⟨a, b, c | abc=b, caaa=1⟩

Up:

Monoid enumeration

#2788 ⟨a, b, c | abc=b, bbbb=1⟩

#2796 ⟨a, b, c | abc=b, caca=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 c is not injective:

c ⋅ a³c = c and c ⋅ 1 = c, however a³c ≠ 1

Not right cancellative, because right multiplication by a³ is not injective:

a³c ⋅ a³ = a³ and 1 ⋅ a³ = a³, however a³c ≠ 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
3. f = ae
Reduction order:
- Left-to-right recursive path with deg(c) = 0; deg(b) = deg(d) = deg(e) = deg(f) = 1, b < d < e < f; deg(a) = 2
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	cb ⇒ bc³	[23]
2.	dc ⇒ b	[6]
3.	cd ⇒ bc²	[19]
4.	ec ⇒ d	[7]
5.	ce ⇒ bc	[18]
6.	fc ⇒ e	[8]
7.	cf ⇒ b	[9]
8.	db ⇒ bf	[15]
9.	eb ⇒ df	[16]
10.	fb ⇒ ef	[17]
11.	ab ⇒ d	[3]
12.	ad ⇒ e	[4]
13.	ae ⇒ f	[5]
14.	caf ⇒ d	[10]
15.	baf ⇒ d²	[20]
16.	daf ⇒ ed	[21]
17.	eaf ⇒ fd	[22]
18.	faf ⇒ afd	[24]
19.	ca²f ⇒ e	[11]
20.	ba²f ⇒ de	[25]
21.	da²f ⇒ e²	[26]
22.	ea²f ⇒ fe	[27]
23.	fa²f ⇒ afe	[29]
24.	ca³ ⇒ 1	[2]
25.	ba³ ⇒ d	[12]
26.	da³ ⇒ e	[13]
27.	ea³ ⇒ f	[14]
28.	fa³ ⇒ af	[28]

# abc:abc=b,caaa=1 c/bdef/a ab=d,ad=e,ae=f morph:2/1,2/2,2/1 cb=bccc dc=b cd=bcc ec=d ce=bc fc=e cf=b db=bf eb=df fb=ef ab=d ad=e ae=f caf=d baf=dd daf=ed eaf=fd faf=afd caaf=e baaf=de daaf=ee eaaf=fe faaf=afe caaa=1 baaa=d daaa=e eaaa=f faaa=af

#2794 ⟨a, b, c | abc=b, caaa=1⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
Prev:	#2788 ⟨a, b, c \| abc=b, bbbb=1⟩
Next:	#2796 ⟨a, b, c \| abc=b, caca=1⟩