⟨a, b, c | aba=bb, acb=1⟩

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Monoid enumeration

#1850 ⟨a, b, c | aba=bb, aca=1⟩

#1852 ⟨a, b, c | aba=bb, acc=1⟩

Properties

Sum of relation sides is 8

Infinite non-cancellative non-commutative monoid

Element a has infinite order

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

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

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

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

#	Rule	Proof
1.	b³d ⇒ db³	[11]
2.	ab ⇒ d	[3]
3.	b³a ⇒ db²	[10]
4.	cb³ ⇒ ba	[15]
5.	b²cb ⇒ d	[5]
6.	d² ⇒ b³	[6]
7.	da ⇒ b²	[4]
8.	ad ⇒ dbcb	[8]
9.	acb ⇒ 1	[2]
10.	cdb² ⇒ ba²	[17]
11.	cbd ⇒ b	[14]
12.	cb²d ⇒ bdcb	[16]
13.	b²cd ⇒ dbcb	[9]
14.	cba ⇒ 1	[13]
15.	acd ⇒ bcb	[7]
16.	cdbd ⇒ bdb(cb)²	[18]

# abc:aba=bb,acb=1 b/dac ab=d morph:2/0 bbbd=dbbb ab=d bbba=dbb cbbb=ba bbcb=d dd=bbb da=bb ad=dbcb acb=1 cdbb=baa cbd=b cbbd=bdcb bbcd=dbcb cba=1 acd=bcb cdbd=bdbcbcb

#1851 ⟨a, b, c | aba=bb, acb=1⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
Prev:	#1850 ⟨a, b, c \| aba=bb, aca=1⟩
Next:	#1852 ⟨a, b, c \| aba=bb, acc=1⟩