#746 ⟨a, b, c | aa=b, abbc=1⟩

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 ⋅ a⁴ca = a and a ⋅ 1 = a, however a⁴ca ≠ 1
Not right cancellative, because right multiplication by a⁴c is not injective:
- a⁴ca ⋅ a⁴c = a⁴c and 1 ⋅ a⁴c = a⁴c, however a⁴ca ≠ 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.	a⁵c ⇒ 1	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,abbc=1 ac/b - -
aaaaac=1
b=aa

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

16 total

Σ	#	Presentation	Mapping
7	758	⟨a, b, c \| aa=b, babc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
7	761	⟨a, b, c \| aa=b, bbac=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	2314	⟨a, b, c \| aaa=b, aabc=1⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
8	2318	⟨a, b, c \| aaa=b, abac=1⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
8	2330	⟨a, b, c \| aaa=b, baac=1⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
8	2480	⟨a, b, c \| aab=c, aaac=1⟩	φ(a) = a, φ(b) = c, φ(c) = aac
8	4158	⟨a, b, c \| aab=1, aaac=b⟩	φ(a) = a, φ(b) = aaac, φ(c) = c
8	4482	⟨a, b, c \| abc=1, aaaa=b⟩	φ(a) = a, φ(b) = aaaa, φ(c) = c
8	4576	⟨a, b, c \| abc=1, bbbb=a⟩	φ(a) = aaaa, φ(b) = a, φ(c) = c
8	4658	⟨a, b, c \| aa=b, aaabc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	4662	⟨a, b, c \| aa=b, aabac=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	4676	⟨a, b, c \| aa=b, abaac=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	4715	⟨a, b, c \| aa=b, baaac=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	4985	⟨a, b, c \| ab=c, aaaac=1⟩	φ(a) = a, φ(b) = c, φ(c) = ac
8	6783	⟨a, b, c \| ab=1, aaaac=b⟩	φ(a) = a, φ(b) = aaaac, φ(c) = c
8	7115	⟨a, b, c \| ab=1, ccccc=a⟩	φ(a) = aaaaa, φ(b) = c, φ(c) = a