#4153 ⟨a, b | aabbaaaab=ba⟩

Contents

Properties
Rewriting system
Other submonoids of same group

Properties

Sum of relation sides is 11
Infinite cancellative non-commutative monoid
Element a has infinite order
Submonoid of enveloping group: ⟨a, b | aaaabbaab^-1a^-1b⟩
- Anti-isomorphic embedding: φ(a) = a, φ(b) = b

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 = aaab
2. d = abba
Reduction order:
- Right-to-left recursive path with deg(c) = deg(d) = 0, c < d; deg(a) = 1; deg(b) = 2
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	cadc²dc ⇒ d	[10]
2.	cadc²d² ⇒ dadc²dc	[11]
3.	a²d ⇒ cadc	[9]
4.	a²cadc² ⇒ ca	[12]
5.	a²cadcd ⇒ c²adc³dc	[13]
6.	a³b ⇒ c	[2]
7.	bc ⇒ adca²b	[7]
8.	bd ⇒ (adc)²dc	[8]
9.	ba ⇒ adc	[4]

# ab:aabbaaaab=ba reversed:cd/a/b aaab=c,abba=d morph:4/0,4/0
cadccdc=d
cadccdd=dadccdc
aad=cadc
aacadcc=ca
aacadcd=ccadcccdc
aaab=c
bc=adcaab
bd=adcadcdc
ba=adc

Other submonoids of same group

1 unique, 1 total

Σ	#	Presentation	Properties
11	4857	⟨a, b \| abbaaaab=baa⟩	Can Inf