#241 ⟨a, b | aa=a, abb=a⟩

Properties

Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element ba has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ a = a and a ⋅ 1 = a, however a ≠ 1
Not right cancellative, because right multiplication by a is not injective:
- a ⋅ a = a and 1 ⋅ a = a, however a ≠ 1

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

#	Rule	Proof
1.	a² ⇒ a	[1]
2.	ab² ⇒ a	[2]

# ab:aa=a,abb=a ab - -
aa=a
abb=a

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

23 total

Σ	#	Presentation	Mapping
8	886	⟨a, b \| aa=a, aabb=a⟩	φ(a) = a, φ(b) = b
8	962	⟨a, b \| aa=a, abb=aa⟩	φ(a) = a, φ(b) = b
9	1624	⟨a, b \| aab=ab, abb=a⟩	φ(a) = a, φ(b) = b
9	2862	⟨a, b \| aa=a, aaabb=a⟩	φ(a) = a, φ(b) = b
9	3011	⟨a, b \| aa=a, aabb=aa⟩	φ(a) = a, φ(b) = b
9	3139	⟨a, b \| aa=a, abb=aaa⟩	φ(a) = a, φ(b) = b
10	5028	⟨a, b \| aaa=aa, aabb=a⟩	φ(a) = a, φ(b) = b
10	5132	⟨a, b \| aab=ab, aabb=a⟩	φ(a) = a, φ(b) = b
10	8574	⟨a, b \| aa=a, aaaabb=a⟩	φ(a) = a, φ(b) = b
10	8851	⟨a, b \| aa=a, aaabb=aa⟩	φ(a) = a, φ(b) = b
10	9115	⟨a, b \| aa=a, aaaa=abb⟩	φ(a) = a, φ(b) = b
10	9134	⟨a, b \| aa=a, aabb=aaa⟩	φ(a) = a, φ(b) = b
11	14671	⟨a, b \| aabb=a, aaabb=a⟩	φ(a) = a, φ(b) = b
11	14677	⟨a, b \| aabb=a, aabba=a⟩	φ(a) = a, φ(b) = b
11	14689	⟨a, b \| aabb=a, abbaa=a⟩	φ(a) = a, φ(b) = b
11	14799	⟨a, b \| abba=a, aaabb=a⟩	φ(a) = a, φ(b) = b
11	15407	⟨a, b \| aaa=aa, aaabb=a⟩	φ(a) = a, φ(b) = b
11	15615	⟨a, b \| aab=ab, aaabb=a⟩	φ(a) = a, φ(b) = b
11	24050	⟨a, b \| aa=a, aaaaabb=a⟩	φ(a) = a, φ(b) = b
11	24599	⟨a, b \| aa=a, aaaabb=aa⟩	φ(a) = a, φ(b) = b
11	25119	⟨a, b \| aa=a, aaaaa=abb⟩	φ(a) = a, φ(b) = b
11	25138	⟨a, b \| aa=a, aaabb=aaa⟩	φ(a) = a, φ(b) = b
11	25663	⟨a, b \| aa=a, aabb=aaaa⟩	φ(a) = a, φ(b) = b

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

17 total

Σ	#	Presentation	Mapping
9	2059	⟨a, b \| aab=b, aabb=b⟩	φ(a) = b, φ(b) = a
9	2071	⟨a, b \| aab=b, baab=b⟩	φ(a) = b, φ(b) = a
10	5150	⟨a, b \| aab=ab, bbaa=a⟩	φ(a) = a, φ(b) = b
11	12245	⟨a, b \| aaab=ab, aabb=b⟩	φ(a) = b, φ(b) = a
11	12379	⟨a, b \| aaba=ba, aabb=b⟩	φ(a) = b, φ(b) = a
11	12391	⟨a, b \| aaba=ba, baab=b⟩	φ(a) = b, φ(b) = a
11	12564	⟨a, b \| abba=aa, bbaa=a⟩	φ(a) = a, φ(b) = b
11	15657	⟨a, b \| aab=ab, bbaaa=a⟩	φ(a) = a, φ(b) = b
11	18967	⟨a, b \| aab=b, aaaabb=b⟩	φ(a) = b, φ(b) = a
11	18979	⟨a, b \| aab=b, aabaab=b⟩	φ(a) = b, φ(b) = a
11	19027	⟨a, b \| aab=b, baaaab=b⟩	φ(a) = b, φ(b) = a
11	19511	⟨a, b \| aab=b, aaaab=bb⟩	φ(a) = b, φ(b) = a
11	19517	⟨a, b \| aab=b, aaabb=ab⟩	φ(a) = b, φ(b) = a
11	19541	⟨a, b \| aab=b, abaab=ab⟩	φ(a) = b, φ(b) = a
11	20059	⟨a, b \| aab=b, aaab=abb⟩	φ(a) = b, φ(b) = a
11	20073	⟨a, b \| aab=b, aabb=aab⟩	φ(a) = b, φ(b) = a
11	20121	⟨a, b \| aab=b, baab=aab⟩	φ(a) = b, φ(b) = a