⟨a, b, c | abc=b, caa=1⟩

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Monoid enumeration

#413 ⟨a, b, c | abc=b, bbc=1⟩

#417 ⟨a, b, c | abc=b, cba=1⟩

Properties

Sum of relation sides is 7

Infinite non-cancellative non-commutative monoid

Element a has infinite order

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

c ⋅ a²c = c and c ⋅ 1 = c, however a²c ≠ 1

Not right cancellative, because right multiplication by a² is not injective:

a²c ⋅ a² = a² and 1 ⋅ a² = a², however a²c ≠ 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
2. e = ad
Reduction order:
- Left-to-right recursive path with deg(c) = 0; deg(b) = deg(d) = deg(a) = deg(e) = 1, b < d < a < e
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	cb ⇒ bc²	[14]
2.	dc ⇒ b	[5]
3.	cd ⇒ bc	[13]
4.	ec ⇒ d	[6]
5.	ce ⇒ b	[7]
6.	db ⇒ be	[11]
7.	ab ⇒ d	[3]
8.	ad ⇒ e	[4]
9.	ca² ⇒ 1	[2]
10.	cae ⇒ d	[8]
11.	eb ⇒ de	[12]
12.	ba² ⇒ d	[9]
13.	bae ⇒ d²	[15]
14.	da² ⇒ e	[10]
15.	dae ⇒ ed	[16]
16.	ea² ⇒ ae	[17]
17.	eae ⇒ aed	[18]

# abc:abc=b,caa=1 c/bdae ab=d,ad=e morph:2/0,2/0 cb=bcc dc=b cd=bc ec=d ce=b db=be ab=d ad=e caa=1 cae=d eb=de baa=d bae=dd daa=e dae=ed eaa=ae eae=aed

#416 ⟨a, b, c | abc=b, caa=1⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
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