⟨a, b, c | aa=b, bcc=1⟩

Up:

Monoid enumeration

#94 ⟨a, b, c | aa=b, bbc=1⟩

#99 ⟨a, b, c | aa=b, ccc=1⟩

Properties

Sum of relation sides is 6

Infinite non-cancellative non-commutative monoid

Element a has infinite order

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

a ⋅ ac²a = a and a ⋅ 1 = a, however ac²a ≠ 1

Not right cancellative, because right multiplication by ac² is not injective:

ac²a ⋅ ac² = ac² and 1 ⋅ ac² = ac², however ac²a ≠ 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.	a²c² ⇒ 1	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,bcc=1 ac/b - - aacc=1 b=aa

Isomorphic instances

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

Σ	#	Presentation	Mapping
6	125	⟨a, b, c \| ab=c, acb=1⟩	φ(a) = a, φ(b) = c, φ(c) = ac
6	242	⟨a, b, c \| ab=1, acc=b⟩	φ(a) = a, φ(b) = acc, φ(c) = c
7	744	⟨a, b, c \| aa=b, aacc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
7	831	⟨a, b, c \| ab=a, ccaa=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
7	844	⟨a, b, c \| ab=c, aabb=1⟩	φ(a) = a, φ(b) = c, φ(c) = ac
7	870	⟨a, b, c \| ab=c, bbaa=1⟩	φ(a) = c, φ(b) = a, φ(c) = ca
8	1685	⟨a, b, c \| aab=bb, bcc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	1757	⟨a, b, c \| aab=cb, bbc=1⟩	φ(a) = c, φ(b) = a, φ(c) = cc
8	1836	⟨a, b, c \| aba=ac, bca=1⟩	φ(a) = c, φ(b) = a, φ(c) = ac
8	2317	⟨a, b, c \| aaa=b, aacc=1⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
8	2482	⟨a, b, c \| aab=c, aabb=1⟩	φ(a) = a, φ(b) = c, φ(c) = aac
8	2514	⟨a, b, c \| aab=c, bbaa=1⟩	φ(a) = c, φ(b) = a, φ(c) = cca
8	2633	⟨a, b, c \| aba=c, aabb=1⟩	φ(a) = a, φ(b) = c, φ(c) = aca
8	3161	⟨a, b, c \| ac=ab, bbaa=1⟩	φ(a) = c, φ(b) = a, φ(c) = a
8	3170	⟨a, b, c \| ac=ab, bcaa=1⟩	φ(a) = c, φ(b) = a, φ(c) = a
8	4622	⟨a, b, c \| aa=a, abbcc=1⟩	φ(a) = 1, φ(b) = a, φ(c) = c
8	4893	⟨a, b, c \| ab=a, bccaa=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4947	⟨a, b, c \| ab=a, cbcaa=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4957	⟨a, b, c \| ab=a, ccaab=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4959	⟨a, b, c \| ab=a, ccaba=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4965	⟨a, b, c \| ab=a, ccbaa=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	6815	⟨a, b, c \| ab=1, aabcc=b⟩	φ(a) = a, φ(b) = acc, φ(c) = c
8	6859	⟨a, b, c \| ab=1, abacc=b⟩	φ(a) = a, φ(b) = acc, φ(c) = c
8	6895	⟨a, b, c \| ab=1, abccb=a⟩	φ(a) = aac, φ(b) = c, φ(c) = a
8	6911	⟨a, b, c \| ab=1, acabc=b⟩	φ(a) = a, φ(b) = acc, φ(c) = c
8	7437	⟨a, b, c \| ab=1, acca=ba⟩	φ(a) = a, φ(b) = acc, φ(c) = c
8	7497	⟨a, b, c \| ab=1, bacc=bb⟩	φ(a) = a, φ(b) = acc, φ(c) = c
8	7732	⟨a, b, c \| ab=1, acc=abb⟩	φ(a) = a, φ(b) = acc, φ(c) = c
8	7744	⟨a, b, c \| ab=1, bab=acc⟩	φ(a) = a, φ(b) = acc, φ(c) = c

