#1686 ⟨a, b | abb=aa, bbb=a⟩

Properties
Commutative structure
Finite structure
Staircase diagram
Cayley table
Right Cayley graph
Rewriting system
Same cardinality
Isomorphic instances

Properties

Sum of relation sides is 9
Isomorphic to ℕ_{(6 = 5)}
Finite commutative monoid with 6 elements
Not cancellative, because multiplication by b⁵ is not injective:
- b⁵ ⋅ b = b⁵ and b⁵ ⋅ 1 = b⁵, however b ≠ 1

Commutative structure

Gröbner basis: ⟨a, b | b⁶=b⁵, a=b³⟩
Cancellative quotient is isomorphic to ℤ₁
Enveloping group is isomorphic to ℤ₁
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

Finite structure

The zero element z satisfies zy = yz = z for all y.
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.

Zero element: b⁵
1 non-trivial idempotent:
- b⁵
Order of generators:
- a: index 2, period 1
- b: index 5, period 1
Histogram:

index 1, period 1 1 element b⁵

index 2, period 1 2 elements b³, b⁴

index 3, period 1 1 element b²

index 5, period 1 1 element b

Staircase diagram

Cayley table

Idempotents are shown in bold.

	1	b	b²	b³	b⁴	b⁵
1	1	b	b²	b³	b⁴	b⁵
b	b	b²	b³	b⁴	b⁵	b⁵
b²	b²	b³	b⁴	b⁵	b⁵	b⁵
b³	b³	b⁴	b⁵	b⁵	b⁵	b⁵
b⁴	b⁴	b⁵	b⁵	b⁵	b⁵	b⁵
b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵

Right Cayley graph

Idempotents are shown in bold.

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

Reduction order:
- Left-to-right recursive path with deg(b) = 0; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b⁶ ⇒ b⁵	[4]
2.	a ⇒ b³	[2]

# ab:abb=aa,bbb=a b/a - -
bbbbbb=bbbbb
a=bbb

Same cardinality

19 unique, 1876 total

Σ	#	Presentation	Properties	Description	φ
6	22	⟨a, b \| ab=aa, bb=1⟩	Noc 6		107
6	26	⟨a, b \| aaa=1, abb=1⟩	Grp Com 6	ℤ₆	1373
7	160	⟨a, b \| ab=aa, bb=b⟩	Noc 6		4
7	245	⟨a, b \| aa=a, bbb=a⟩	Noc Com 6	ℕ_{(6 = 3)}	49
7	257	⟨a, b \| aa=b, bbb=a⟩	Noc Com 6	ℕ_{(6 = 1)}	61
7	258	⟨a, b \| aa=b, bbb=b⟩	Noc Com 6	ℕ_{(6 = 2)}	55
8	639	⟨a, b \| ab=aa, baa=b⟩	Noc 6		13
8	644	⟨a, b \| ab=aa, bbb=a⟩	Noc Com 6	ℕ_{(6 = 4)}	46
8	893	⟨a, b \| aa=a, abbb=b⟩	Noc 6		19
8	1011	⟨a, b \| ab=a, baa=bb⟩	Noc 6		13
9	1427	⟨a, b \| abba=b, baba=1⟩	Grp 6		66
9	1581	⟨a, b \| aaa=aa, abb=b⟩	Noc 6		5
9	1648	⟨a, b \| aab=bb, abb=a⟩	Noc Com 6		13
9	3075	⟨a, b \| ab=a, aaaa=bb⟩	Noc Com 6		14
9	3132	⟨a, b \| ab=a, bbbb=aa⟩	Noc Com 6		5
9	3134	⟨a, b \| ab=a, bbbb=ba⟩	Noc 6		2
9	3193	⟨a, b \| ab=a, bbb=aaa⟩	Noc Com 6		9
9	3199	⟨a, b \| ab=a, bbb=bba⟩	Noc 6		3
11	24145	⟨a, b \| aa=a, abbbbba=b⟩	Noc Com 6

