⟨a, b, c | aa=b, accc=1⟩

Up:

Monoid enumeration

#746 ⟨a, b, c | aa=b, abbc=1⟩

#764 ⟨a, b, c | aa=b, bbcc=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 ⋅ c³a = a and a ⋅ 1 = a, however c³a ≠ 1

Not right cancellative, because right multiplication by c³ is not injective:

c³a ⋅ c³ = c³ and 1 ⋅ c³ = c³, however c³a ≠ 1

Rewriting system

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(c) = 0, a < c; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	ac³ ⇒ 1	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,accc=1 ac/b - - accc=1 b=aa

Isomorphic instances

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

Σ	#	Presentation	Mapping
7	813	⟨a, b, c \| ab=a, caaa=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
7	864	⟨a, b, c \| ab=c, baaa=1⟩	φ(a) = c, φ(b) = a, φ(c) = ca
8	1560	⟨a, b, c \| aaa=ab, cab=1⟩	φ(a) = c, φ(b) = cc, φ(c) = a
8	1562	⟨a, b, c \| aaa=ab, cba=1⟩	φ(a) = c, φ(b) = cc, φ(c) = a
8	1754	⟨a, b, c \| aab=cb, bac=1⟩	φ(a) = c, φ(b) = a, φ(c) = cc
8	1758	⟨a, b, c \| aab=cb, bca=1⟩	φ(a) = c, φ(b) = a, φ(c) = cc
8	1839	⟨a, b, c \| aba=ac, caa=1⟩	φ(a) = c, φ(b) = a, φ(c) = ac
8	2329	⟨a, b, c \| aaa=b, accc=1⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
8	2491	⟨a, b, c \| aab=c, abbb=1⟩	φ(a) = a, φ(b) = c, φ(c) = aac
8	2505	⟨a, b, c \| aab=c, baaa=1⟩	φ(a) = c, φ(b) = a, φ(c) = cca
8	2641	⟨a, b, c \| aba=c, abbb=1⟩	φ(a) = a, φ(b) = c, φ(c) = aca
8	3152	⟨a, b, c \| ac=ab, baaa=1⟩	φ(a) = c, φ(b) = a, φ(c) = a
8	4629	⟨a, b, c \| aa=a, abccc=1⟩	φ(a) = 1, φ(b) = a, φ(c) = c
8	4875	⟨a, b, c \| ab=a, bcaaa=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4903	⟨a, b, c \| ab=a, caaab=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4905	⟨a, b, c \| ab=a, caaba=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4911	⟨a, b, c \| ab=a, cabaa=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4929	⟨a, b, c \| ab=a, cbaaa=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	6899	⟨a, b, c \| ab=1, abccc=b⟩	φ(a) = a, φ(b) = ccc, φ(c) = c
8	7088	⟨a, b, c \| ab=1, cabcc=b⟩	φ(a) = a, φ(b) = ccc, φ(c) = c
8	7569	⟨a, b, c \| ab=1, bccc=bb⟩	φ(a) = a, φ(b) = ccc, φ(c) = c

Presentation

Mapping

813

⟨a, b, c | ab=a, caaa=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

864

⟨a, b, c | ab=c, baaa=1⟩

φ(a) = c, φ(b) = a, φ(c) = ca

1560

⟨a, b, c | aaa=ab, cab=1⟩

φ(a) = c, φ(b) = cc, φ(c) = a

1562

⟨a, b, c | aaa=ab, cba=1⟩

φ(a) = c, φ(b) = cc, φ(c) = a

1754

⟨a, b, c | aab=cb, bac=1⟩

φ(a) = c, φ(b) = a, φ(c) = cc

1758

⟨a, b, c | aab=cb, bca=1⟩

φ(a) = c, φ(b) = a, φ(c) = cc

1839

⟨a, b, c | aba=ac, caa=1⟩

φ(a) = c, φ(b) = a, φ(c) = ac

2329

⟨a, b, c | aaa=b, accc=1⟩

φ(a) = a, φ(b) = aaa, φ(c) = c

2491

⟨a, b, c | aab=c, abbb=1⟩

φ(a) = a, φ(b) = c, φ(c) = aac

2505

⟨a, b, c | aab=c, baaa=1⟩

φ(a) = c, φ(b) = a, φ(c) = cca

2641

⟨a, b, c | aba=c, abbb=1⟩

φ(a) = a, φ(b) = c, φ(c) = aca

3152

⟨a, b, c | ac=ab, baaa=1⟩

φ(a) = c, φ(b) = a, φ(c) = a

4629

⟨a, b, c | aa=a, abccc=1⟩

φ(a) = 1, φ(b) = a, φ(c) = c

4875

⟨a, b, c | ab=a, bcaaa=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4903

⟨a, b, c | ab=a, caaab=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4905

⟨a, b, c | ab=a, caaba=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4911

⟨a, b, c | ab=a, cabaa=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4929

⟨a, b, c | ab=a, cbaaa=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

6899

⟨a, b, c | ab=1, abccc=b⟩

φ(a) = a, φ(b) = ccc, φ(c) = c

7088

⟨a, b, c | ab=1, cabcc=b⟩

φ(a) = a, φ(b) = ccc, φ(c) = c

7569

⟨a, b, c | ab=1, bccc=bb⟩

φ(a) = a, φ(b) = ccc, φ(c) = c

Anti-isomorphic instances

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

Σ	#	Presentation	Mapping
8	3135	⟨a, b, c \| ab=aa, ccca=1⟩	φ(a) = a, φ(b) = a, φ(c) = c

Presentation

Mapping

3135

⟨a, b, c | ab=aa, ccca=1⟩

φ(a) = a, φ(b) = a, φ(c) = c

#756 ⟨a, b, c | aa=b, accc=1⟩

Contents

Properties

Rewriting system

Isomorphic instances

Anti-isomorphic instances

Up:	Monoid enumeration
Prev:	#746 ⟨a, b, c \| aa=b, abbc=1⟩
Next:	#764 ⟨a, b, c \| aa=b, bbcc=1⟩