#860 ⟨a, b, c | ab=c, acbc=1⟩

Properties

Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ ab(ba)² = a and a ⋅ 1 = a, however ab(ba)² ≠ 1
Not right cancellative, because right multiplication by ab²ab is not injective:
- ab(ba)² ⋅ ab²ab = ab²ab and 1 ⋅ ab²ab = ab²ab, however ab(ba)² ≠ 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.	a²b²ab ⇒ 1	[2]
2.	c ⇒ ab	[1]

# abc:ab=c,acbc=1 ab/c - -
aabbab=1
c=ab

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

9 total

Σ	#	Presentation	Mapping
8	2520	⟨a, b, c \| aab=c, bbca=1⟩	φ(a) = b, φ(b) = a, φ(c) = bba
8	2542	⟨a, b, c \| aab=c, cbab=1⟩	φ(a) = a, φ(b) = b, φ(c) = aab
8	2659	⟨a, b, c \| aba=c, bbac=1⟩	φ(a) = b, φ(b) = a, φ(c) = bab
8	4346	⟨a, b, c \| aab=1, ccac=b⟩	φ(a) = a, φ(b) = bbab, φ(c) = b
8	4545	⟨a, b, c \| abc=1, acca=b⟩	φ(a) = a, φ(b) = abba, φ(c) = b
8	4750	⟨a, b, c \| aa=b, bccac=1⟩	φ(a) = a, φ(b) = aa, φ(c) = b
8	4996	⟨a, b, c \| ab=c, aabbc=1⟩	φ(a) = a, φ(b) = b, φ(c) = ab
8	5072	⟨a, b, c \| ab=c, bbaca=1⟩	φ(a) = b, φ(b) = a, φ(c) = ba
8	6944	⟨a, b, c \| ab=1, accac=b⟩	φ(a) = a, φ(b) = abbab, φ(c) = b