Isomorphic instances

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

49 total

Σ	#	Presentation	Mapping
9	1694	⟨a, b \| bab=aa, bbb=a⟩	φ(a) = bbb, φ(b) = b
9	2998	⟨a, b \| ab=a, bbbbb=a⟩	φ(a) = bbbbb, φ(b) = b
9	3170	⟨a, b \| aa=b, bbb=abb⟩	φ(a) = b, φ(b) = bb
9	3171	⟨a, b \| aa=b, bbb=bab⟩	φ(a) = b, φ(b) = bb
10	5127	⟨a, b \| aab=ab, aaaa=b⟩	φ(a) = b, φ(b) = bbbb
10	5159	⟨a, b \| aab=ba, aaaa=b⟩	φ(a) = b, φ(b) = bbbb
10	5243	⟨a, b \| aba=ab, aaaa=b⟩	φ(a) = b, φ(b) = bbbb
10	9109	⟨a, b \| ab=a, bbbbb=ab⟩	φ(a) = bbbbb, φ(b) = b
10	9197	⟨a, b \| aa=b, aaab=bbb⟩	φ(a) = b, φ(b) = bb
10	9205	⟨a, b \| aa=b, aaba=bbb⟩	φ(a) = b, φ(b) = bb
10	9209	⟨a, b \| aa=b, aabb=abb⟩	φ(a) = b, φ(b) = bb
10	9211	⟨a, b \| aa=b, aabb=bab⟩	φ(a) = b, φ(b) = bb
10	9212	⟨a, b \| aa=b, aabb=bba⟩	φ(a) = b, φ(b) = bb
10	9217	⟨a, b \| aa=b, abab=abb⟩	φ(a) = b, φ(b) = bb
10	9219	⟨a, b \| aa=b, abab=bab⟩	φ(a) = b, φ(b) = bb
10	9220	⟨a, b \| aa=b, abab=bba⟩	φ(a) = b, φ(b) = bb
10	9225	⟨a, b \| aa=b, abba=abb⟩	φ(a) = b, φ(b) = bb
10	9226	⟨a, b \| aa=b, abba=bab⟩	φ(a) = b, φ(b) = bb
10	9239	⟨a, b \| aa=b, baab=abb⟩	φ(a) = b, φ(b) = bb
10	9240	⟨a, b \| aa=b, baab=bab⟩	φ(a) = b, φ(b) = bb
11	19306	⟨a, b \| aaa=b, aaaaa=bb⟩	φ(a) = b, φ(b) = bbb
11	19855	⟨a, b \| aaa=b, aaab=aab⟩	φ(a) = b, φ(b) = bbb
11	19856	⟨a, b \| aaa=b, aaab=aba⟩	φ(a) = b, φ(b) = bbb
11	19858	⟨a, b \| aaa=b, aaab=baa⟩	φ(a) = b, φ(b) = bbb
11	19863	⟨a, b \| aaa=b, aaba=aab⟩	φ(a) = b, φ(b) = bbb
11	19864	⟨a, b \| aaa=b, aaba=aba⟩	φ(a) = b, φ(b) = bbb
11	19866	⟨a, b \| aaa=b, aaba=baa⟩	φ(a) = b, φ(b) = bbb
11	25265	⟨a, b \| aa=b, aaaaa=bbb⟩	φ(a) = b, φ(b) = bb
11	25269	⟨a, b \| aa=b, aaaab=abb⟩	φ(a) = b, φ(b) = bb
11	25271	⟨a, b \| aa=b, aaaab=bab⟩	φ(a) = b, φ(b) = bb
11	25272	⟨a, b \| aa=b, aaaab=bba⟩	φ(a) = b, φ(b) = bb
11	25277	⟨a, b \| aa=b, aaaba=abb⟩	φ(a) = b, φ(b) = bb
11	25279	⟨a, b \| aa=b, aaaba=bab⟩	φ(a) = b, φ(b) = bb
11	25280	⟨a, b \| aa=b, aaaba=bba⟩	φ(a) = b, φ(b) = bb
11	25293	⟨a, b \| aa=b, aabaa=abb⟩	φ(a) = b, φ(b) = bb
11	25294	⟨a, b \| aa=b, aabaa=bab⟩	φ(a) = b, φ(b) = bb
11	25655	⟨a, b \| ab=a, bbbbb=abb⟩	φ(a) = bbbbb, φ(b) = b
11	25730	⟨a, b \| aa=b, aabb=aaab⟩	φ(a) = b, φ(b) = bb
11	25731	⟨a, b \| aa=b, aabb=aaba⟩	φ(a) = b, φ(b) = bb
11	25734	⟨a, b \| aa=b, abaa=aabb⟩	φ(a) = b, φ(b) = bb
11	25736	⟨a, b \| aa=b, abab=aaab⟩	φ(a) = b, φ(b) = bb
11	25737	⟨a, b \| aa=b, abab=aaba⟩	φ(a) = b, φ(b) = bb
11	25739	⟨a, b \| aa=b, abab=abaa⟩	φ(a) = b, φ(b) = bb
11	25741	⟨a, b \| aa=b, abba=aaab⟩	φ(a) = b, φ(b) = bb
11	25742	⟨a, b \| aa=b, abba=aaba⟩	φ(a) = b, φ(b) = bb
11	25753	⟨a, b \| aa=b, baaa=aabb⟩	φ(a) = b, φ(b) = bb
11	25754	⟨a, b \| aa=b, baaa=abab⟩	φ(a) = b, φ(b) = bb
11	25757	⟨a, b \| aa=b, baab=aaab⟩	φ(a) = b, φ(b) = bb
11	25758	⟨a, b \| aa=b, baab=aaba⟩	φ(a) = b, φ(b) = bb

Up:	Monoid enumeration
Prev:	#1685 ⟨a, b \| aba=bb, bbb=b⟩
Next:	#1687 ⟨a, b \| abb=aa, bbb=b⟩

index 1, period 1	1 element	b⁵
index 2, period 1	2 elements	b³, b⁴
index 3, period 1	1 element	b²
index 5, period 1	1 element	b

#1686 ⟨a, b | abb=aa, bbb=a⟩

Contents