#4648 ⟨a, b | aaaa=b, bbbb=a⟩

Properties

Sum of relation sides is 10
Isomorphic to ℕ_{(16 = 1)}
Finite commutative monoid with 16 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ a¹⁵ = a and a ⋅ 1 = a, however a¹⁵ ≠ 1

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.

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

#	Rule	Proof
1.	a¹⁶ ⇒ a	[2]
2.	b ⇒ a⁴	[1]

# ab:aaaa=b,bbbb=a a/b - -
aaaaaaaaaaaaaaaa=a
b=aaaa

20 unique, 170 total

Σ	#	Presentation	Properties	Description	φ
8	628	⟨a, b \| bb=aa, abab=1⟩	Grp 16		58
9	1331	⟨a, b \| aaaa=b, bbbb=1⟩	Grp Com 16	ℤ₁₆	67
9	2051	⟨a, b \| aab=a, bbbb=b⟩	Noc 16		4
10	3808	⟨a, b \| aaab=ba, abab=1⟩	Grp 16		7
10	4630	⟨a, b \| aaaa=a, bbbb=a⟩	Noc Com 16	ℕ_{(16 = 4)}	1
10	6205	⟨a, b \| aba=b, aaaabb=1⟩	Grp 16		3
11	12164	⟨a, b \| aaaa=aa, bbbb=a⟩	Noc Com 16	ℕ_{(16 = 8)}
11	12194	⟨a, b \| aaaa=ab, bbbb=a⟩	Noc Com 16	ℕ_{(16 = 5)}	2
11	12212	⟨a, b \| aaaa=bb, bbbb=a⟩	Noc Com 16	ℕ_{(16 = 2)}
11	12306	⟨a, b \| aaab=bb, abba=a⟩	Noc 16		6
11	13251	⟨a, b \| bab=aab, bbb=aa⟩	Noc 16
11	13259	⟨a, b \| bab=aba, bbb=aa⟩	Noc 16
11	16028	⟨a, b \| aaa=ab, babb=bb⟩	Noc 16
11	16032	⟨a, b \| aaa=ab, bbaa=bb⟩	Noc 16
11	16060	⟨a, b \| aaa=bb, abab=aa⟩	Noc 16
11	16371	⟨a, b \| aba=aa, bbbb=ab⟩	Noc 16
11	18811	⟨a, b \| aaa=b, abbbbb=b⟩	Noc Com 16	ℕ_{(16 = 3)}	2
11	20251	⟨a, b \| aba=b, aaaa=abb⟩	Noc 16
11	21039	⟨a, b \| ab=aa, bbba=bbb⟩	Noc 16
11	24660	⟨a, b \| aa=a, ababab=bb⟩	Noc 16

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

5 total

Σ	#	Presentation	Mapping
11	14535	⟨a, b \| aaab=b, bbbbb=a⟩	φ(a) = bbbbb, φ(b) = b
11	14663	⟨a, b \| aaba=b, bbbbb=a⟩	φ(a) = bbbbb, φ(b) = b
11	18810	⟨a, b \| aaa=b, abbbbb=a⟩	φ(a) = b, φ(b) = bbb
11	18824	⟨a, b \| aaa=b, babbbb=a⟩	φ(a) = b, φ(b) = bbb
11	18828	⟨a, b \| aaa=b, bbabbb=a⟩	φ(a) = b, φ(b) = bbb