#2051 ⟨a, b | aab=a, bbbb=b⟩

Properties

Presentation has sum-of-sides 9
Finite non-commutative monoid with 16 elements

Element profile

1 element center:
- 1
6 idempotent elements:
- 1, ba, a³, b³, b²a², b³a³

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(a) = 0; deg(b) = 1

a⁴ ⇒ a
ab ⇒ a³
b⁴ ⇒ b

# ab:aab=a,bbbb=b a/b
aaaa=a
ab=aaa
bbbb=b

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	b²	a³	ba²	b²a	b³	ba³	b²a²	b³a	b²a³	b³a²	b³a³
1	1	a	b	a²	ba	b²	a³	ba²	b²a	b³	ba³	b²a²	b³a	b²a³	b³a²	b³a³
a	a	a²	a³	a³	a	a²	a	a²	a³	a	a³	a	a²	a²	a³	a
b	b	ba	b²	ba²	b²a	b³	ba³	b²a²	b³a	b	b²a³	b³a²	ba	b³a³	ba²	ba³
a²	a²	a³	a	a	a²	a³	a²	a³	a	a²	a	a²	a³	a³	a	a²
ba	ba	ba²	ba³	ba³	ba	ba²	ba	ba²	ba³	ba	ba³	ba	ba²	ba²	ba³	ba
b²	b²	b²a	b³	b²a²	b³a	b	b²a³	b³a²	ba	b²	b³a³	ba²	b²a	ba³	b²a²	b²a³
a³	a³	a	a²	a²	a³	a	a³	a	a²	a³	a²	a³	a	a	a²	a³
ba²	ba²	ba³	ba	ba	ba²	ba³	ba²	ba³	ba	ba²	ba	ba²	ba³	ba³	ba	ba²
b²a	b²a	b²a²	b²a³	b²a³	b²a	b²a²	b²a	b²a²	b²a³	b²a	b²a³	b²a	b²a²	b²a²	b²a³	b²a
b³	b³	b³a	b	b³a²	ba	b²	b³a³	ba²	b²a	b³	ba³	b²a²	b³a	b²a³	b³a²	b³a³
ba³	ba³	ba	ba²	ba²	ba³	ba	ba³	ba	ba²	ba³	ba²	ba³	ba	ba	ba²	ba³
b²a²	b²a²	b²a³	b²a	b²a	b²a²	b²a³	b²a²	b²a³	b²a	b²a²	b²a	b²a²	b²a³	b²a³	b²a	b²a²
b³a	b³a	b³a²	b³a³	b³a³	b³a	b³a²	b³a	b³a²	b³a³	b³a	b³a³	b³a	b³a²	b³a²	b³a³	b³a
b²a³	b²a³	b²a	b²a²	b²a²	b²a³	b²a	b²a³	b²a	b²a²	b²a³	b²a²	b²a³	b²a	b²a	b²a²	b²a³
b³a²	b³a²	b³a³	b³a	b³a	b³a²	b³a³	b³a²	b³a³	b³a	b³a²	b³a	b³a²	b³a³	b³a³	b³a	b³a²
b³a³	b³a³	b³a	b³a²	b³a²	b³a³	b³a	b³a³	b³a	b³a²	b³a³	b³a²	b³a³	b³a	b³a	b³a²	b³a³

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

20 unique, 171 total

Σ	#	Presentation	Description	Related
8	628	⟨a, b \| bb=aa, abab=1⟩	Finite non-Abelian group with 16 elements	58 iso
9	1331	⟨a, b \| aaaa=b, bbbb=1⟩	Isomorphic to ℤ₁₆	67 iso
10	3808	⟨a, b \| aaab=ba, abab=1⟩	Finite non-Abelian group with 16 elements	7 iso
10	4630	⟨a, b \| aaaa=a, bbbb=a⟩	Isomorphic to ℕ_{(16 = 4)}	1 iso
10	4648	⟨a, b \| aaaa=b, bbbb=a⟩	Isomorphic to ℕ_{(16 = 1)}	5 iso
10	6205	⟨a, b \| aba=b, aaaabb=1⟩	Finite non-Abelian group with 16 elements	3 iso
11	12164	⟨a, b \| aaaa=aa, bbbb=a⟩	Isomorphic to ℕ_{(16 = 8)}
11	12194	⟨a, b \| aaaa=ab, bbbb=a⟩	Isomorphic to ℕ_{(16 = 5)}	2 iso
11	12212	⟨a, b \| aaaa=bb, bbbb=a⟩	Isomorphic to ℕ_{(16 = 2)}
11	12306	⟨a, b \| aaab=bb, abba=a⟩	Finite non-commutative monoid with 16 elements	6 iso
11	13251	⟨a, b \| bab=aab, bbb=aa⟩	Finite non-commutative monoid with 16 elements
11	13259	⟨a, b \| bab=aba, bbb=aa⟩	Finite non-commutative monoid with 16 elements
11	16028	⟨a, b \| aaa=ab, babb=bb⟩	Finite non-commutative monoid with 16 elements
11	16032	⟨a, b \| aaa=ab, bbaa=bb⟩	Finite non-commutative monoid with 16 elements
11	16060	⟨a, b \| aaa=bb, abab=aa⟩	Finite non-commutative monoid with 16 elements
11	16371	⟨a, b \| aba=aa, bbbb=ab⟩	Finite non-commutative monoid with 16 elements
11	18811	⟨a, b \| aaa=b, abbbbb=b⟩	Isomorphic to ℕ_{(16 = 3)}	2 iso
11	20251	⟨a, b \| aba=b, aaaa=abb⟩	Finite non-commutative monoid with 16 elements
11	21039	⟨a, b \| ab=aa, bbba=bbb⟩	Finite non-commutative monoid with 16 elements
11	24660	⟨a, b \| aa=a, ababab=bb⟩	Finite non-commutative monoid with 16 elements

Other anti-isomorphic instances

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

4 total

Σ	#	Presentation	Mapping
10	4619	⟨a, b \| aaaa=a, aabb=b⟩	φ(a) = b, φ(b) = aa
10	4621	⟨a, b \| aaaa=a, abab=b⟩	φ(a) = b, φ(b) = aa
11	14340	⟨a, b \| aaaa=a, aabab=b⟩	φ(a) = b, φ(b) = aaa
11	14346	⟨a, b \| aaaa=a, abaab=b⟩	φ(a) = b, φ(b) = aaa

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