#2265 ⟨a, b | aabbaba=baa⟩

Contents

Properties
Rewriting system

Properties

Sum of relation sides is 10
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not right cancellative, because right multiplication by c is not injective:
- ba ⋅ c = dc and d ⋅ c = dc, however ba ≠ d

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

#	Rule	Proof
1.	bd ⇒ dcb	[14]
2.	bc² ⇒ d²cbc	[16]
3.	bcd ⇒ d(dc)²b	[18]
4.	bca ⇒ d²cba	[20]
5.	bac ⇒ dc	[11]
6.	bad ⇒ d²	[6]
7.	acdc ⇒ dac	[13]
8.	acd² ⇒ dad	[7]
9.	a(dc)² ⇒ cac	[23]
10.	adcd² ⇒ cad	[22]
11.	acb ⇒ d	[3]
12.	adcbc ⇒ c²	[15]
13.	ab²c ⇒ cb²a	[8]
14.	ba² ⇒ da	[4]
15.	acda ⇒ da²	[5]
16.	adcda ⇒ ca²	[21]
17.	adcba ⇒ ca	[19]
18.	ab²a ⇒ c	[2]

# ab:aabbaba=baa reversed:cd/b/a abba=c,acb=d morph:4/1,3/0
bd=dcb
bcc=ddcbc
bcd=ddcdcb
bca=ddcba
bac=dc
bad=dd
acdc=dac
acdd=dad
adcdc=cac
adcdd=cad
acb=d
adcbc=cc
abbc=cbba
baa=da
acda=daa
adcda=caa
adcba=ca
abba=c