⟨a, b | aaaaababaa=a⟩

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Monoid enumeration

#3344 ⟨a, b | aaaaabaabb=b⟩

#3346 ⟨a, b | aaaaababaa=b⟩

Properties

Sum of relation sides is 11

Infinite non-cancellative non-commutative monoid

Element a³ba² has infinite order

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

c ⋅ bcba = c and c ⋅ 1 = c, however bcba ≠ 1

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

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

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

#	Rule	Proof
1.	ac ⇒ ca	[3]
2.	a⁶ ⇒ c	[2]
3.	abc ⇒ cba	[16]
4.	c⁶bc ⇒ cbc⁶	[26]
5.	c⁶ba ⇒ cbc⁵a	[25]
6.	c⁵aba ⇒ cbc⁴a²	[24]
7.	c⁴a²ba ⇒ cbc³a³	[23]
8.	c³a³ba ⇒ cbc²a⁴	[22]
9.	c²a⁴ba ⇒ cbca⁵	[20]
10.	ca⁵ba ⇒ cbc	[18]
11.	c(bc)² ⇒ ca⁵	[21]
12.	(cb)²a ⇒ c	[19]
13.	c(ba)² ⇒ a	[10]
14.	ca(ba)² ⇒ a²	[9]
15.	a²(ba)² ⇒ (ab)²a²	[17]

# ab:aaaaababaa=a ca/b aaaaaa=c morph:6/0 ac=ca aaaaaa=c abc=cba ccccccbc=cbcccccc ccccccba=cbccccca cccccaba=cbccccaa ccccaaba=cbcccaaa cccaaaba=cbccaaaa ccaaaaba=cbcaaaaa caaaaaba=cbc cbcbc=caaaaa cbcba=c cbaba=a cababa=aa aababa=ababaa

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

Σ	#	Presentation	Mapping
11	3699	⟨a, b \| abaaaaaaba=a⟩	φ(a) = a, φ(b) = b

Presentation

Mapping

3699

⟨a, b | abaaaaaaba=a⟩

φ(a) = a, φ(b) = b

#3345 ⟨a, b | aaaaababaa=a⟩

Contents

Properties

Rewriting system

Isomorphic instances

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