#19507 ⟨a, b | aab=b, aaaaa=bb⟩

Properties

Sum of relation sides is 11
Finite commutative monoid with 9 elements
Not cancellative, because multiplication by ab is not injective:
- ab ⋅ b² = b and ab ⋅ a = b, however b² ≠ a

An idempotent element x satisfies x² = x.
The index and period of x is the least m (index) and n (period) such that x^(m+n) = x^m.

Histogram:

Idempotents are shown in bold.

Idempotents are shown in bold.

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

# ab:aab=b,aaaaa=bb ab - -
ba=ab
aab=b
bbb=ab
aaaaa=bb

26 unique, 942 total

Σ	#	Presentation	Properties	Description	φ
7	118	⟨a, b \| aaa=b, bbb=1⟩	Grp Com 9	ℤ₉	562
8	568	⟨a, b \| aaa=a, abb=b⟩	Noc 9		5
8	571	⟨a, b \| aaa=a, bbb=a⟩	Noc Com 9	ℕ_{(9 = 3)}	48
8	581	⟨a, b \| aaa=b, bbb=a⟩	Noc Com 9	ℕ_{(9 = 1)}	62
8	601	⟨a, b \| aba=b, bab=a⟩	Noc 9		3
9	1584	⟨a, b \| aaa=aa, bbb=a⟩	Noc Com 9	ℕ_{(9 = 6)}	16
9	1595	⟨a, b \| aaa=ab, bab=b⟩	Noc 9		9
9	1598	⟨a, b \| aaa=ab, bbb=a⟩	Noc Com 9	ℕ_{(9 = 4)}	40
9	1608	⟨a, b \| aaa=bb, bbb=a⟩	Noc Com 9	ℕ_{(9 = 2)}	70
9	1991	⟨a, b \| aaa=a, abba=b⟩	Noc 9		3
9	2064	⟨a, b \| aab=b, abba=a⟩	Noc 9		38
9	2255	⟨a, b \| ab=aa, bba=bb⟩	Noc 9
10	4125	⟨a, b \| aab=aaa, baa=b⟩	Noc 9		2
10	4130	⟨a, b \| aab=aaa, bbb=a⟩	Noc Com 9	ℕ_{(9 = 7)}	1
10	4156	⟨a, b \| abb=aaa, bbb=a⟩	Noc Com 9	ℕ_{(9 = 5)}	12
10	5035	⟨a, b \| aaa=aa, abbb=b⟩	Noc 9		3
10	5079	⟨a, b \| aaa=bb, aaba=b⟩	Noc Com 9		8
10	5124	⟨a, b \| aab=aa, bbbb=a⟩	Noc Com 9	ℕ_{(9 = 8)}	5
10	5339	⟨a, b \| aaa=bb, aab=ab⟩	Noc 9		1
10	6824	⟨a, b \| aab=b, bbb=aaa⟩	Noc Com 9		5
10	7139	⟨a, b \| bb=aa, aaaa=ab⟩	Noc 9		3
10	9327	⟨a, b \| ab=a, baaa=bbb⟩	Noc 9		3
10	9359	⟨a, b \| ab=a, bbaa=bbb⟩	Noc 9		2
11	19506	⟨a, b \| aab=b, aaaaa=ba⟩	Noc 9
11	20723	⟨a, b \| ab=aa, aaaaa=bb⟩	Noc 9		15
11	25656	⟨a, b \| ab=a, bbbbb=baa⟩	Noc 9