⟨a, b | aaa=1, abbbba=bb⟩

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Monoid enumeration

#22284 ⟨a, b | aaa=1, abbbab=bb⟩

#22290 ⟨a, b | aaa=1, abbbbb=ba⟩

Properties

Sum of relation sides is 11

Infinite non-cancellative non-commutative monoid

Element a²b has infinite order

Not left cancellative, because left multiplication by c is not injective:

c ⋅ c³ = c and c ⋅ 1 = c, however c³ ≠ 1

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

c³ ⋅ c = c and 1 ⋅ c = c, however c³ ≠ 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 = abb
Reduction order:
- Left-to-right recursive path with deg(c) = 0; deg(a) = 1; deg(b) = 2
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	c⁴ ⇒ c	[20]
2.	cac³ ⇒ ca	[18]
3.	c³a ⇒ ac³	[31]
4.	ac²a ⇒ c²ac²	[28]
5.	ca²c³ ⇒ ca²	[16]
6.	(ca)² ⇒ (ac)²	[17]
7.	a³ ⇒ 1	[1]
8.	a²ca ⇒ c²a²c	[14]
9.	aca² ⇒ ca²c²	[13]
10.	cbc³ ⇒ cb	[21]
11.	cbac³ ⇒ cba	[19]
12.	cba²c³ ⇒ cba²	[24]
13.	c³b ⇒ bc³	[42]
14.	abc ⇒ cbc²ac	[30]
15.	abac ⇒ cbc²a²c	[22]
16.	aba²c ⇒ cb	[6]
17.	ac²b ⇒ c²bcac²	[43]
18.	cab ⇒ c²bc²a	[41]
19.	cacb ⇒ acbca²c²	[44]
20.	a²cb ⇒ ba²c	[7]
21.	ca²b ⇒ acba²c	[35]
22.	b² ⇒ a²c	[5]
23.	cbab ⇒ (cb)²c²a	[36]
24.	cba²b ⇒ cbacbc²a	[45]

# ab:aaa=1,abbbba=bb c/a/b abb=c morph:3/0 cccc=c caccc=ca ccca=accc acca=ccacc caaccc=caa caca=acac aaa=1 aaca=ccaac acaa=caacc cbccc=cb cbaccc=cba cbaaccc=cbaa cccb=bccc abc=cbccac abac=cbccaac abaac=cb accb=ccbcacc cab=ccbcca cacb=acbcaacc aacb=baac caab=acbaac bb=aac cbab=cbcbcca cbaab=cbacbcca

#22287 ⟨a, b | aaa=1, abbbba=bb⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
Prev:	#22284 ⟨a, b \| aaa=1, abbbab=bb⟩
Next:	#22290 ⟨a, b \| aaa=1, abbbbb=ba⟩