#94 ⟨a, b, c | aa=b, bbc=1⟩

Properties

Sum of relation sides is 6
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ a³ca = a and a ⋅ 1 = a, however a³ca ≠ 1
Not right cancellative, because right multiplication by a³c is not injective:
- a³ca ⋅ a³c = a³c and 1 ⋅ a³c = a³c, however a³ca ≠ 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(a) = deg(c) = 0, a < c; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a⁴c ⇒ 1	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,bbc=1 ac/b - -
aaaac=1
b=aa

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

13 total

Σ	#	Presentation	Mapping
7	270	⟨a, b, c \| aaa=b, abc=1⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
7	274	⟨a, b, c \| aaa=b, bac=1⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
7	319	⟨a, b, c \| aab=c, aac=1⟩	φ(a) = a, φ(b) = c, φ(c) = aac
7	741	⟨a, b, c \| aa=b, aabc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
7	745	⟨a, b, c \| aa=b, abac=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
7	757	⟨a, b, c \| aa=b, baac=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
7	842	⟨a, b, c \| ab=c, aaac=1⟩	φ(a) = a, φ(b) = c, φ(c) = ac
7	1278	⟨a, b, c \| ab=1, aaac=b⟩	φ(a) = a, φ(b) = aaac, φ(c) = c
7	1383	⟨a, b, c \| ab=1, cccc=a⟩	φ(a) = aaaa, φ(b) = c, φ(c) = a
8	1682	⟨a, b, c \| aab=bb, bbc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	4657	⟨a, b, c \| aa=b, aaaac=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	4980	⟨a, b, c \| ab=a, cccca=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4984	⟨a, b, c \| ab=c, aaaab=1⟩	φ(a) = a, φ(b) = c, φ(c) = ac