#159 ⟨a, b, c | ab=c, ba=c⟩

Properties

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.	ba ⇒ ab	[2]
2.	c ⇒ ab	[1]

# abc:ab=c,ba=c ab/c - -
ba=ab
c=ab

3 unique, 9 total

Σ	#	Presentation	Properties	Description	φ
6	83	⟨a, b, c \| abc=1, cba=1⟩	Grp Com Inf	ℤ ⊕ ℤ	4
7	461	⟨a, b, c \| ba=ab, aca=1⟩	Can Com Inf	ℕ ⊕ ℤ	2
8	3181	⟨a, b, c \| ba=ab, aaca=1⟩	Can Com Inf	ℕ ⊕ ℤ

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

5 total

Σ	#	Presentation	Mapping
7	529	⟨a, b, c \| ba=ab, bb=c⟩	φ(a) = a, φ(b) = b, φ(c) = bb
7	1060	⟨a, b, c \| ab=c, ba=ab⟩	φ(a) = a, φ(b) = b, φ(c) = ab
8	3413	⟨a, b, c \| ba=ab, aaa=c⟩	φ(a) = b, φ(b) = a, φ(c) = bbb
8	3414	⟨a, b, c \| ba=ab, aab=c⟩	φ(a) = a, φ(b) = b, φ(c) = aab
8	3418	⟨a, b, c \| ba=ab, aba=c⟩	φ(a) = a, φ(b) = b, φ(c) = aab