#908 ⟨a, b, c | aa=b, abc=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 ⋅ a²c = a and a ⋅ 1 = a, however a²c ≠ 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.	a³c ⇒ a	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,abc=a ac/b - -
aaac=a
b=aa

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

10 total

Σ	#	Presentation	Mapping
7	921	⟨a, b, c \| aa=b, bac=a⟩	φ(a) = a, φ(b) = aa, φ(c) = c
7	946	⟨a, b, c \| ab=a, aac=b⟩	φ(a) = a, φ(b) = aac, φ(c) = c
7	1014	⟨a, b, c \| ab=c, aac=a⟩	φ(a) = a, φ(b) = c, φ(c) = ac
7	1048	⟨a, b, c \| ab=c, ccc=a⟩	φ(a) = aaa, φ(b) = c, φ(c) = a
8	5161	⟨a, b, c \| aa=b, aaac=a⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	5282	⟨a, b, c \| ab=a, aabc=b⟩	φ(a) = a, φ(b) = aac, φ(c) = c
8	5296	⟨a, b, c \| ab=a, abac=b⟩	φ(a) = a, φ(b) = aac, φ(c) = c
8	5488	⟨a, b, c \| ab=c, aaab=a⟩	φ(a) = a, φ(b) = c, φ(c) = ac
8	5531	⟨a, b, c \| ab=c, abcc=a⟩	φ(a) = aaa, φ(b) = c, φ(c) = a
8	5599	⟨a, b, c \| ab=c, cabc=a⟩	φ(a) = aaa, φ(b) = c, φ(c) = a