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

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

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

9 total

Σ	#	Presentation	Mapping
8	2822	⟨a, b, c \| aaa=b, abc=a⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
8	2835	⟨a, b, c \| aaa=b, bac=a⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
8	2856	⟨a, b, c \| aab=a, aac=b⟩	φ(a) = a, φ(b) = aac, φ(c) = c
8	5166	⟨a, b, c \| aa=b, aabc=a⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	5179	⟨a, b, c \| aa=b, abac=a⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	5218	⟨a, b, c \| aa=b, baac=a⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	5277	⟨a, b, c \| ab=a, aaac=b⟩	φ(a) = a, φ(b) = aaac, φ(c) = c
8	5491	⟨a, b, c \| ab=c, aaac=a⟩	φ(a) = a, φ(b) = c, φ(c) = ac
8	5609	⟨a, b, c \| ab=c, cccc=a⟩	φ(a) = aaaa, φ(b) = c, φ(c) = a