⟨a, b, c | ab=1, bbbca=c⟩

Up:

Monoid enumeration

#7022 ⟨a, b, c | ab=1, bbbbc=c⟩

#7028 ⟨a, b, c | ab=1, bbbcc=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 = ca
2. e = bbc
3. f = bbe
Reduction order:
- Right-to-left recursive path with deg(b) = 0; deg(f) = deg(e) = deg(c) = deg(d) = 1, f < e < c < d; deg(a) = 2
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	fb ⇒ b³f	[16]
2.	b²e ⇒ f	[5]
3.	eb ⇒ bf	[15]
4.	b²c ⇒ e	[4]
5.	cb ⇒ be	[13]
6.	b³d ⇒ c	[11]
7.	db ⇒ c	[7]
8.	fe ⇒ bdf	[20]
9.	be² ⇒ df	[17]
10.	fc ⇒ bde	[22]
11.	bec ⇒ de	[18]
12.	b²fd ⇒ dc	[19]
13.	e³ ⇒ b²d²f	[27]
14.	ce² ⇒ d²f	[21]
15.	e²c ⇒ b²d²e	[28]
16.	cec ⇒ d²e	[23]
17.	f²d ⇒ bd²c	[29]
18.	befd ⇒ d²c	[26]
19.	e²fd ⇒ b²d³c	[31]
20.	cefd ⇒ d³c	[30]
21.	fa ⇒ bc	[24]
22.	ea ⇒ b²d	[10]
23.	ca ⇒ d	[3]
24.	ab ⇒ 1	[1]
25.	af ⇒ be	[8]
26.	ae ⇒ bc	[9]
27.	ac ⇒ b²d	[12]
28.	ad ⇒ b²da	[25]

# abc:ab=1,bbbca=c reversed:b/fecd/a ca=d,bbc=e,bbe=f morph:2/1,3/2,3/2 fb=bbbf bbe=f eb=bf bbc=e cb=be bbbd=c db=c fe=bdf bee=df fc=bde bec=de bbfd=dc eee=bbddf cee=ddf eec=bbdde cec=dde ffd=bddc befd=ddc eefd=bbdddc cefd=dddc fa=bc ea=bbd ca=d ab=1 af=be ae=bc ac=bbd ad=bbda

#7025 ⟨a, b, c | ab=1, bbbca=c⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
Prev:	#7022 ⟨a, b, c \| ab=1, bbbbc=c⟩
Next:	#7028 ⟨a, b, c \| ab=1, bbbcc=c⟩