⟨a, b | baaab=a, bbbbb=1⟩

Up:

Monoid enumeration

#11569 ⟨a, b | abbba=a, bbbbb=1⟩

#11604 ⟨a, b | baabb=a, bbbbb=1⟩

Properties

Sum of relation sides is 11

Infinite non-cancellative non-commutative monoid

Element a has infinite order

Not left cancellative, because left multiplication by c is not injective:

c ⋅ ca² = c and c ⋅ 1 = c, however ca² ≠ 1

Not right cancellative, because right multiplication by a is not injective:

ca² ⋅ a = a and 1 ⋅ a = a, however ca² ≠ 1

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

Auxiliary generators:
1. c = baabaabaabaab
Reduction order:
- Left-to-right recursive path with deg(c) = deg(a) = 0, c < a; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	ac ⇒ ca	[21]
2.	c²a² ⇒ c	[33]
3.	ca³ ⇒ a	[13]
4.	cbca² ⇒ cb	[35]
5.	abca² ⇒ ab	[25]
6.	c²b ⇒ bc²	[39]
7.	a²b ⇒ cba⁴	[20]
8.	c²ab ⇒ abc²	[40]
9.	bcab ⇒ c²a	[27]
10.	cb²ca² ⇒ cb²	[36]
11.	ab²ca² ⇒ ab²	[18]
12.	b³c ⇒ cabcba⁵	[34]
13.	b³a ⇒ cabcba⁸	[30]
14.	b(cb)² ⇒ ab²c⁶a	[45]
15.	cb³ ⇒ abcba³	[44]
16.	ab³ ⇒ bcba⁷	[31]
17.	(ab²)² ⇒ (bcba)²a¹⁴	[32]
18.	(abcb)² ⇒ cb²ab²c⁶a	[46]
19.	b⁵ ⇒ 1	[2]

# ab:baaab=a,bbbbb=1 ca/b baabaabaabaab=c morph:13/0 ac=ca ccaa=c caaa=a cbcaa=cb abcaa=ab ccb=bcc aab=cbaaaa ccab=abcc bcab=cca cbbcaa=cbb abbcaa=abb bbbc=cabcbaaaaa bbba=cabcbaaaaaaaa bcbcb=abbcccccca cbbb=abcbaaa abbb=bcbaaaaaaa abbabb=bcbabcbaaaaaaaaaaaaaaa abcbabcb=cbbabbcccccca bbbbb=1

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

Σ	#	Presentation	Mapping
11	12859	⟨a, b \| bab=aaa, bbbbb=1⟩	φ(a) = a, φ(b) = bbbb

Presentation

Mapping

12859

⟨a, b | bab=aaa, bbbbb=1⟩

φ(a) = a, φ(b) = bbbb

#11590 ⟨a, b | baaab=a, bbbbb=1⟩

Contents

Properties

Rewriting system

Isomorphic instances

Up:	Monoid enumeration
Prev:	#11569 ⟨a, b \| abbba=a, bbbbb=1⟩
Next:	#11604 ⟨a, b \| baabb=a, bbbbb=1⟩