#914 ⟨a, b, c | aa=b, acb=a⟩

Properties

Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ ca² = a and a ⋅ 1 = a, however ca² ≠ 1
Not right cancellative, because right multiplication by a is not injective:
- aca ⋅ a = a and 1 ⋅ a = a, however aca ≠ 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.	aca² ⇒ a	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,acb=a ac/b - -
acaa=a
b=aa

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

5 total

Σ	#	Presentation	Mapping
7	982	⟨a, b, c \| ab=a, caa=b⟩	φ(a) = a, φ(b) = caa, φ(c) = c
8	5264	⟨a, b, c \| aa=b, cacc=c⟩	φ(a) = c, φ(b) = cc, φ(c) = a
8	5408	⟨a, b, c \| ab=a, caab=b⟩	φ(a) = a, φ(b) = caa, φ(c) = c
8	5414	⟨a, b, c \| ab=a, caba=b⟩	φ(a) = a, φ(b) = caa, φ(c) = c
8	5511	⟨a, b, c \| ab=c, abaa=a⟩	φ(a) = a, φ(b) = c, φ(c) = ac

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

7 total

Σ	#	Presentation	Mapping
7	954	⟨a, b, c \| ab=a, aca=b⟩	φ(a) = a, φ(b) = aca, φ(c) = c
7	1023	⟨a, b, c \| ab=c, aca=a⟩	φ(a) = a, φ(b) = c, φ(c) = ac
7	1038	⟨a, b, c \| ab=c, bbc=b⟩	φ(a) = c, φ(b) = a, φ(c) = ca
8	5169	⟨a, b, c \| aa=b, aaca=a⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	5304	⟨a, b, c \| ab=a, abca=b⟩	φ(a) = a, φ(b) = aca, φ(c) = c
8	5316	⟨a, b, c \| ab=a, acab=b⟩	φ(a) = a, φ(b) = aca, φ(c) = c
8	5494	⟨a, b, c \| ab=c, aaba=a⟩	φ(a) = a, φ(b) = c, φ(c) = ac