#8617 ⟨a, b | aa=a, abbbba=b⟩

Properties

Sum of relation sides is 10
Finite commutative monoid with 5 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ a = a and a ⋅ 1 = a, however a ≠ 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.

2 non-trivial idempotents:
- a, b³
Order of generators:
- a: index 1, period 1
- b: index 1, period 3
Histogram:

index 1, period 1 2 elements a, b³

index 1, period 3 2 elements b, b²

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

# ab:aa=a,abbbba=b ab - -
aa=a
ab=b
ba=b
bbbb=b

11 unique, 1459 total

Σ	#	Presentation	Properties	Description	φ
6	44	⟨a, b \| aa=b, abb=1⟩	Grp Com 5	ℤ₅	1132
7	253	⟨a, b \| aa=b, abb=a⟩	Noc Com 5	ℕ_{(5 = 1)}	71
7	254	⟨a, b \| aa=b, abb=b⟩	Noc Com 5	ℕ_{(5 = 2)}	43
7	268	⟨a, b \| ab=a, baa=b⟩	Noc 5		86
8	574	⟨a, b \| aaa=b, aab=b⟩	Noc Com 5	ℕ_{(5 = 3)}	27
8	950	⟨a, b \| ab=a, bbbb=a⟩	Noc Com 5	ℕ_{(5 = 4)}	32
8	995	⟨a, b \| ab=a, aaa=bb⟩	Noc Com 5		19
8	1019	⟨a, b \| ab=a, bba=bb⟩	Noc 5		25
8	1020	⟨a, b \| ab=a, bbb=aa⟩	Noc Com 5		9
8	1022	⟨a, b \| ab=a, bbb=ba⟩	Noc 5		4
11	15426	⟨a, b \| aaa=aa, abbba=b⟩	Noc Com 5

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

3 total

Σ	#	Presentation	Mapping
11	24101	⟨a, b \| aa=a, aabbbba=b⟩	φ(a) = a, φ(b) = b
11	24125	⟨a, b \| aa=a, ababbba=b⟩	φ(a) = a, φ(b) = b
11	24135	⟨a, b \| aa=a, abbabba=b⟩	φ(a) = a, φ(b) = b

index 1, period 1	2 elements	a, b³
index 1, period 3	2 elements	b, b²