⟨a, b, c | aa=1, abbc=1⟩

Up:

Monoid enumeration

#162 ⟨a, b, c | aa=1, abac=1⟩

#166 ⟨a, b, c | aa=1, abcc=1⟩

Properties

Sum of relation sides is 6

Infinite non-cancellative non-commutative monoid

Element b has infinite order

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

b ⋅ (bcb)² = b and b ⋅ 1 = b, however (bcb)² ≠ 1

Not right cancellative, because right multiplication by bcb²c is not injective:

(bcb)² ⋅ bcb²c = bcb²c and 1 ⋅ bcb²c = bcb²c, however (bcb)² ≠ 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(b) = deg(c) = 0, b < c; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	(b²c)² ⇒ 1	[4]
2.	a ⇒ b²c	[3]

# abc:aa=1,abbc=1 bc/a - - bbcbbc=1 a=bbc

Isomorphic instances

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

Σ	#	Presentation	Mapping
6	221	⟨a, b, c \| aa=1, bbc=a⟩	φ(a) = bbc, φ(b) = b, φ(c) = c
7	766	⟨a, b, c \| aa=b, bcbc=1⟩	φ(a) = b, φ(b) = bb, φ(c) = c
7	858	⟨a, b, c \| ab=c, acac=1⟩	φ(a) = b, φ(b) = c, φ(c) = bc
8	2300	⟨a, b, c \| aaa=a, abbc=1⟩	φ(a) = bbc, φ(b) = b, φ(c) = c
8	2419	⟨a, b, c \| aab=b, abbc=1⟩	φ(a) = bbc, φ(b) = b, φ(c) = c
8	2439	⟨a, b, c \| aab=b, bbca=1⟩	φ(a) = bbc, φ(b) = b, φ(c) = c
8	2483	⟨a, b, c \| aab=c, aabc=1⟩	φ(a) = b, φ(b) = c, φ(c) = bbc
8	2533	⟨a, b, c \| aab=c, caab=1⟩	φ(a) = b, φ(b) = c, φ(c) = bbc
8	2646	⟨a, b, c \| aba=c, acab=1⟩	φ(a) = b, φ(b) = c, φ(c) = bcb
8	4292	⟨a, b, c \| aab=1, caac=b⟩	φ(a) = b, φ(b) = cbbc, φ(c) = c
8	4531	⟨a, b, c \| abc=1, acaa=b⟩	φ(a) = b, φ(b) = bcbb, φ(c) = c
8	4672	⟨a, b, c \| aa=b, aacbc=1⟩	φ(a) = b, φ(b) = bb, φ(c) = c
8	4742	⟨a, b, c \| aa=b, bcaac=1⟩	φ(a) = b, φ(b) = bb, φ(c) = c
8	4994	⟨a, b, c \| ab=c, aabac=1⟩	φ(a) = b, φ(b) = c, φ(c) = bc
8	6115	⟨a, b, c \| aa=1, aaabbc=1⟩	φ(a) = bbc, φ(b) = b, φ(c) = c
8	6126	⟨a, b, c \| aa=1, aabbca=1⟩	φ(a) = bbc, φ(b) = b, φ(c) = c
8	6139	⟨a, b, c \| aa=1, abaabc=1⟩	φ(a) = bbc, φ(b) = b, φ(c) = c
8	6156	⟨a, b, c \| aa=1, abbaac=1⟩	φ(a) = bbc, φ(b) = b, φ(c) = c
8	6628	⟨a, b, c \| aa=1, aabbc=a⟩	φ(a) = bbc, φ(b) = b, φ(c) = c
8	6666	⟨a, b, c \| aa=1, abbca=a⟩	φ(a) = bbc, φ(b) = b, φ(c) = c
8	6700	⟨a, b, c \| aa=1, baabc=a⟩	φ(a) = bbc, φ(b) = b, φ(c) = c
8	6905	⟨a, b, c \| ab=1, acaac=b⟩	φ(a) = b, φ(b) = bcbbc, φ(c) = c
8	7164	⟨a, b, c \| aa=1, abbc=aa⟩	φ(a) = bbc, φ(b) = b, φ(c) = c
8	7660	⟨a, b, c \| aa=1, bbc=aaa⟩	φ(a) = bbc, φ(b) = b, φ(c) = c

Presentation

Mapping

221

⟨a, b, c | aa=1, bbc=a⟩

φ(a) = bbc, φ(b) = b, φ(c) = c

766

⟨a, b, c | aa=b, bcbc=1⟩

φ(a) = b, φ(b) = bb, φ(c) = c

858

⟨a, b, c | ab=c, acac=1⟩

φ(a) = b, φ(b) = c, φ(c) = bc

2300

⟨a, b, c | aaa=a, abbc=1⟩

φ(a) = bbc, φ(b) = b, φ(c) = c

2419

⟨a, b, c | aab=b, abbc=1⟩

φ(a) = bbc, φ(b) = b, φ(c) = c

2439

⟨a, b, c | aab=b, bbca=1⟩

φ(a) = bbc, φ(b) = b, φ(c) = c

2483

⟨a, b, c | aab=c, aabc=1⟩

φ(a) = b, φ(b) = c, φ(c) = bbc

2533

⟨a, b, c | aab=c, caab=1⟩

φ(a) = b, φ(b) = c, φ(c) = bbc

2646

⟨a, b, c | aba=c, acab=1⟩

φ(a) = b, φ(b) = c, φ(c) = bcb

4292

⟨a, b, c | aab=1, caac=b⟩

φ(a) = b, φ(b) = cbbc, φ(c) = c

4531

⟨a, b, c | abc=1, acaa=b⟩

φ(a) = b, φ(b) = bcbb, φ(c) = c

4672

⟨a, b, c | aa=b, aacbc=1⟩

φ(a) = b, φ(b) = bb, φ(c) = c

4742

⟨a, b, c | aa=b, bcaac=1⟩

φ(a) = b, φ(b) = bb, φ(c) = c

4994

⟨a, b, c | ab=c, aabac=1⟩

φ(a) = b, φ(b) = c, φ(c) = bc

6115

⟨a, b, c | aa=1, aaabbc=1⟩

φ(a) = bbc, φ(b) = b, φ(c) = c

6126

⟨a, b, c | aa=1, aabbca=1⟩

φ(a) = bbc, φ(b) = b, φ(c) = c

6139

⟨a, b, c | aa=1, abaabc=1⟩

φ(a) = bbc, φ(b) = b, φ(c) = c

6156

⟨a, b, c | aa=1, abbaac=1⟩

φ(a) = bbc, φ(b) = b, φ(c) = c

6628

⟨a, b, c | aa=1, aabbc=a⟩

φ(a) = bbc, φ(b) = b, φ(c) = c

6666

⟨a, b, c | aa=1, abbca=a⟩

φ(a) = bbc, φ(b) = b, φ(c) = c

6700

⟨a, b, c | aa=1, baabc=a⟩

φ(a) = bbc, φ(b) = b, φ(c) = c

6905

⟨a, b, c | ab=1, acaac=b⟩

φ(a) = b, φ(b) = bcbbc, φ(c) = c

7164

⟨a, b, c | aa=1, abbc=aa⟩

φ(a) = bbc, φ(b) = b, φ(c) = c

7660

⟨a, b, c | aa=1, bbc=aaa⟩

φ(a) = bbc, φ(b) = b, φ(c) = c

#163 ⟨a, b, c | aa=1, abbc=1⟩

Contents

Properties

Rewriting system

Isomorphic instances

Up:	Monoid enumeration
Prev:	#162 ⟨a, b, c \| aa=1, abac=1⟩
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