#1077 ⟨a, b, c | aa=1, abbbc=1⟩

Properties

Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element b has infinite order
Not left cancellative, because left multiplication by b is not injective:
- b ⋅ (b²cb)² = b and b ⋅ 1 = b, however (b²cb)² ≠ 1
Not right cancellative, because right multiplication by b²cb³c is not injective:
- (b²cb)² ⋅ b²cb³c = b²cb³c and 1 ⋅ b²cb³c = b²cb³c, however (b²cb)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.	(b³c)² ⇒ 1	[4]
2.	a ⇒ b³c	[3]

# abc:aa=1,abbbc=1 bc/a - -
bbbcbbbc=1
a=bbbc

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

3 total

Σ	#	Presentation	Mapping
7	1262	⟨a, b, c \| aa=1, bbbc=a⟩	φ(a) = bbbc, φ(b) = b, φ(c) = c
8	2339	⟨a, b, c \| aaa=b, bcbc=1⟩	φ(a) = b, φ(b) = bbb, φ(c) = c
8	2498	⟨a, b, c \| aab=c, acac=1⟩	φ(a) = b, φ(b) = c, φ(c) = bbc