#450 ⟨a, b | bab=aa, bbb=1⟩

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Quick links

  1. Properties
  2. Elements
  3. Right Cayley graph
  4. Left Cayley graph
  5. Rewriting system
  6. Same cardinality
  7. Isomorphic instances

Properties

Elements

Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
The order of x is the least n (if it exists) such that xn = 1.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Right Cayley graph

Left Cayley graph

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. b3 ⇒ 1 [2]
2. a2 ⇒ bab [1]
3. (ba)2 ⇒ (ab)2 [3]
4. abab2a ⇒ (b2a)2b [4]
5. ba(b2a)2 ⇒ a(b2a)2b [5]
# ab:bab=aa,bbb=1 b/a
bbb=1
aa=bab
baba=abab
ababba=bbabbab
babbabba=abbabbab

Same cardinality

6 unique, 66 total

Σ#PresentationDescriptionRelated
8752⟨a, b | aaa=1, babb=a⟩Finite non-Abelian group with 27 elements59 iso
91695⟨a, b | bab=aa, bbb=b⟩Finite non-commutative monoid with 27 elements1 iso
105300⟨a, b | aaa=aa, aba=bb⟩Finite non-commutative monoid with 27 elements
1115514⟨a, b | aaa=bb, aabaa=b⟩Finite non-commutative monoid with 27 elements
1118739⟨a, b | aaa=a, abbbbb=b⟩Finite non-commutative monoid with 27 elements
1120314⟨a, b | aba=b, bbbb=aaa⟩Finite non-commutative monoid with 27 elements

Isomorphic instances

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

27 total

Σ#PresentationMapping
8747⟨a, b | aaa=1, abba=b⟩φ(a) = bb, φ(b) = a
91305⟨a, b | baa=abb, bbb=1⟩φ(a) = a, φ(b) = bb
91310⟨a, b | bab=aba, bbb=1⟩φ(a) = abb, φ(b) = b
92435⟨a, b | aaa=1, ababa=b⟩φ(a) = bb, φ(b) = ab
92573⟨a, b | aaa=1, aabb=ba⟩φ(a) = b, φ(b) = a
107503⟨a, b | aaa=1, aabbaa=b⟩φ(a) = b, φ(b) = a
107513⟨a, b | aaa=1, abaaba=b⟩φ(a) = bb, φ(b) = abb
107769⟨a, b | aaa=1, aabaa=bb⟩φ(a) = bb, φ(b) = a
107772⟨a, b | aaa=1, aabab=ba⟩φ(a) = b, φ(b) = abb
107775⟨a, b | aaa=1, aabba=ab⟩φ(a) = bb, φ(b) = a
108041⟨a, b | aaa=1, aaba=abb⟩φ(a) = b, φ(b) = a
108058⟨a, b | aaa=1, abab=baa⟩φ(a) = bb, φ(b) = ab
108078⟨a, b | aaa=1, baab=aba⟩φ(a) = b, φ(b) = ab
1121661⟨a, b | aaa=1, aaaabba=b⟩φ(a) = bb, φ(b) = a
1121687⟨a, b | aaa=1, aababaa=b⟩φ(a) = b, φ(b) = abb
1121711⟨a, b | aaa=1, abaaaba=b⟩φ(a) = bb, φ(b) = a
1122202⟨a, b | aaa=1, aaaaba=bb⟩φ(a) = b, φ(b) = a
1122225⟨a, b | aaa=1, aabaab=ba⟩φ(a) = b, φ(b) = ab
1122228⟨a, b | aaa=1, aababa=ab⟩φ(a) = bb, φ(b) = ab
1122744⟨a, b | aaa=1, aaabb=aba⟩φ(a) = b, φ(b) = a
1122754⟨a, b | aaa=1, aabaa=bab⟩φ(a) = bb, φ(b) = ab
1122768⟨a, b | aaa=1, aabba=baa⟩φ(a) = b, φ(b) = a
1122784⟨a, b | aaa=1, abaab=baa⟩φ(a) = bb, φ(b) = abb
1122826⟨a, b | aaa=1, baaab=aba⟩φ(a) = b, φ(b) = a
1123272⟨a, b | aaa=1, abaa=aabb⟩φ(a) = bb, φ(b) = a
1123275⟨a, b | aaa=1, abab=aaba⟩φ(a) = b, φ(b) = abb
1123279⟨a, b | aaa=1, abba=aaab⟩φ(a) = bb, φ(b) = a