⟨a, b | aab=bb, bba=b⟩

Up:

Monoid enumeration

#1654 ⟨a, b | aab=bb, bba=a⟩

#1657 ⟨a, b | aab=bb, bbb=b⟩

Properties

Sum of relation sides is 9

Infinite non-cancellative non-commutative monoid

Element a has infinite order

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

b ⋅ b³ = b and b ⋅ 1 = b, however b³ ≠ 1

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

b³ ⋅ b = b and 1 ⋅ b = b, however b³ ≠ 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) = 0; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b⁴ ⇒ b	[5]
2.	ba ⇒ b³	[6]
3.	a²b ⇒ b²	[1]

# ab:aab=bb,bba=b b/a - - bbbb=b ba=bbb aab=bb

Isomorphic instances

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

Σ	#	Presentation	Mapping
9	2046	⟨a, b \| aab=a, bbab=a⟩	φ(a) = b, φ(b) = a
10	5195	⟨a, b \| aab=bb, aaba=b⟩	φ(a) = a, φ(b) = b
11	12528	⟨a, b \| abab=ab, bbab=a⟩	φ(a) = b, φ(b) = a
11	12580	⟨a, b \| abba=ab, bbab=a⟩	φ(a) = b, φ(b) = a
11	18934	⟨a, b \| aab=a, bbaabb=a⟩	φ(a) = b, φ(b) = a
11	18938	⟨a, b \| aab=a, bbabab=a⟩	φ(a) = b, φ(b) = a
11	18940	⟨a, b \| aab=a, bbabba=a⟩	φ(a) = b, φ(b) = a
11	19476	⟨a, b \| aab=a, bbaab=aa⟩	φ(a) = b, φ(b) = a
11	19480	⟨a, b \| aab=a, bbaba=aa⟩	φ(a) = b, φ(b) = a
11	19485	⟨a, b \| aab=a, bbabb=ab⟩	φ(a) = b, φ(b) = a
11	19934	⟨a, b \| aab=a, aaab=bba⟩	φ(a) = b, φ(b) = a
11	19942	⟨a, b \| aab=a, aaba=bba⟩	φ(a) = b, φ(b) = a
11	19958	⟨a, b \| aab=a, abaa=bba⟩	φ(a) = b, φ(b) = a
11	20016	⟨a, b \| aab=a, bbaa=aaa⟩	φ(a) = b, φ(b) = a
11	20025	⟨a, b \| aab=a, bbab=aab⟩	φ(a) = b, φ(b) = a
11	20026	⟨a, b \| aab=a, bbab=aba⟩	φ(a) = b, φ(b) = a

Presentation

Mapping

2046

⟨a, b | aab=a, bbab=a⟩

φ(a) = b, φ(b) = a

5195

⟨a, b | aab=bb, aaba=b⟩

φ(a) = a, φ(b) = b

12528

⟨a, b | abab=ab, bbab=a⟩

φ(a) = b, φ(b) = a

12580

⟨a, b | abba=ab, bbab=a⟩

φ(a) = b, φ(b) = a

18934

⟨a, b | aab=a, bbaabb=a⟩

φ(a) = b, φ(b) = a

18938

⟨a, b | aab=a, bbabab=a⟩

φ(a) = b, φ(b) = a

18940

⟨a, b | aab=a, bbabba=a⟩

φ(a) = b, φ(b) = a

19476

⟨a, b | aab=a, bbaab=aa⟩

φ(a) = b, φ(b) = a

19480

⟨a, b | aab=a, bbaba=aa⟩

φ(a) = b, φ(b) = a

19485

⟨a, b | aab=a, bbabb=ab⟩

φ(a) = b, φ(b) = a

19934

⟨a, b | aab=a, aaab=bba⟩

φ(a) = b, φ(b) = a

19942

⟨a, b | aab=a, aaba=bba⟩

φ(a) = b, φ(b) = a

19958

⟨a, b | aab=a, abaa=bba⟩

φ(a) = b, φ(b) = a

20016

⟨a, b | aab=a, bbaa=aaa⟩

φ(a) = b, φ(b) = a

20025

⟨a, b | aab=a, bbab=aab⟩

φ(a) = b, φ(b) = a

20026

⟨a, b | aab=a, bbab=aba⟩

φ(a) = b, φ(b) = a

Anti-isomorphic instances

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

Σ	#	Presentation	Mapping
9	2100	⟨a, b \| aba=a, babb=a⟩	φ(a) = b, φ(b) = a
10	5199	⟨a, b \| aab=bb, abaa=b⟩	φ(a) = a, φ(b) = b
11	12455	⟨a, b \| aabb=ab, abaa=b⟩	φ(a) = a, φ(b) = b
11	12479	⟨a, b \| aabb=ba, babb=a⟩	φ(a) = b, φ(b) = a
11	12525	⟨a, b \| abab=ab, babb=a⟩	φ(a) = b, φ(b) = a
11	12618	⟨a, b \| baab=ab, babb=a⟩	φ(a) = b, φ(b) = a
11	19148	⟨a, b \| aba=a, bababb=a⟩	φ(a) = b, φ(b) = a
11	20236	⟨a, b \| aba=a, babb=aba⟩	φ(a) = b, φ(b) = a

Presentation

Mapping

2100

⟨a, b | aba=a, babb=a⟩

φ(a) = b, φ(b) = a

5199

⟨a, b | aab=bb, abaa=b⟩

φ(a) = a, φ(b) = b

12455

⟨a, b | aabb=ab, abaa=b⟩

φ(a) = a, φ(b) = b

12479

⟨a, b | aabb=ba, babb=a⟩

φ(a) = b, φ(b) = a

12525

⟨a, b | abab=ab, babb=a⟩

φ(a) = b, φ(b) = a

12618

⟨a, b | baab=ab, babb=a⟩

φ(a) = b, φ(b) = a

19148

⟨a, b | aba=a, bababb=a⟩

φ(a) = b, φ(b) = a

20236

⟨a, b | aba=a, babb=aba⟩

φ(a) = b, φ(b) = a

#1655 ⟨a, b | aab=bb, bba=b⟩

Contents

Properties

Rewriting system

Isomorphic instances

Anti-isomorphic instances

Up:	Monoid enumeration
Prev:	#1654 ⟨a, b \| aab=bb, bba=a⟩
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