#2908 ⟨a, b, c | aab=a, cca=b⟩

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 ⋅ ac²a = a and a ⋅ 1 = a, however ac²a ≠ 1
Not right cancellative, because right multiplication by a is not injective:
- a²c² ⋅ a = a and 1 ⋅ a = a, however a²c² ≠ 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.	a²c²a ⇒ a	[3]
2.	b ⇒ c²a	[2]

# abc:aab=a,cca=b ac/b - -
aacca=a
b=cca

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

4 total

Σ	#	Presentation	Mapping
8	2999	⟨a, b, c \| aab=c, cba=a⟩	φ(a) = a, φ(b) = c, φ(c) = aac
8	5331	⟨a, b, c \| ab=a, acca=b⟩	φ(a) = a, φ(b) = acca, φ(c) = c
8	5540	⟨a, b, c \| ab=c, acba=a⟩	φ(a) = a, φ(b) = c, φ(c) = ac
8	5574	⟨a, b, c \| ab=c, bbac=b⟩	φ(a) = c, φ(b) = a, φ(c) = ca

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

4 total

Σ	#	Presentation	Mapping
8	2989	⟨a, b, c \| aab=c, bcb=b⟩	φ(a) = c, φ(b) = a, φ(c) = cca
8	5211	⟨a, b, c \| aa=b, accb=a⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	5270	⟨a, b, c \| aa=b, cbcc=c⟩	φ(a) = c, φ(b) = cc, φ(c) = a
8	5459	⟨a, b, c \| ab=a, ccaa=b⟩	φ(a) = a, φ(b) = ccaa, φ(c) = c