#145 ⟨a, b | aababba=1⟩

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Quick links

  1. Properties
  2. Rewriting system
  3. Other submonoids of same group
  4. Other submonoids of anti-isomorphic group
  5. Isomorphic instances
  6. Anti-isomorphic instances

Properties

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
#RuleProof
1. dc ⇒ 1 [11]
2. cd ⇒ 1 [8]
3. ca ⇒ ac [5]
4. da ⇒ ad [10]
5. a3 ⇒ c [2]
6. b2c ⇒ a(ab)2 [15]
7. babd ⇒ adb2 [12]
8. bcba ⇒ cbab [18]
9. b2ac ⇒ a2(ba)2 [17]
10. (ba)2d ⇒ adb2a [13]
11. b2a2 ⇒ a2dbcb [23]
12. baba2 ⇒ d(bc)2 [21]
13. bab2 ⇒ d [3]
# ab:aababba=1 reversed:cda/b aaa=c,babb=d magic:0
dc=1
cd=1
ca=ac
da=ad
aaa=c
bbc=aabab
babd=adbb
bcba=cbab
bbac=aababa
babad=adbba
bbaa=aadbcb
babaa=dbcbc
babb=d

Other submonoids of same group

11 unique, 11 total

Σ#PresentationDescriptionRelated
6104⟨a, b | aaba=bb⟩Infinite cancellative non-commutative monoid
7193⟨a, b | ababba=b⟩Infinite cancellative non-commutative monoid
7252⟨a, b | aaba=bab⟩Infinite cancellative non-commutative monoid
9842⟨a, b | abaababa=b⟩Infinite cancellative non-commutative monoid
91126⟨a, b | ababba=bab⟩Infinite cancellative non-commutative monoid
91273⟨a, b | abbba=babb⟩Infinite cancellative non-commutative monoid
102633⟨a, b | abbbba=babb⟩Infinite cancellative non-commutative monoid
113716⟨a, b | abaaabaaba=b⟩Infinite cancellative non-commutative monoid
114876⟨a, b | abbabbba=bab⟩Infinite cancellative non-commutative monoid
115415⟨a, b | abbbbba=babb⟩Infinite cancellative non-commutative monoid
115898⟨a, b | ababba=babab⟩Infinite cancellative non-commutative monoid

Other submonoids of anti-isomorphic group

7 unique, 7 total

Σ#PresentationDescriptionRelated
687⟨a, b | aabba=b⟩Infinite cancellative non-commutative monoid
7179⟨a, b | aababa=b⟩Infinite cancellative non-commutative monoid
8374⟨a, b | aabaaba=b⟩Infinite cancellative non-commutative monoid
8586⟨a, b | baab=aaba⟩Infinite cancellative non-commutative monoid
9792⟨a, b | aabaaaba=b⟩Infinite cancellative non-commutative monoid
101666⟨a, b | aabaaaaba=b⟩Infinite cancellative non-commutative monoid
113536⟨a, b | aabaaaaaba=b⟩Infinite cancellative non-commutative monoid

