⟨a, b, c | ab=1, bbcca=c⟩

Up:

Monoid enumeration

#7036 ⟨a, b, c | ab=1, bbcbc=c⟩

#7042 ⟨a, b, c | ab=1, bbccc=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 a is not injective:

a ⋅ ba = a and a ⋅ 1 = a, however ba ≠ 1

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

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

#	Rule	Proof
1.	eb ⇒ bde	[22]
2.	db² ⇒ be	[17]
3.	ab ⇒ 1	[1]
4.	dadb ⇒ ae	[24]
5.	ea ⇒ adb	[23]
6.	edad ⇒ ad²b	[21]
7.	dba ⇒ d	[20]
8.	bad ⇒ d	[14]
9.	bdad ⇒ adb	[11]
10.	a²d ⇒ (da)²	[16]
11.	bae ⇒ e	[25]
12.	eda²e ⇒ ad³b	[29]
13.	ba²e ⇒ ae	[27]
14.	bda²e ⇒ ad²b	[26]
15.	a³e ⇒ (dad)²	[28]
16.	c ⇒ da	[18]

# abc:ab=1,bbcca=c reversed:d/eba/c cb=d,adbb=e morph:2/0,4/1 eb=bde dbb=be ab=1 dadb=ae ea=adb edad=addb dba=d bad=d bdad=adb aad=dada bae=e edaae=adddb baae=ae bdaae=addb aaae=daddad c=da

#7039 ⟨a, b, c | ab=1, bbcca=c⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
Prev:	#7036 ⟨a, b, c \| ab=1, bbcbc=c⟩
Next:	#7042 ⟨a, b, c \| ab=1, bbccc=c⟩