#5819 ⟨a, b, c | ab=a, aca=bc⟩

Contents

Properties
Rewriting system

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 ⋅ b = a and a ⋅ 1 = a, however b ≠ 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 = acc
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.	ab ⇒ a	[1]
2.	db ⇒ d	[7]
3.	ad ⇒ da	[13]
4.	acb ⇒ ac	[5]
5.	bc ⇒ aca	[2]
6.	a²ca ⇒ ac	[4]
7.	dcb ⇒ dc	[9]
8.	daca ⇒ dc	[8]
9.	acd ⇒ dca	[14]
10.	ac² ⇒ d	[3]
11.	a(ca)² ⇒ d	[6]
12.	dcd ⇒ d²ca²	[17]
13.	dc²a ⇒ d²	[15]
14.	dc²b ⇒ dc²	[12]
15.	d(ca)² ⇒ dc²	[11]
16.	dc²d ⇒ d²c²	[18]
17.	dc³ ⇒ d²ca	[16]

# abc:ab=a,aca=bc ab/dc acc=d morph:3/0
ab=a
db=d
ad=da
acb=ac
bc=aca
aaca=ac
dcb=dc
daca=dc
acd=dca
acc=d
acaca=d
dcd=ddcaa
dcca=dd
dccb=dcc
dcaca=dcc
dccd=ddcc
dccc=ddca