#143 ⟨a, b, c | aa=b, bc=a⟩

Properties

Sum of relation sides is 6
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ ac = a and a ⋅ 1 = a, however ac ≠ 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.	a²c ⇒ a	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,bc=a ac/b - -
aac=a
b=aa

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

14 total

Σ	#	Presentation	Mapping
6	148	⟨a, b, c \| ab=a, ac=b⟩	φ(a) = a, φ(b) = ac, φ(c) = c
7	903	⟨a, b, c \| aa=b, aac=a⟩	φ(a) = a, φ(b) = aa, φ(c) = c
7	951	⟨a, b, c \| ab=a, abc=b⟩	φ(a) = a, φ(b) = ac, φ(c) = c
7	1011	⟨a, b, c \| ab=c, aab=a⟩	φ(a) = a, φ(b) = c, φ(c) = ac
7	1020	⟨a, b, c \| ab=c, abc=a⟩	φ(a) = aa, φ(b) = c, φ(c) = a
8	2817	⟨a, b, c \| aaa=b, aac=a⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
8	2858	⟨a, b, c \| aab=a, aba=c⟩	φ(a) = a, φ(b) = c, φ(c) = aca
8	2859	⟨a, b, c \| aab=a, abb=c⟩	φ(a) = a, φ(b) = c, φ(c) = acc
8	2861	⟨a, b, c \| aab=a, abc=b⟩	φ(a) = a, φ(b) = acc, φ(c) = c
8	2872	⟨a, b, c \| aab=a, baa=c⟩	φ(a) = a, φ(b) = c, φ(c) = caa
8	2873	⟨a, b, c \| aab=a, bab=c⟩	φ(a) = a, φ(b) = c, φ(c) = cac
8	2877	⟨a, b, c \| aab=a, bba=c⟩	φ(a) = a, φ(b) = c, φ(c) = cca
8	5301	⟨a, b, c \| ab=a, abbc=b⟩	φ(a) = a, φ(b) = ac, φ(c) = c
8	5514	⟨a, b, c \| ab=c, abab=a⟩	φ(a) = aa, φ(b) = c, φ(c) = a