⟨a, b, c | aab=1, baca=c⟩

Up:

Monoid enumeration

#4229 ⟨a, b, c | aab=1, babc=c⟩

#4235 ⟨a, b, c | aab=1, bacb=c⟩

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 e is not injective:

e ⋅ ae = e and e ⋅ 1 = e, however ae ≠ 1

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

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

#	Rule	Proof
1.	ce ⇒ ec	[16]
2.	de ⇒ c	[14]
3.	ed ⇒ c	[5]
4.	cd ⇒ dc	[17]
5.	eae ⇒ e	[10]
6.	eac ⇒ c	[12]
7.	ca ⇒ d	[3]
8.	cb ⇒ e²c	[19]
9.	dae ⇒ d	[9]
10.	dac ⇒ d²	[15]
11.	db ⇒ ec	[18]
12.	ead ⇒ d	[13]
13.	a²e ⇒ a	[6]
14.	a²c ⇒ ad	[11]
15.	eab ⇒ b	[8]
16.	ba ⇒ e	[4]
17.	dad ⇒ d²a	[20]
18.	dab ⇒ c	[7]
19.	a²d ⇒ ada	[21]
20.	a²b ⇒ 1	[1]

# abc:aab=1,baca=c ec/dab ca=d,ba=e morph:2/0,2/0 ce=ec de=c ed=c cd=dc eae=e eac=c ca=d cb=eec dae=d dac=dd db=ec ead=d aae=a aac=ad eab=b ba=e dad=dda dab=c aad=ada aab=1

#4232 ⟨a, b, c | aab=1, baca=c⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
Prev:	#4229 ⟨a, b, c \| aab=1, babc=c⟩
Next:	#4235 ⟨a, b, c \| aab=1, bacb=c⟩