⟨a, b, c | aab=cc, baa=1⟩

Up:

Monoid enumeration

#1776 ⟨a, b, c | aab=cc, acc=1⟩

#1780 ⟨a, b, c | aab=cc, bba=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 b is not injective:

b ⋅ a²b = b and b ⋅ 1 = b, however a²b ≠ 1

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

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

#	Rule	Proof
1.	ba² ⇒ 1	[2]
2.	bada ⇒ abda	[27]
3.	a²bda ⇒ da	[29]
4.	abad ⇒ d	[12]
5.	b²ad ⇒ babdaba	[20]
6.	(ab)²da ⇒ bda	[22]
7.	ca ⇒ bad	[4]
8.	bc ⇒ babdab	[26]
9.	dbda ⇒ a³	[15]
10.	bad² ⇒ abd²	[28]
11.	a²bd² ⇒ d²	[30]
12.	(ab)²d² ⇒ bd²	[24]
13.	cd ⇒ (bad)²	[5]
14.	cbda ⇒ badbabda	[23]
15.	cbad ⇒ a²ba	[7]
16.	c² ⇒ a²b	[1]
17.	dbd² ⇒ a²d	[16]
18.	cbd² ⇒ badbabd²	[25]

# abc:aab=cc,baa=1 ab/dc aca=d morph:3/0 baa=1 bada=abda aabda=da abad=d bbad=babdaba ababda=bda ca=bad bc=babdab dbda=aaa badd=abdd aabdd=dd ababdd=bdd cd=badbad cbda=badbabda cbad=aaba cc=aab dbdd=aad cbdd=badbabdd

#1777 ⟨a, b, c | aab=cc, baa=1⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
Prev:	#1776 ⟨a, b, c \| aab=cc, acc=1⟩
Next:	#1780 ⟨a, b, c \| aab=cc, bba=1⟩