⟨a, b | aaaa=bab, bbbb=1⟩

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Monoid enumeration

#10651 ⟨a, b | aaaa=abb, bbbb=1⟩

#10664 ⟨a, b | aaaa=bbb, abab=1⟩

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 d is not injective:

d ⋅ da³ = d and d ⋅ 1 = d, however da³ ≠ 1

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

da³ ⋅ a = a and 1 ⋅ a = a, however da³ ≠ 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 = bbb
2. d = caaacaaacaaac
Reduction order:
- Left-to-right recursive path with deg(a) = 0; deg(d) = 1; deg(b) = deg(c) = 2, b < c
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	da⁴ ⇒ a	[27]
2.	ad ⇒ da	[29]
3.	d²a³ ⇒ d	[39]
4.	abda³ ⇒ ab	[18]
5.	a⁵b ⇒ ba⁵	[8]
6.	db ⇒ a²bd²a	[37]
7.	dab ⇒ a³bd²a	[41]
8.	da²b ⇒ a⁴bd²a	[44]
9.	da³b ⇒ bda³	[36]
10.	ca ⇒ abd	[17]
11.	cd ⇒ abd³a²	[42]
12.	ac ⇒ a²bd²a²	[38]
13.	dc ⇒ a³bd³	[43]
14.	bab ⇒ a⁴	[1]
15.	ab²da³ ⇒ ab²	[19]
16.	aba²b ⇒ b²a⁹	[13]
17.	a⁴b² ⇒ ba²bd²a²	[45]
18.	bc ⇒ 1	[6]
19.	cb ⇒ 1	[5]
20.	c² ⇒ b²	[7]
21.	b³ ⇒ c	[3]

# ab:aaaa=bab,bbbb=1 a/d/bc bbb=c,caaacaaacaaac=d morph:3/0,13/0 daaaa=a ad=da ddaaa=d abdaaa=ab aaaaab=baaaaa db=aabdda dab=aaabdda daab=aaaabdda daaab=bdaaa ca=abd cd=abdddaa ac=aabddaa dc=aaabddd bab=aaaa abbdaaa=abb abaab=bbaaaaaaaaa aaaabb=baabddaa bc=1 cb=1 cc=bb bbb=c

#10660 ⟨a, b | aaaa=bab, bbbb=1⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
Prev:	#10651 ⟨a, b \| aaaa=abb, bbbb=1⟩
Next:	#10664 ⟨a, b \| aaaa=bbb, abab=1⟩