#12306 ⟨a, b | aaab=bb, abba=a⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 16 elements

Element profile

1 element center:
- 1
3 idempotent elements:
- 1, a⁴b, a⁷

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

a⁸ ⇒ a
ba ⇒ a⁴
b² ⇒ a³b

# ab:aaab=bb,abba=a a/b
aaaaaaaa=a
ba=aaaa
bb=aaab

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a⁷b
1	1	a	b	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a⁷b
a	a	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a	a⁷b	ab
b	b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a	a⁷b	a²	ab	a³	a²b	a³b
a²	a²	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a	a⁷b	a²	ab	a²b
ab	ab	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a	a⁷b	a²	ab	a³	a²b	a⁴	a³b	a⁴b
a³	a³	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a	a⁷b	a²	ab	a³	a²b	a³b
a²b	a²b	a⁶	a⁵b	a⁷	a⁶b	a	a⁷b	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁵b
a⁴	a⁴	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a	a⁷b	a²	ab	a³	a²b	a⁴	a³b	a⁴b
a³b	a³b	a⁷	a⁶b	a	a⁷b	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁶b
a⁵	a⁵	a⁶	a⁵b	a⁷	a⁶b	a	a⁷b	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁵b
a⁴b	a⁴b	a	a⁷b	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a⁷b
a⁶	a⁶	a⁷	a⁶b	a	a⁷b	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁶b
a⁵b	a⁵b	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a	a⁷b	ab
a⁷	a⁷	a	a⁷b	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a⁷b
a⁶b	a⁶b	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a	a⁷b	a²	ab	a²b
a⁷b	a⁷b	a⁴	a³b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a	a⁷b	a²	ab	a³	a²b	a³b

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

20 unique, 169 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
9	2051	⟨a, b \| aab=a, bbbb=b⟩	Finite non-commutative monoid with 16 elements	4 anti-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	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 isomorphic instances

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

6 total

Σ	#	Presentation	Mapping
11	12314	⟨a, b \| aaab=bb, baba=a⟩	φ(a) = a, φ(b) = b
11	12318	⟨a, b \| aaab=bb, bbaa=a⟩	φ(a) = a, φ(b) = b
11	15765	⟨a, b \| aab=bb, abbba=a⟩	φ(a) = aaaaa, φ(b) = b
11	15781	⟨a, b \| aab=bb, babba=a⟩	φ(a) = aaaaa, φ(b) = b
11	15789	⟨a, b \| aab=bb, bbaba=a⟩	φ(a) = aaaaa, φ(b) = b
11	15793	⟨a, b \| aab=bb, bbbaa=a⟩	φ(a) = aaaaa, φ(b) = b

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