⟨a, b | aaaa=1, abbbba=b⟩

Up:

Monoid enumeration

#17135 ⟨a, b | aaaa=1, abbbab=b⟩

#17139 ⟨a, b | aaaa=1, abbbbb=b⟩

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 b is not injective:

b ⋅ b³d = b and b ⋅ 1 = b, however b³d ≠ 1

Not right cancellative, because right multiplication by d is not injective:

b³d ⋅ d = d and 1 ⋅ d = d, however b³d ≠ 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 = bbba
2. d = accc
Reduction order:
- Left-to-right recursive path with deg(b) = deg(d) = 0, b < d; deg(c) = deg(a) = 1, c < a
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	db ⇒ bd	[20]
2.	b⁴d ⇒ b	[11]
3.	b³d² ⇒ d	[28]
4.	cb³d ⇒ c	[43]
5.	b²c ⇒ dcb⁵	[30]
6.	b²dc ⇒ d²cb⁵	[50]
7.	bd²c ⇒ cbd²	[53]
8.	d³c ⇒ bcb²d⁴	[54]
9.	ab ⇒ dcb	[21]
10.	ad ⇒ dcd	[29]
11.	ba ⇒ bdc	[41]
12.	da ⇒ d²c	[42]
13.	cbc ⇒ b⁴	[7]
14.	bcdc ⇒ c²b⁴	[22]
15.	(dc)² ⇒ bc²b²d	[45]
16.	d²c² ⇒ cdcd³	[55]
17.	ca ⇒ cdc	[46]
18.	ac ⇒ dc²	[25]
19.	c³ ⇒ bcb²d²	[44]
20.	a⁴ ⇒ 1	[1]

# ab:aaaa=1,abbbba=b bd/ca bbba=c,accc=d morph:4/0,4/0 db=bd bbbbd=b bbbdd=d cbbbd=c bbc=dcbbbbb bbdc=ddcbbbbb bddc=cbdd dddc=bcbbdddd ab=dcb ad=dcd ba=bdc da=ddc cbc=bbbb bcdc=ccbbbb dcdc=bccbbd ddcc=cdcddd ca=cdc ac=dcc ccc=bcbbdd aaaa=1

#17137 ⟨a, b | aaaa=1, abbbba=b⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
Prev:	#17135 ⟨a, b \| aaaa=1, abbbab=b⟩
Next:	#17139 ⟨a, b \| aaaa=1, abbbbb=b⟩