⟨a, b | aaa=1, ababbbb=1⟩

Up:

Monoid enumeration

#7206 ⟨a, b | bb=aa, bab=aba⟩

#7279 ⟨a, b | aaa=1, bbbbbbb=1⟩

Finite structure

10 element center:

1, b³, b⁶, b⁹, b¹², b¹⁵, b¹⁸, b²¹, b²⁴, b²⁷

Order of generators:

a: order 3
b: order 30

Histogram:

order 2	1 element	b¹⁵
order 3	8 elements	a, ab⁵, bab⁴, b²ab³, b¹⁰, ...
order 4	6 elements	ab¹⁰, bab⁹, b²ab⁸, ab²⁵, bab²⁴, ...
order 5	4 elements	b⁶, b¹², b¹⁸, b²⁴
order 6	8 elements	b⁵, ab¹⁵, bab¹⁴, b²ab¹³, ab²⁰, ...
order 10	4 elements	b³, b⁹, b²¹, b²⁷
order 15	32 elements	b², b⁴, ab⁶, bab⁵, b²ab⁴, ...
order 20	24 elements	ab, ba, ab⁴, bab³, b²ab², ...
order 30	32 elements	b, ab², bab, b²a, ab³, ...

Rewriting system

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	1 1

Reduction order:
- Left-to-right recursive path with deg(b) = 0; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b³⁰ ⇒ 1	[20]
2.	b³a ⇒ ab³	[9]
3.	a² ⇒ bab⁴	[3]
4.	aba ⇒ b²⁶	[18]
5.	ab²a ⇒ b²ab²⁰	[19]

# ab:aaa=1,ababbbb=1 b/a - - bbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=1 bbba=abbb aa=babbbb aba=bbbbbbbbbbbbbbbbbbbbbbbbbb abba=bbabbbbbbbbbbbbbbbbbbbb

Isomorphic instances

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

Σ	#	Presentation	Mapping
10	7257	⟨a, b \| aaa=1, abbbbab=1⟩	φ(a) = a, φ(b) = b
10	7270	⟨a, b \| aaa=1, bababbb=1⟩	φ(a) = a, φ(b) = b
10	7276	⟨a, b \| aaa=1, bbababb=1⟩	φ(a) = a, φ(b) = b
11	14991	⟨a, b \| aaa=bb, ababab=1⟩	φ(a) = babbbbbbbb, φ(b) = ab
11	21194	⟨a, b \| aaa=1, abababbb=1⟩	φ(a) = abbbbb, φ(b) = ab
11	21198	⟨a, b \| aaa=1, ababbbab=1⟩	φ(a) = abbbbb, φ(b) = ab
11	21213	⟨a, b \| aaa=1, abbbabab=1⟩	φ(a) = abbbbb, φ(b) = ab
11	21238	⟨a, b \| aaa=1, babababb=1⟩	φ(a) = abbbbb, φ(b) = ab
11	21766	⟨a, b \| aaa=1, baabbbb=a⟩	φ(a) = abbbbb, φ(b) = b
11	21776	⟨a, b \| aaa=1, babbbab=a⟩	φ(a) = abbbbb, φ(b) = b
11	21784	⟨a, b \| aaa=1, bbababb=a⟩	φ(a) = a, φ(b) = bb
11	22314	⟨a, b \| aaa=1, babbbb=aa⟩	φ(a) = a, φ(b) = b

Presentation

Mapping

7257

⟨a, b | aaa=1, abbbbab=1⟩

φ(a) = a, φ(b) = b

7270

⟨a, b | aaa=1, bababbb=1⟩

φ(a) = a, φ(b) = b

7276

⟨a, b | aaa=1, bbababb=1⟩

φ(a) = a, φ(b) = b

14991

⟨a, b | aaa=bb, ababab=1⟩

φ(a) = babbbbbbbb, φ(b) = ab

21194

⟨a, b | aaa=1, abababbb=1⟩

φ(a) = abbbbb, φ(b) = ab

21198

⟨a, b | aaa=1, ababbbab=1⟩

φ(a) = abbbbb, φ(b) = ab

21213

⟨a, b | aaa=1, abbbabab=1⟩

φ(a) = abbbbb, φ(b) = ab

21238

⟨a, b | aaa=1, babababb=1⟩

φ(a) = abbbbb, φ(b) = ab

21766

⟨a, b | aaa=1, baabbbb=a⟩

φ(a) = abbbbb, φ(b) = b

21776

⟨a, b | aaa=1, babbbab=a⟩

φ(a) = abbbbb, φ(b) = b

21784

⟨a, b | aaa=1, bbababb=a⟩

φ(a) = a, φ(b) = bb

22314

⟨a, b | aaa=1, babbbb=aa⟩

φ(a) = a, φ(b) = b

#7249 ⟨a, b | aaa=1, ababbbb=1⟩

Contents

Properties

Finite structure

Right Cayley graph

Left Cayley graph

Rewriting system

Isomorphic instances

Up:	Monoid enumeration
Prev:	#7206 ⟨a, b \| bb=aa, bab=aba⟩
Next:	#7279 ⟨a, b \| aaa=1, bbbbbbb=1⟩