#271 ⟨a, b | ab=a, bba=a⟩

Properties

Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ b = a and a ⋅ 1 = a, however b ≠ 1
Not right cancellative, because right multiplication by a is not injective:
- b² ⋅ a = a and 1 ⋅ a = a, however b² ≠ 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.	ab ⇒ a	[1]
2.	b²a ⇒ a	[2]

# ab:ab=a,bba=a ab - -
ab=a
bba=a

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

24 total

Σ	#	Presentation	Mapping
8	946	⟨a, b \| ab=a, bbab=a⟩	φ(a) = a, φ(b) = b
8	1017	⟨a, b \| ab=a, bba=ab⟩	φ(a) = a, φ(b) = b
9	2057	⟨a, b \| aab=b, aaba=b⟩	φ(a) = b, φ(b) = a
9	2990	⟨a, b \| ab=a, bbabb=a⟩	φ(a) = a, φ(b) = b
9	3125	⟨a, b \| ab=a, bbab=ab⟩	φ(a) = a, φ(b) = b
9	3190	⟨a, b \| ab=a, bba=abb⟩	φ(a) = a, φ(b) = b
10	5247	⟨a, b \| aba=ab, aaba=b⟩	φ(a) = b, φ(b) = a
10	8822	⟨a, b \| ab=a, bbabbb=a⟩	φ(a) = a, φ(b) = b
10	9093	⟨a, b \| ab=a, bbabb=ab⟩	φ(a) = a, φ(b) = b
10	9318	⟨a, b \| ab=a, abbb=bba⟩	φ(a) = a, φ(b) = b
10	9363	⟨a, b \| ab=a, bbab=abb⟩	φ(a) = a, φ(b) = b
11	12243	⟨a, b \| aaab=ab, aaba=b⟩	φ(a) = b, φ(b) = a
11	14610	⟨a, b \| aaba=b, aabaa=b⟩	φ(a) = b, φ(b) = a
11	15850	⟨a, b \| aba=ab, aabaa=b⟩	φ(a) = b, φ(b) = a
11	18965	⟨a, b \| aab=b, aaaaba=b⟩	φ(a) = b, φ(b) = a
11	19510	⟨a, b \| aab=b, aaaab=ba⟩	φ(a) = b, φ(b) = a
11	19513	⟨a, b \| aab=b, aaaba=ab⟩	φ(a) = b, φ(b) = a
11	20058	⟨a, b \| aab=b, aaab=aba⟩	φ(a) = b, φ(b) = a
11	20065	⟨a, b \| aab=b, aaba=aab⟩	φ(a) = b, φ(b) = a
11	24554	⟨a, b \| ab=a, bbabbbb=a⟩	φ(a) = a, φ(b) = b
11	25081	⟨a, b \| ab=a, bbabbb=ab⟩	φ(a) = a, φ(b) = b
11	25530	⟨a, b \| ab=a, abbbb=bba⟩	φ(a) = a, φ(b) = b
11	25623	⟨a, b \| ab=a, bbabb=abb⟩	φ(a) = a, φ(b) = b
11	25877	⟨a, b \| ab=a, bbab=abbb⟩	φ(a) = a, φ(b) = b