#3448 ⟨a, b, c | ba=ac, abb=c⟩

Properties

Sum of relation sides is 8
Infinite cancellative non-commutative monoid
Element a has infinite order
Submonoid of enveloping group: ⟨a, b | aabba^-1b^-1⟩
- Embedding: φ(a) = a, φ(b) = aba^-1, φ(c) = b

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.	a²b² ⇒ ba	[3]
2.	c ⇒ ab²	[2]

# abc:ba=ac,abb=c ab/c - -
aabb=ba
c=abb

8 unique, 22 total

Σ	#	Presentation	Properties	φ
8	2228	⟨a, b, c \| aabc=1, cbac=1⟩	Grp Inf	11
8	2983	⟨a, b, c \| aab=c, bba=c⟩	Can Inf
8	2987	⟨a, b, c \| aab=c, bca=c⟩	Can Inf
8	3537	⟨a, b, c \| bb=ac, caa=b⟩	Can Inf
8	3593	⟨a, b, c \| ba=ac, ca=bb⟩	Can Inf
8	5581	⟨a, b, c \| ab=c, bbca=c⟩	Can Inf	1
8	5717	⟨a, b, c \| aa=b, acc=cb⟩	Can Inf	1
8	6064	⟨a, b, c \| ab=c, baa=cc⟩	Can Inf	1

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

3 total

Σ	#	Presentation	Mapping
8	5572	⟨a, b, c \| ab=c, bbaa=c⟩	φ(a) = b, φ(b) = a, φ(c) = ba
8	5755	⟨a, b, c \| aa=b, bcc=ca⟩	φ(a) = a, φ(b) = aa, φ(c) = b
8	6044	⟨a, b, c \| ab=c, acb=ba⟩	φ(a) = a, φ(b) = b, φ(c) = ab