Presentation

Mapping

125

⟨a, b, c | ab=c, acb=1⟩

φ(a) = a, φ(b) = c, φ(c) = ac

242

⟨a, b, c | ab=1, acc=b⟩

φ(a) = a, φ(b) = acc, φ(c) = c

744

⟨a, b, c | aa=b, aacc=1⟩

φ(a) = a, φ(b) = aa, φ(c) = c

831

⟨a, b, c | ab=a, ccaa=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

844

⟨a, b, c | ab=c, aabb=1⟩

φ(a) = a, φ(b) = c, φ(c) = ac

870

⟨a, b, c | ab=c, bbaa=1⟩

φ(a) = c, φ(b) = a, φ(c) = ca

1685

⟨a, b, c | aab=bb, bcc=1⟩

φ(a) = a, φ(b) = aa, φ(c) = c

1757

⟨a, b, c | aab=cb, bbc=1⟩

φ(a) = c, φ(b) = a, φ(c) = cc

1836

⟨a, b, c | aba=ac, bca=1⟩

φ(a) = c, φ(b) = a, φ(c) = ac

2317

⟨a, b, c | aaa=b, aacc=1⟩

φ(a) = a, φ(b) = aaa, φ(c) = c

2482

⟨a, b, c | aab=c, aabb=1⟩

φ(a) = a, φ(b) = c, φ(c) = aac

2514

⟨a, b, c | aab=c, bbaa=1⟩

φ(a) = c, φ(b) = a, φ(c) = cca

2633

⟨a, b, c | aba=c, aabb=1⟩

φ(a) = a, φ(b) = c, φ(c) = aca

3161

⟨a, b, c | ac=ab, bbaa=1⟩

φ(a) = c, φ(b) = a, φ(c) = a

3170

⟨a, b, c | ac=ab, bcaa=1⟩

φ(a) = c, φ(b) = a, φ(c) = a

4622

⟨a, b, c | aa=a, abbcc=1⟩

φ(a) = 1, φ(b) = a, φ(c) = c

4893

⟨a, b, c | ab=a, bccaa=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4947

⟨a, b, c | ab=a, cbcaa=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4957

⟨a, b, c | ab=a, ccaab=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4959

⟨a, b, c | ab=a, ccaba=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4965

⟨a, b, c | ab=a, ccbaa=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

6815

⟨a, b, c | ab=1, aabcc=b⟩

φ(a) = a, φ(b) = acc, φ(c) = c

6859

⟨a, b, c | ab=1, abacc=b⟩

φ(a) = a, φ(b) = acc, φ(c) = c

6895

⟨a, b, c | ab=1, abccb=a⟩

φ(a) = aac, φ(b) = c, φ(c) = a

6911

⟨a, b, c | ab=1, acabc=b⟩

φ(a) = a, φ(b) = acc, φ(c) = c

7437

⟨a, b, c | ab=1, acca=ba⟩

φ(a) = a, φ(b) = acc, φ(c) = c

7497

⟨a, b, c | ab=1, bacc=bb⟩

φ(a) = a, φ(b) = acc, φ(c) = c

7732

⟨a, b, c | ab=1, acc=abb⟩

φ(a) = a, φ(b) = acc, φ(c) = c

7744

⟨a, b, c | ab=1, bab=acc⟩

φ(a) = a, φ(b) = acc, 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	3129	⟨a, b, c \| ab=aa, ccaa=1⟩	φ(a) = a, φ(b) = a, φ(c) = c
8	3130	⟨a, b, c \| ab=aa, ccab=1⟩	φ(a) = a, φ(b) = a, φ(c) = c
8	3132	⟨a, b, c \| ab=aa, ccba=1⟩	φ(a) = a, φ(b) = a, φ(c) = c

Presentation

Mapping

3129

⟨a, b, c | ab=aa, ccaa=1⟩

φ(a) = a, φ(b) = a, φ(c) = c

3130

⟨a, b, c | ab=aa, ccab=1⟩

φ(a) = a, φ(b) = a, φ(c) = c

3132

⟨a, b, c | ab=aa, ccba=1⟩

φ(a) = a, φ(b) = a, φ(c) = c

#96 ⟨a, b, c | aa=b, bcc=1⟩

Contents

Properties

Rewriting system

Isomorphic instances

Anti-isomorphic instances

Up:	Monoid enumeration
Prev:	#94 ⟨a, b, c \| aa=b, bbc=1⟩
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