Isomorphic instances

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

22 total

Σ#PresentationMapping
7158⟨a, b | abbaaab=1⟩φ(a) = a, φ(b) = aadab
8312⟨a, b | aabbaaba=1⟩φ(a) = aad, φ(b) = aaba
8324⟨a, b | abaaabba=1⟩φ(a) = aad, φ(b) = aaba
8338⟨a, b | abbabbba=1⟩φ(a) = aaba, φ(b) = aad
9671⟨a, b | aababbaab=1⟩φ(a) = cbabaada, φ(b) = aadabab
9706⟨a, b | abaababba=1⟩φ(a) = cbabaada, φ(b) = abaadab
9723⟨a, b | abbaabaab=1⟩φ(a) = cbabaada, φ(b) = aadabab
101346⟨a, b | aaabaababa=1⟩φ(a) = a, φ(b) = aadabaad
101406⟨a, b | aabaababaa=1⟩φ(a) = a, φ(b) = aadabaad
101415⟨a, b | aababaaaab=1⟩φ(a) = a, φ(b) = aadaadab
101479⟨a, b | abaaaabaab=1⟩φ(a) = a, φ(b) = aadaadab
112928⟨a, b | aaababaaaba=1⟩φ(a) = aad, φ(b) = aaaba
113012⟨a, b | aabaaaababa=1⟩φ(a) = aad, φ(b) = aaaba
113064⟨a, b | aabababaaba=1⟩φ(a) = aadab, φ(b) = acbabaada
113077⟨a, b | aababbaabab=1⟩φ(a) = cbabaadcbabaada, φ(b) = aadababab
113185⟨a, b | abaaabaaaab=1⟩φ(a) = aad, φ(b) = aaaba
113190⟨a, b | abaaabababa=1⟩φ(a) = aadab, φ(b) = acbabaada
113201⟨a, b | abaabaaabab=1⟩φ(a) = aadab, φ(b) = cbabaadaa
113211⟨a, b | abaababbaab=1⟩φ(a) = cbabaadacbabaad, φ(b) = abaadabab
113233⟨a, b | ababaabaaab=1⟩φ(a) = aadab, φ(b) = cbabaadaa
113235⟨a, b | ababaababba=1⟩φ(a) = cbabaadcbabaada, φ(b) = abaadabab
113279⟨a, b | abbaababaab=1⟩φ(a) = cbabaadacbabaad, φ(b) = abaadabab

Anti-isomorphic instances

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

22 total

Σ#PresentationMapping
7147⟨a, b | aabbaba=1⟩φ(a) = a, φ(b) = aadab
7157⟨a, b | ababbba=1⟩φ(a) = aadab, φ(b) = a
8304⟨a, b | aabaabba=1⟩φ(a) = aad, φ(b) = aaba
8311⟨a, b | aabbaaab=1⟩φ(a) = aad, φ(b) = aaab
9681⟨a, b | aabbabaab=1⟩φ(a) = cbabaada, φ(b) = aadabab
9708⟨a, b | abaabbaba=1⟩φ(a) = cbabaada, φ(b) = abaadab
9717⟨a, b | ababbabba=1⟩φ(a) = abaadab, φ(b) = cbabaada
101352⟨a, b | aaababaaba=1⟩φ(a) = a, φ(b) = aadabaad
101480⟨a, b | abaaaababa=1⟩φ(a) = a, φ(b) = aadabaad
101491⟨a, b | abaabaaaab=1⟩φ(a) = a, φ(b) = aadaadab
112908⟨a, b | aaabaaababa=1⟩φ(a) = aad, φ(b) = aabaa
112927⟨a, b | aaababaaaab=1⟩φ(a) = aad, φ(b) = aaaab
113020⟨a, b | aabaaababaa=1⟩φ(a) = aad, φ(b) = aabaa
113036⟨a, b | aabaabababa=1⟩φ(a) = aadab, φ(b) = acbabaada
113053⟨a, b | aababaaaaba=1⟩φ(a) = aad, φ(b) = aabaa
113063⟨a, b | aabababaaab=1⟩φ(a) = aadab, φ(b) = cbabaadaa
113177⟨a, b | abaaaabaaab=1⟩φ(a) = aad, φ(b) = aaaba
113187⟨a, b | abaaabaabab=1⟩φ(a) = aadab, φ(b) = cbabaadaa
113217⟨a, b | abaabbabaab=1⟩φ(a) = cbabaadacbabaad, φ(b) = abaadabab
113230⟨a, b | ababaaabaab=1⟩φ(a) = aadab, φ(b) = cbabaadaa
113237⟨a, b | ababaabbaba=1⟩φ(a) = cbabaadcbabaada, φ(b) = abaadabab
113256⟨a, b | ababbababba=1⟩φ(a) = ababaadab, φ(b) = cbabaadacbabaad