1, abbb=aba⟩φ(a) = a, φ(b) = b
1126469⟨a, b | aa=1, aaaabbba=b⟩φ(a) = a, φ(b) = b
1126513⟨a, b | aa=1, aabaabba=b⟩φ(a) = a, φ(b) = b
1126519⟨a, b | aa=1, aabababa=b⟩φ(a) = a, φ(b) = b
1126533⟨a, b | aa=1, aabbaaba=b⟩φ(a) = a, φ(b) = b
1126997⟨a, b | aa=1, aaaabbb=ba⟩φ(a) = a, φ(b) = b
1127023⟨a, b | aa=1, aaabbba=ab⟩φ(a) = a, φ(b) = b
1127040⟨a, b | aa=1, aabaabb=ba⟩φ(a) = a, φ(b) = b
1127047⟨a, b | aa=1, aababab=ba⟩φ(a) = a, φ(b) = b
1127059⟨a, b | aa=1, aabbaab=ba⟩φ(a) = a, φ(b) = b
1127100⟨a, b | aa=1, abaabba=ab⟩φ(a) = a, φ(b) = b
1127112⟨a, b | aa=1, abababa=ab⟩φ(a) = a, φ(b) = b
1127173⟨a, b | aa=1, baaaabb=ba⟩φ(a) = a, φ(b) = b
1127177⟨a, b | aa=1, baaabab=ba⟩φ(a) = a, φ(b) = b
1127517⟨a, b | aa=1, aaaaba=bbb⟩φ(a) = a, φ(b) = b
1127552⟨a, b | aa=1, aaabbb=aba⟩φ(a) = a, φ(b) = b
1127597⟨a, b | aa=1, aabbba=aab⟩φ(a) = a, φ(b) = b
1127600⟨a, b | aa=1, aabbba=baa⟩φ(a) = a, φ(b) = b
1127628⟨a, b | aa=1, abaabb=aba⟩φ(a) = a, φ(b) = b
1127636⟨a, b | aa=1, ababab=aba⟩φ(a) = a, φ(b) = b
1127660⟨a, b | aa=1, abbaab=aba⟩φ(a) = a, φ(b) = b
1128048⟨a, b | aa=1, aaaab=bbba⟩φ(a) = a, φ(b) = b
1128057⟨a, b | aa=1, aaaba=abbb⟩φ(a) = a, φ(b) = b
1128126⟨a, b | aa=1, aabbb=aaba⟩φ(a) = a, φ(b) = b
1128132⟨a, b | aa=1, aabbb=baaa⟩φ(a) = a, φ(b) = b
1128199⟨a, b | aa=1, abbba=aaab⟩φ(a) = a, φ(b) = b
1128236⟨a, b | aa=1, baabb=aaba⟩φ(a) = a, φ(b) = b
1128242⟨a, b | aa=1, baabb=baaa⟩φ(a) = a, φ(b) = b
1128252⟨a, b | aa=1, babab=aaba⟩φ(a) = a, φ(b) = b