#20096 ⟨a, b | aab=b, abba=aaa⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 11 elements

Element profile

1 element center:
- 1
3 idempotent elements:
- 1, b²a, b⁴

Complete rewriting system

Format:	Pretty Plain
Word to reduce:
	Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Reduction strategy:	Leftmost Rightmost
Path to normal form:	1 1

Reduction order:
- Left-to-right recursive path with deg(b) = 0; deg(a) = 1

b⁵ ⇒ b
ab ⇒ b³
ba² ⇒ b³a
a³ ⇒ b⁴a

# ab:aab=b,abba=aaa b/a
bbbbb=b
ab=bbb
baa=bbba
aaa=bbbba

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	b²	b²a	b³	b³a	b⁴	b⁴a
1	1	a	b	a²	ba	b²	b²a	b³	b³a	b⁴	b⁴a
a	a	a²	b³	b⁴a	b³a	b⁴	b⁴a	b	ba	b²	b²a
b	b	ba	b²	b³a	b²a	b³	b³a	b⁴	b⁴a	b	ba
a²	a²	b⁴a	b	b²a	ba	b²	b²a	b³	b³a	b⁴	b⁴a
ba	ba	b³a	b⁴	ba	b⁴a	b	ba	b²	b²a	b³	b³a
b²	b²	b²a	b³	b⁴a	b³a	b⁴	b⁴a	b	ba	b²	b²a
b²a	b²a	b⁴a	b	b²a	ba	b²	b²a	b³	b³a	b⁴	b⁴a
b³	b³	b³a	b⁴	ba	b⁴a	b	ba	b²	b²a	b³	b³a
b³a	b³a	ba	b²	b³a	b²a	b³	b³a	b⁴	b⁴a	b	ba
b⁴	b⁴	b⁴a	b	b²a	ba	b²	b²a	b³	b³a	b⁴	b⁴a
b⁴a	b⁴a	b²a	b³	b⁴a	b³a	b⁴	b⁴a	b	ba	b²	b²a

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

29 unique, 479 total

Σ	#	Presentation	Description	Related
9	1472	⟨a, b \| aaa=bb, abbb=1⟩	Isomorphic to ℤ₁₁	259 iso
10	4155	⟨a, b \| abb=aaa, bba=b⟩	Finite non-commutative monoid with 11 elements	4 iso
10	5086	⟨a, b \| aaa=bb, abbb=a⟩	Finite commutative monoid with 11 elements	11 iso
10	5087	⟨a, b \| aaa=bb, abbb=b⟩	Finite commutative monoid with 11 elements	3 iso
10	5111	⟨a, b \| aab=aa, baaa=b⟩	Finite non-commutative monoid with 11 elements	7 iso
10	5322	⟨a, b \| aaa=ab, abb=bb⟩	Finite non-commutative monoid with 11 elements	3 iso
10	5335	⟨a, b \| aaa=ab, bbb=aa⟩	Finite non-commutative monoid with 11 elements	2 iso, 1 anti-iso
10	5410	⟨a, b \| aab=bb, bab=aa⟩	Finite non-commutative monoid with 11 elements	2 iso
10	6292	⟨a, b \| aaa=b, aabbb=a⟩	Isomorphic to ℕ_{(11 = 1)}	35 iso
10	6293	⟨a, b \| aaa=b, aabbb=b⟩	Isomorphic to ℕ_{(11 = 3)}	11 iso
10	6380	⟨a, b \| aab=a, bbbbb=a⟩	Isomorphic to ℕ_{(11 = 5)}	1 iso
10	6834	⟨a, b \| aba=b, abb=aaa⟩	Finite non-commutative monoid with 11 elements
10	7114	⟨a, b \| ab=aa, bbbb=aa⟩	Finite non-commutative monoid with 11 elements	1 iso
10	7116	⟨a, b \| ab=aa, bbbb=ba⟩	Finite non-commutative monoid with 11 elements
10	8691	⟨a, b \| aa=b, abbbbb=b⟩	Isomorphic to ℕ_{(11 = 2)}	32 iso
10	8777	⟨a, b \| ab=a, baaaaa=b⟩	Finite non-commutative monoid with 11 elements	20 iso, 5 anti-iso
10	9083	⟨a, b \| ab=a, bbaaa=bb⟩	Finite non-commutative monoid with 11 elements	4 iso
11	12179	⟨a, b \| aaaa=ab, abbb=b⟩	Isomorphic to ℕ_{(11 = 4)}	20 iso
11	15607	⟨a, b \| aab=aa, bbbbb=a⟩	Isomorphic to ℕ_{(11 = 10)}	1 iso
11	15671	⟨a, b \| aab=ab, bbbbb=a⟩	Isomorphic to ℕ_{(11 = 6)}	8 iso
11	16148	⟨a, b \| aab=ab, aaaa=bb⟩	Finite non-commutative monoid with 11 elements	1 iso
11	16212	⟨a, b \| aab=ba, aaaa=bb⟩	Finite non-commutative monoid with 11 elements
11	18715	⟨a, b \| aaa=a, aabbba=b⟩	Finite non-commutative monoid with 11 elements
11	19070	⟨a, b \| aab=b, bbabbb=a⟩	Finite commutative monoid with 11 elements	4 iso
11	20112	⟨a, b \| aab=b, baaa=aaa⟩	Finite non-commutative monoid with 11 elements
11	20168	⟨a, b \| aab=b, bbbb=aaa⟩	Finite commutative monoid with 11 elements
11	20885	⟨a, b \| bb=aa, aaaaa=ab⟩	Finite non-commutative monoid with 11 elements	7 iso, 8 anti-iso
11	25539	⟨a, b \| ab=a, baaaa=bbb⟩	Finite non-commutative monoid with 11 elements
11	25635	⟨a, b \| ab=a, bbbaa=bbb⟩	Finite non-commutative monoid with 11 elements

Other isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	20128	⟨a, b \| aab=b, baba=aaa⟩	φ(a) = a, φ(b) = b

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