⟨a, b | aaa=1, abbbbba=b⟩

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Monoid enumeration

#21741 ⟨a, b | aaa=1, abbabbb=b⟩

#21751 ⟨a, b | aaa=1, abbbbbb=b⟩

Properties

Sum of relation sides is 11

Infinite non-cancellative non-commutative monoid

Element bab has infinite order

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

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

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

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

#	Rule	Proof
1.	bc ⇒ cb	[6]
2.	b³ ⇒ c	[3]
3.	dc² ⇒ b²	[24]
4.	dcb² ⇒ b	[50]
5.	cd ⇒ dc	[28]
6.	bd ⇒ db	[55]
7.	d²cb ⇒ d	[63]
8.	cadcb ⇒ ca	[45]
9.	badcb ⇒ ba	[51]
10.	c²a ⇒ ac²	[9]
11.	c²ba ⇒ bac²	[10]
12.	c²b²a ⇒ b²ac²	[15]
13.	da ⇒ b²ad²b²	[62]
14.	dca ⇒ cb²ad²b²	[59]
15.	dba ⇒ cad²b²	[52]
16.	dcba ⇒ adcb	[37]
17.	db²a ⇒ cbad²b²	[53]
18.	a²c ⇒ cb²ab²	[12]
19.	a²b ⇒ cb²a	[7]
20.	a²d ⇒ cb²ad²c	[61]
21.	acba ⇒ bad²c	[60]
22.	acb²a ⇒ b	[5]
23.	ca² ⇒ b²acb²	[13]
24.	ba² ⇒ acb²	[8]
25.	(ba)² ⇒ acac³b²	[65]
26.	b²aca ⇒ cabad³b	[69]
27.	a³ ⇒ 1	[1]
28.	a(ca)² ⇒ bab²ad⁵c	[73]

# ab:aaa=1,abbbbba=b cb/d/a bbb=c,acbacba=d morph:3/0,7/0 bc=cb bbb=c dcc=bb dcbb=b cd=dc bd=db ddcb=d cadcb=ca badcb=ba cca=acc ccba=bacc ccbba=bbacc da=bbaddbb dca=cbbaddbb dba=caddbb dcba=adcb dbba=cbaddbb aac=cbbabb aab=cbba aad=cbbaddc acba=baddc acbba=b caa=bbacbb baa=acbb baba=acacccbb bbaca=cabadddb aaa=1 acaca=babbadddddc

#21749 ⟨a, b | aaa=1, abbbbba=b⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
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