#995 ⟨a, b | abbaaab=ba⟩

Contents

Properties
Rewriting system
Other submonoids of same group

Properties

Sum of relation sides is 9
Infinite cancellative non-commutative monoid
Element a has infinite order
Submonoid of enveloping group: ⟨a, b | aaabbab^-1a^-1b⟩
- Embedding: φ(a) = a^-1, φ(b) = abaa

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 = baaa
2. d = ab
Reduction order:
- Left-to-right 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.	dc²b ⇒ cd	[10]
2.	(dc)²bd ⇒ cb	[9]
3.	(dc)³b ⇒ c	[5]
4.	d(dc)²d²cbd² ⇒ cdb	[19]
5.	dc(dcd)²cbc ⇒ c²(dc)²b	[16]
6.	cbc²b ⇒ (dc)²bcd	[14]
7.	cbc(dc)²b ⇒ (dc)²bc	[11]
8.	c²b² ⇒ (dc)²d²cbd²	[15]
9.	cdbc²b ⇒ d(dc)²d²cbdcd	[24]
10.	cdbc(dc)²b ⇒ d(dc)²d²cbdc	[21]
11.	dcb² ⇒ bd	[7]
12.	c³bcdcb ⇒ dc(dcd)³cbc	[29]
13.	c²dcbdb ⇒ (dc)²(d²c)²bd²	[28]
14.	dcd(cb)² ⇒ cbcdcbd	[13]
15.	(dc)²bcdb ⇒ cb(dc)²d²cbd²	[20]
16.	(dc)²bc³b ⇒ cbc(dcd)²cbcd	[18]
17.	(dc)²bc²(dc)²b ⇒ cbc(dcd)²cbc	[17]
18.	d(dc)²d²cbcdb ⇒ cdbcdcd²cbd²	[25]
19.	d²(cd)²(dcb)² ⇒ cdbcdcbd	[23]
20.	d(dc)²d²cbdcdb ⇒ cdb(dc)²d²cbd²	[26]
21.	d(dc)²d²cbdc³b ⇒ cdbc(dcd)²cbcd	[32]
22.	d(dc)²d²cbdc²(dc)²b ⇒ cdbc(dcd)²cbc	[31]
23.	cbcb² ⇒ dcbd²	[12]
24.	cbcdcbdb ⇒ dcd²cbd²	[27]
25.	cdbcb² ⇒ d(dc)²d²c(bd)²	[22]
26.	cdbcdcbdb ⇒ d²(cd)²bd²	[33]
27.	c³bcbdb ⇒ (dc)²(d²c)³bd²	[30]
28.	ab ⇒ d	[3]
29.	ca ⇒ dc²	[8]
30.	da ⇒ adcb	[6]
31.	ba ⇒ dcb	[4]

# ab:abbaaab=ba cd/b/a baaa=c,ab=d morph:4/2,2/4
dccb=cd
dcdcbd=cb
dcdcdcb=c
ddcdcddcbdd=cdb
dcdcddcdcbc=ccdcdcb
cbccb=dcdcbcd
cbcdcdcb=dcdcbc
ccbb=dcdcddcbdd
cdbccb=ddcdcddcbdcd
cdbcdcdcb=ddcdcddcbdc
dcbb=bd
cccbcdcb=dcdcddcddcdcbc
ccdcbdb=dcdcddcddcbdd
dcdcbcb=cbcdcbd
dcdcbcdb=cbdcdcddcbdd
dcdcbcccb=cbcdcddcdcbcd
dcdcbccdcdcb=cbcdcddcdcbc
ddcdcddcbcdb=cdbcdcddcbdd
ddcdcddcbdcb=cdbcdcbd
ddcdcddcbdcdb=cdbdcdcddcbdd
ddcdcddcbdcccb=cdbcdcddcdcbcd
ddcdcddcbdccdcdcb=cdbcdcddcdcbc
cbcbb=dcbdd
cbcdcbdb=dcddcbdd
cdbcbb=ddcdcddcbdbd
cdbcdcbdb=ddcdcdbdd
cccbcbdb=dcdcddcddcddcbdd
ab=d
ca=dcc
da=adcb
ba=dcb

Other submonoids of same group

2 unique, 2 total

Σ	#	Presentation	Properties
11	4285	⟨a, b \| ababaaaab=ba⟩	Can Inf
11	4334	⟨a, b \| abbaabaab=ba⟩	Can Inf