⟨a, b | aaa=1, abbabba=b⟩

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Monoid enumeration

#21731 ⟨a, b | aaa=1, ababbbb=b⟩

#21741 ⟨a, b | aaa=1, abbabbb=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 ⋅ b²d = b and b ⋅ 1 = b, however b²d ≠ 1

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

b²d ⋅ d = d and 1 ⋅ d = d, however b²d ≠ 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 = bbaba
2. d = abacc
Reduction order:
- Left-to-right recursive path with deg(b) = deg(d) = 0, b < d; deg(a) = deg(c) = 1, a < c
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	db ⇒ bd	[47]
2.	b³d ⇒ b	[31]
3.	b²d² ⇒ d	[55]
4.	bab²d ⇒ ba	[50]
5.	da ⇒ babd²	[57]
6.	b³a ⇒ ab³	[13]
7.	cb²d ⇒ c	[43]
8.	dc ⇒ bcbd²	[36]
9.	b³c ⇒ cb³	[18]
10.	aba ⇒ bcbd²	[53]
11.	ba² ⇒ (ab²)²	[12]
12.	(b²a)² ⇒ a²b	[11]
13.	ca ⇒ bacb⁵	[22]
14.	cba ⇒ b²ab²cbd²	[51]
15.	cb²a ⇒ abcbd³	[59]
16.	(bc)² ⇒ c²b¹⁴	[38]
17.	b²c² ⇒ (cb)²d⁶	[64]
18.	a³ ⇒ 1	[1]
19.	a²b²a ⇒ b²ac	[20]
20.	a²bc ⇒ bab³	[17]
21.	ab²ac ⇒ b²a	[14]
22.	ac² ⇒ (bab)²d²	[63]
23.	c³ ⇒ b²d	[6]

# ab:aaa=1,abbabba=b bd/ac bbaba=c,abacc=d morph:5/0,5/0 db=bd bbbd=b bbdd=d babbd=ba da=babdd bbba=abbb cbbd=c dc=bcbdd bbbc=cbbb aba=bcbdd baa=abbabb bbabba=aab ca=bacbbbbb cba=bbabbcbdd cbba=abcbddd bcbc=ccbbbbbbbbbbbbbb bbcc=cbcbdddddd aaa=1 aabba=bbac aabc=babbb abbac=bba acc=babbabdd ccc=bbd

#21739 ⟨a, b | aaa=1, abbabba=b⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
Prev:	#21731 ⟨a, b \| aaa=1, ababbbb=b⟩
Next:	#21741 ⟨a, b \| aaa=1, abbabbb=b⟩