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〈a, b | aa=1, ababba=b〉

Monoid presentation of length 9

Properties

Completion parameters

Complete rewriting system

  1. a2 ⇒ 1
  2. bab2 ⇒ aba
  3. b3a ⇒ (ab)3
  4. b(ba)2 ⇒ ab2ab
  5. (ba)3 ⇒ ab3
  6. b5 ⇒ b

Idempotents

2 elements

Cayley table

Idempotents are shown in bold.

1ababbab2abaab2babb2ab3(ab)2ab2a(ba)2bab2b3ab4a(ba)2b(ab)2bab2ab4a(ba)3
11ababbab2abaab2babb2ab3(ab)2ab2a(ba)2bab2b3ab4a(ba)2b(ab)2bab2ab4a(ba)3
aa1abbabaab2bab2(ab)2ab2a(ba)3babb2aa(ba)2bab2ab(ab)2b4a(ba)2b3abab2b4b3
bbbab2babb2ab3(ba)2bab2abab3ab4b(ab)2bab2aab(ab)2b4ab(ba)3ab2a(ba)2baab2a
abababaab2(ab)2ab2a(ba)3a(ba)2bab2abab(ab)2b4ab3abab2bbabb4abb3b2(ba)2abab2a
bababbabb2(ba)2bab2b2ab3b(ab)2bab2aab2aabab3a(ba)3a(ba)2ab2baabb4a(ab)2bb4
b2b2b2ab3abab3ab4ab(ab)2(ba)2b4abab2a(ba)2babb(ab)2bab2ab2abab2(ba)3b2abab2a
abaabaab(ab)2ab2a(ba)2bab2aab2a(ba)3b3abab2b2abab(ab)2b3(ba)2b2ababb4bababb4a
ab2ab2ab2a(ba)3bab(ab)2b4abbaba(ba)2b4abb2(ba)2(ab)2b3aabaab2b2abab2ab3ab2abab2
babbab(ba)2bab2b(ab)2bab2aab2a(ba)3a(ba)2b2aab2bab4a(ab)2b2ababbabb4b3ab(ba)2b3a
b2ab2ab2abab3ab(ab)2b3ab4ab2a(ba)2bab2a(ba)2b4aab2a(ba)3bab2b2ababbab(ab)2b2b
b3b3b3ab4(ba)2b4abbabb(ab)2abbab2bab2(ba)3abaab2b2ab3bab2a(ab)2ab2ab3aa(ba)2
(ab)2(ab)2a(ba)2bab2ab3abab2b2ab3(ba)2ab2ab2abab4babab2baab(ab)2b4a(ba)3ba(ba)2b(ab)2
ab2aab2aab2ba(ba)3bbabb(ab)2b4ab2(ba)2bab2a(ba)2b4b2ab3bab2aab2a(ab)2abab3aab2ab
(ba)2(ba)2babb(ab)2bab2(ba)3a(ba)2bab2aab2ab4a(ab)2b3ab2aab2b4abb3(ba)2b2babababba
bab2bab2bab2aab2ab2aab2bab2aba(ba)3bbabb3abb(ab)2b4a(ba)2bab2b3aa(ba)2b4bab2a(ab)2
b3ab3ab3(ba)2b4babb(ab)2b4abbab2(ba)3a(ba)2abbabab2aab2a(ab)2b3aabab2aab2b3b2
b4b4b4ababbab2abaab2babb2ab3(ab)2ab2a(ba)2bab2b3ab4a(ba)2b(ab)2bab2ab4a(ba)3
a(ba)2a(ba)2(ab)2b3abab2ab3(ba)2bab2b2ab4babb(ab)2ab2ab2b4ab(ba)3a(ba)2ab2abba(ab)2aba
b(ab)2b(ab)2(ba)3a(ba)2b4a(ab)2b3ab4abbab2ab3(ba)2bababab2b2ababb(ab)2baab2ab2(ba)3ab2
bab2abab2abab2b2aab2ab2abaab2bab3ab(ab)2(ba)3bb3ab4a(ba)2bab2ab(ab)2(ba)2b4abab2bab
b4ab4ab4abbabaab2bab2(ab)2ab2a(ba)3babb2aa(ba)2bab2ab(ab)2b4a(ba)2b3abab2b4b3
(ba)3(ba)3b(ab)2b4aa(ba)2b4ab(ab)2b3ababaab2bab2ab3bab2ab2a(ba)3bab2babb2ab(ab)2(ba)2

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

4 unique, 8 total

Length:Presentation:Description:Related:
10 〈a, b | aa=1, abbbbbb=b〉 Finite non-commutative monoid with 22 elements 1 isomorphic
11 〈a, b | aba=b, baaab=aa〉 Finite non-commutative monoid with 22 elements
11 〈a, b | aa=a, bbbbbb=ab〉 Finite non-commutative monoid with 22 elements
11 〈a, b | aa=1, abbabbbb=b〉 Finite non-commutative monoid with 22 elements 3 isomorphic

Other isomorphic instances

15 total

Length:Presentation:
9〈a, b | aa=1, ababb=ba〉
9〈a, b | aa=1, babb=aba〉
11〈a, b | aa=1, aaababba=b〉
11〈a, b | aa=1, abaaabba=b〉
11〈a, b | aa=1, aaababb=ba〉
11〈a, b | aa=1, aababba=ab〉
11〈a, b | aa=1, abaaabb=ba〉
11〈a, b | aa=1, ababaab=ba〉
11〈a, b | aa=1, aababb=aba〉
11〈a, b | aa=1, ababba=aab〉
11〈a, b | aa=1, ababba=baa〉
11〈a, b | aa=1, baaabb=aba〉
11〈a, b | aa=1, aaaba=babb〉
11〈a, b | aa=1, ababb=aaba〉
11〈a, b | aa=1, ababb=baaa〉

Other anti-isomorphic instances

12 total

Length:Presentation:
9〈a, b | aa=1, abbab=ba〉
11〈a, b | aa=1, aaabbaba=b〉
11〈a, b | aa=1, abaababa=b〉
11〈a, b | aa=1, aaabbab=ba〉
11〈a, b | aa=1, aabbaba=ab〉
11〈a, b | aa=1, abaabab=ba〉
11〈a, b | aa=1, abbaaab=ba〉
11〈a, b | aa=1, aabbab=aba〉
11〈a, b | aa=1, baabab=aba〉
11〈a, b | aa=1, aaaba=bbab〉
11〈a, b | aa=1, abbab=aaba〉
11〈a, b | aa=1, abbab=baaa〉