⟨a, b, c | aab=ca, abc=1⟩

Up:

Sum of relation sides is 8

Infinite non-cancellative non-commutative monoid

Element b has infinite order

Not left cancellative, because left multiplication by d is not injective:

Not right cancellative, because right multiplication by b is not injective:

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

# abc:aab=ca,abc=1 bcd/a ac=d morph:2/0 cdc=d dbc=cdb dcc=dbd ddb=ccd ddc=cdd ccdb=1 cdbd=dc dcbc=db dcbd=dbdc dcdb=cd a=cd

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

Σ	#	Presentation	Mapping
8	1840	⟨a, b, c \| aba=ac, cab=1⟩	φ(a) = cd, φ(b) = b, φ(c) = c
8	2601	⟨a, b, c \| aba=b, abca=1⟩	φ(a) = c, φ(b) = d, φ(c) = b

Presentation

Mapping

1840

⟨a, b, c | aba=ac, cab=1⟩

φ(a) = cd, φ(b) = b, φ(c) = c

2601

⟨a, b, c | aba=b, abca=1⟩

φ(a) = c, φ(b) = d, φ(c) = b

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

Σ	#	Presentation	Mapping
8	2597	⟨a, b, c \| aba=b, aacb=1⟩	φ(a) = c, φ(b) = d, φ(c) = b
8	2604	⟨a, b, c \| aba=b, acab=1⟩	φ(a) = c, φ(b) = dcc, φ(c) = b

Presentation

Mapping

2597

⟨a, b, c | aba=b, aacb=1⟩

φ(a) = c, φ(b) = d, φ(c) = b

2604

⟨a, b, c | aba=b, acab=1⟩

φ(a) = c, φ(b) = dcc, φ(c) = b

#1723 ⟨a, b, c | aab=ca, abc=1⟩