#3805 ⟨a, b | abbaabaaab=a⟩

Contents

Properties
Rewriting system

Properties

Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element b has infinite order
Not left cancellative, because left multiplication by c is not injective:
- c ⋅ bd²eb² = c and c ⋅ 1 = c, however bd²eb² ≠ 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 = ab
2. d = ac
3. e = bdada
Reduction order:
- Right-to-left recursive path with deg(b) = deg(d) = 0, b < d; deg(c) = 1; deg(e) = deg(a) = 2, e < a
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	d³eb ⇒ dbd²e	[26]
2.	dbd²eb ⇒ ce	[25]
3.	ceb ⇒ d	[23]
4.	cd²eb ⇒ cbd²e	[24]
5.	ed ⇒ bdcd²e	[33]
6.	cbd²eb² ⇒ c	[16]
7.	cbd²ebd ⇒ deb	[15]
8.	ec ⇒ bd²ebd	[28]
9.	bd²ebc ⇒ eb	[20]
10.	a ⇒ cbd²eb	[13]
11.	e²b ⇒ eb(bdc)²d²e	[34]
12.	(eb)² ⇒ bd²ebd	[22]
13.	(deb)² ⇒ cd²e	[30]
14.	ebd²eb ⇒ eb²d²e	[27]
15.	eb²d²eb ⇒ e	[21]

# ab:abbaabaaab=a reversed:bd/c/ea ab=c,ac=d,bdada=e morph:2/1,2/0,5/1
dddeb=dbdde
dbddeb=ce
ceb=d
cddeb=cbdde
ed=bdcdde
cbddebb=c
cbddebd=deb
ec=bddebd
bddebc=eb
a=cbddeb
eeb=ebbdcbdcdde
ebeb=bddebd
debdeb=cdde
ebddeb=ebbdde
ebbddeb=e