#3180 ⟨a, b, c | ba=ab, aabc=1⟩

Properties

Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ dca = a and a ⋅ 1 = a, however dca ≠ 1
Not right cancellative, because right multiplication by dc is not injective:
- dca ⋅ dc = dc and 1 ⋅ dc = dc, however dca ≠ 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

Auxiliary generators:
1. d = ba
Reduction order:
- Left-to-right recursive path with deg(a) = deg(c) = deg(d) = 0, a < c < d; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

# abc:ba=ab,aabc=1 acd/b ba=d morph:2/1
da=ad
adc=1
addc=d
ddca=d
dddc=ddcd
b=ddc

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

2 total

Σ	#	Presentation	Mapping
8	3184	⟨a, b, c \| ba=ab, abac=1⟩	φ(a) = a, φ(b) = ddc, φ(c) = c
8	3185	⟨a, b, c \| ba=ab, abbc=1⟩	φ(a) = ddc, φ(b) = a, φ(c) = c