#5345 ⟨a, b | ababaab=abba⟩

Contents

Properties
Rewriting system

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 ⋅ aba²b = cba and c ⋅ ba = cba, however aba²b ≠ ba

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

#	Rule	Proof
1.	cbab ⇒ ab²c	[3]
2.	ab²ab ⇒ c	[2]
3.	cabac ⇒ ab²cab	[8]
4.	a(ba)²c ⇒ cab	[5]
5.	caba²b ⇒ cba	[6]
6.	c(ab)³ ⇒ ca²bac	[13]
7.	cba³b ⇒ ca(ba)²	[12]
8.	cba²bac ⇒ ab²c(ab)²	[10]
9.	cba²cab ⇒ cab²a²c	[16]
10.	cba(ac)² ⇒ c(ab)²cab	[20]
11.	(ab)²a²b ⇒ ab²a	[1]
12.	ab²a²bac ⇒ c(ab)²	[7]
13.	c(a²b)² ⇒ caba	[11]
14.	ca²bacab ⇒ cba²c	[15]
15.	ca²b(ac)² ⇒ (cab)²	[19]
16.	cab²a³b ⇒ ca²baca	[14]
17.	cab²a²cab ⇒ cabab²a²c	[22]
18.	cab²a(ac)² ⇒ ca²bac²ab	[24]
19.	cb(a²b)² ⇒ ab²ca	[9]
20.	cba²ca²b ⇒ ca²bacba	[18]
21.	ab²(a²b)² ⇒ ca	[4]
22.	ca²baca²b ⇒ cabab²a	[17]
23.	cabab²a³b ⇒ cba²ca	[21]
24.	cabab²a²cab ⇒ ca²bacba²c	[25]
25.	cabab²a(ac)² ⇒ cba²c²ab	[27]
26.	cab²a²ca²b ⇒ cba²cba	[23]
27.	cabab²a²ca²b ⇒ cab²a²cba	[26]

# ab:ababaab=abba bc/a abbab=c morph:5/2
cbab=abbc
abbab=c
cabac=abbcab
ababac=cab
cabaab=cba
cababab=caabac
cbaaab=cababa
cbaabac=abbcabab
cbaacab=cabbaac
cbaacac=cababcab
ababaab=abba
abbaabac=cabab
caabaab=caba
caabacab=cbaac
caabacac=cabcab
cabbaaab=caabaca
cabbaacab=cababbaac
cabbaacac=caabaccab
cbaabaab=abbca
cbaacaab=caabacba
abbaabaab=ca
caabacaab=cababba
cababbaaab=cbaaca
cababbaacab=caabacbaac
cababbaacac=cbaaccab
cabbaacaab=cbaacba
cababbaacaab=cabbaacba