#774 ⟨a, b, c | aa=b, cccc=1⟩

Properties

Sum of relation sides is 7
Isomorphic to ℤ₄ ∗ ℕ
Infinite cancellative non-commutative monoid
Element a has infinite order
Submonoid of enveloping group: ⟨a, b | aaaa⟩
- Embedding: φ(a) = b, φ(b) = bb, φ(c) = a

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.	c⁴ ⇒ 1	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,cccc=1 ac/b - -
cccc=1
b=aa

2 unique, 13 total

Σ	#	Presentation	Properties	Description	φ
7	679	⟨a, b, c \| aba=1, cccc=1⟩	Grp Inf	ℤ₄ ∗ ℤ	11
8	3138	⟨a, b, c \| ac=ab, aaaa=1⟩	Can Inf	ℤ₄ ∗ ℕ

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

8 total

Σ	#	Presentation	Mapping
7	840	⟨a, b, c \| ab=c, aaaa=1⟩	φ(a) = c, φ(b) = a, φ(c) = ca
8	2347	⟨a, b, c \| aaa=b, cccc=1⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
8	2478	⟨a, b, c \| aab=c, aaaa=1⟩	φ(a) = c, φ(b) = a, φ(c) = cca
8	2518	⟨a, b, c \| aab=c, bbbb=1⟩	φ(a) = a, φ(b) = c, φ(c) = aac
8	2629	⟨a, b, c \| aba=c, aaaa=1⟩	φ(a) = c, φ(b) = a, φ(c) = cac
8	2660	⟨a, b, c \| aba=c, bbbb=1⟩	φ(a) = a, φ(b) = c, φ(c) = aca
8	3198	⟨a, b, c \| ba=ac, aaaa=1⟩	φ(a) = c, φ(b) = a, φ(c) = cccac
8	3262	⟨a, b, c \| bb=ac, aaaa=1⟩	φ(a) = c, φ(b) = a, φ(c) = cccaa