#166 ⟨a, b, c | aa=1, abcc=1⟩

Properties

Sum of relation sides is 6
Infinite non-cancellative non-commutative monoid
Element b has infinite order
Not left cancellative, because left multiplication by b is not injective:
- b ⋅ (c²b)² = b and b ⋅ 1 = b, however (c²b)² ≠ 1
Not right cancellative, because right multiplication by c²bc² is not injective:
- (c²b)² ⋅ c²bc² = c²bc² and 1 ⋅ c²bc² = c²bc², however (c²b)² ≠ 1

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

Reduction order:
- Left-to-right recursive path with deg(b) = deg(c) = 0, b < c; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	(bc²)² ⇒ 1	[4]
2.	a ⇒ bc²	[3]

# abc:aa=1,abcc=1 bc/a - -
bccbcc=1
a=bcc

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

15 total

Σ	#	Presentation	Mapping
8	2303	⟨a, b, c \| aaa=a, abcc=1⟩	φ(a) = bcc, φ(b) = b, φ(c) = c
8	2422	⟨a, b, c \| aab=b, abcc=1⟩	φ(a) = bcc, φ(b) = b, φ(c) = c
8	2448	⟨a, b, c \| aab=b, bcca=1⟩	φ(a) = bcc, φ(b) = b, φ(c) = c
8	2523	⟨a, b, c \| aab=c, bcaa=1⟩	φ(a) = c, φ(b) = b, φ(c) = ccb
8	4561	⟨a, b, c \| abc=1, babb=c⟩	φ(a) = b, φ(b) = c, φ(c) = cbcc
8	5022	⟨a, b, c \| ab=c, abbcb=1⟩	φ(a) = b, φ(b) = c, φ(c) = bc
8	5063	⟨a, b, c \| ab=c, bacaa=1⟩	φ(a) = c, φ(b) = b, φ(c) = cb
8	6118	⟨a, b, c \| aa=1, aaabcc=1⟩	φ(a) = bcc, φ(b) = b, φ(c) = c
8	6135	⟨a, b, c \| aa=1, aabcca=1⟩	φ(a) = bcc, φ(b) = b, φ(c) = c
8	6142	⟨a, b, c \| aa=1, abaacc=1⟩	φ(a) = bcc, φ(b) = b, φ(c) = c
8	6174	⟨a, b, c \| aa=1, abcaac=1⟩	φ(a) = bcc, φ(b) = b, φ(c) = c
8	6637	⟨a, b, c \| aa=1, aabcc=a⟩	φ(a) = bcc, φ(b) = b, φ(c) = c
8	6706	⟨a, b, c \| aa=1, baacc=a⟩	φ(a) = bcc, φ(b) = b, φ(c) = c
8	7113	⟨a, b, c \| ab=1, ccacc=b⟩	φ(a) = b, φ(b) = ccbcc, φ(c) = c
8	7188	⟨a, b, c \| aa=1, abcc=aa⟩	φ(a) = bcc, φ(b) = b, φ(c) = c