#368 ⟨a, b | aa=1, abb=bb⟩

Properties

Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element ba has infinite order
Not right cancellative, because right multiplication by b is not injective:
- ab ⋅ b = b² and b ⋅ b = b², however ab ≠ b

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 ⇒ 1	[1]
2.	ab² ⇒ b²	[2]

# ab:aa=1,abb=bb ab - -
aa=1
abb=bb

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

25 total

Σ	#	Presentation	Mapping
9	3486	⟨a, b \| aa=1, aaabb=bb⟩	φ(a) = a, φ(b) = b
9	3505	⟨a, b \| aa=1, abaab=bb⟩	φ(a) = a, φ(b) = b
9	3633	⟨a, b \| aa=1, aabb=abb⟩	φ(a) = a, φ(b) = b
9	3643	⟨a, b \| aa=1, abab=bab⟩	φ(a) = a, φ(b) = ab
9	3663	⟨a, b \| aa=1, baab=abb⟩	φ(a) = a, φ(b) = b
11	26982	⟨a, b \| aa=1, aaaaabb=bb⟩	φ(a) = a, φ(b) = b
11	27005	⟨a, b \| aa=1, aaabaab=bb⟩	φ(a) = a, φ(b) = b
11	27017	⟨a, b \| aa=1, aaabbaa=bb⟩	φ(a) = a, φ(b) = b
11	27087	⟨a, b \| aa=1, abaaaab=bb⟩	φ(a) = a, φ(b) = b
11	27521	⟨a, b \| aa=1, aaaabb=abb⟩	φ(a) = a, φ(b) = b
11	27539	⟨a, b \| aa=1, aaabab=bab⟩	φ(a) = a, φ(b) = ab
11	27548	⟨a, b \| aa=1, aaabba=bba⟩	φ(a) = a, φ(b) = b
11	27561	⟨a, b \| aa=1, aabaab=abb⟩	φ(a) = a, φ(b) = b
11	27585	⟨a, b \| aa=1, aabbaa=abb⟩	φ(a) = a, φ(b) = b
11	27617	⟨a, b \| aa=1, abaaab=bab⟩	φ(a) = a, φ(b) = ab
11	27699	⟨a, b \| aa=1, baaaab=abb⟩	φ(a) = a, φ(b) = b
11	28069	⟨a, b \| aa=1, aaabb=aabb⟩	φ(a) = a, φ(b) = b
11	28075	⟨a, b \| aa=1, aaabb=baab⟩	φ(a) = a, φ(b) = b
11	28078	⟨a, b \| aa=1, aaabb=bbaa⟩	φ(a) = a, φ(b) = b
11	28097	⟨a, b \| aa=1, aabab=abab⟩	φ(a) = a, φ(b) = ab
11	28114	⟨a, b \| aa=1, aabba=abba⟩	φ(a) = a, φ(b) = b
11	28143	⟨a, b \| aa=1, abaab=aabb⟩	φ(a) = a, φ(b) = b
11	28149	⟨a, b \| aa=1, abaab=baab⟩	φ(a) = a, φ(b) = b
11	28152	⟨a, b \| aa=1, abaab=bbaa⟩	φ(a) = a, φ(b) = b
11	28228	⟨a, b \| aa=1, baaab=abab⟩	φ(a) = a, φ(b) = ab

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

17 total

Σ	#	Presentation	Mapping
9	3497	⟨a, b \| aa=1, aabba=bb⟩	φ(a) = a, φ(b) = b
9	3636	⟨a, b \| aa=1, aabb=bba⟩	φ(a) = a, φ(b) = b
9	3649	⟨a, b \| aa=1, abba=abb⟩	φ(a) = a, φ(b) = b
11	26994	⟨a, b \| aa=1, aaaabba=bb⟩	φ(a) = a, φ(b) = b
11	27037	⟨a, b \| aa=1, aabaaba=bb⟩	φ(a) = a, φ(b) = b
11	27524	⟨a, b \| aa=1, aaaabb=bba⟩	φ(a) = a, φ(b) = b
11	27545	⟨a, b \| aa=1, aaabba=abb⟩	φ(a) = a, φ(b) = b
11	27564	⟨a, b \| aa=1, aabaab=bba⟩	φ(a) = a, φ(b) = b
11	27571	⟨a, b \| aa=1, aababa=bab⟩	φ(a) = a, φ(b) = ba
11	27623	⟨a, b \| aa=1, abaaba=abb⟩	φ(a) = a, φ(b) = b
11	28072	⟨a, b \| aa=1, aaabb=abba⟩	φ(a) = a, φ(b) = b
11	28102	⟨a, b \| aa=1, aabab=baba⟩	φ(a) = a, φ(b) = ba
11	28111	⟨a, b \| aa=1, aabba=aabb⟩	φ(a) = a, φ(b) = b
11	28117	⟨a, b \| aa=1, aabba=baab⟩	φ(a) = a, φ(b) = b
11	28120	⟨a, b \| aa=1, aabba=bbaa⟩	φ(a) = a, φ(b) = b
11	28146	⟨a, b \| aa=1, abaab=abba⟩	φ(a) = a, φ(b) = b
11	28160	⟨a, b \| aa=1, ababa=abab⟩	φ(a) = a, φ(b) = ba