#134 ⟨a, b, c | aa=a, ab=c⟩

Properties

Sum of relation sides is 6
Isomorphic to ℕ_{(2 = 1)} ∗ ℕ
Infinite non-cancellative non-commutative monoid
Element ab has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ a = a and a ⋅ 1 = a, however a ≠ 1
Not right cancellative, because right multiplication by a is not injective:
- a ⋅ a = a and 1 ⋅ a = a, however a ≠ 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(b) = 0, a < b; deg(c) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a² ⇒ a	[1]
2.	c ⇒ ab	[2]

# abc:aa=a,ab=c ab/c - -
aa=a
c=ab

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

15 total

Σ	#	Presentation	Mapping
6	135	⟨a, b, c \| aa=a, bb=c⟩	φ(a) = a, φ(b) = b, φ(c) = bb
7	885	⟨a, b, c \| aa=a, aab=c⟩	φ(a) = a, φ(b) = b, φ(c) = ab
7	886	⟨a, b, c \| aa=a, aba=c⟩	φ(a) = a, φ(b) = b, φ(c) = aba
7	887	⟨a, b, c \| aa=a, abb=c⟩	φ(a) = a, φ(b) = b, φ(c) = abb
7	891	⟨a, b, c \| aa=a, bab=c⟩	φ(a) = a, φ(b) = b, φ(c) = bab
7	894	⟨a, b, c \| aa=a, bbb=c⟩	φ(a) = a, φ(b) = b, φ(c) = bbb
8	5109	⟨a, b, c \| aa=a, aaab=c⟩	φ(a) = a, φ(b) = b, φ(c) = ab
8	5110	⟨a, b, c \| aa=a, aaba=c⟩	φ(a) = a, φ(b) = b, φ(c) = aba
8	5111	⟨a, b, c \| aa=a, aabb=c⟩	φ(a) = a, φ(b) = b, φ(c) = abb
8	5115	⟨a, b, c \| aa=a, abab=c⟩	φ(a) = a, φ(b) = b, φ(c) = abab
8	5119	⟨a, b, c \| aa=a, abba=c⟩	φ(a) = a, φ(b) = b, φ(c) = abba
8	5120	⟨a, b, c \| aa=a, abbb=c⟩	φ(a) = a, φ(b) = b, φ(c) = abbb
8	5132	⟨a, b, c \| aa=a, baab=c⟩	φ(a) = a, φ(b) = b, φ(c) = bab
8	5135	⟨a, b, c \| aa=a, babb=c⟩	φ(a) = a, φ(b) = b, φ(c) = babb
8	5145	⟨a, b, c \| aa=a, bbbb=c⟩	φ(a) = a, φ(b) = b, φ(c) = bbbb