#63 ⟨a, b | ab=a, ba=b⟩

Properties

Presentation has sum-of-sides 6
Finite non-commutative monoid with 3 elements

Elements

Elements in the center commute with all other elements.
A left zero element x satisfies xy = x for all y.
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.

1 element center:
- 1
2 left zero elements:
- a, b
2 non-trivial idempotents:
- a, b
Order of generators:
- a: index 1, period 1
- b: index 1, period 1
Histogram:

index 1, period 1 2 elements a, b

index 1, period 1	2 elements	a, b

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 shortlex with a < b

a² ⇒ a
ab ⇒ a
ba ⇒ b
b² ⇒ b

# ab:ab=a,ba=b ab
aa=a
ab=a
ba=b
bb=b

Cayley table

Idempotents are shown in bold.

	1	a	b
1	1	a	b
a	a	a	a
b	b	b	b

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

4 unique, 2129 total

Σ	#	Presentation	Description	Related
5	6	⟨a, b \| aa=b, ab=1⟩	Isomorphic to ℤ₃	2029 iso
6	59	⟨a, b \| aa=b, ab=a⟩	Isomorphic to ℕ_{(3 = 1)}	61 iso
6	60	⟨a, b \| aa=b, ab=b⟩	Isomorphic to ℕ_{(3 = 2)}	23 iso
8	891	⟨a, b \| aa=a, abba=b⟩	Finite commutative monoid with 3 elements	12 iso

Other isomorphic instances

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

61 total

Σ	#	Presentation	Mapping
7	270	⟨a, b \| ab=a, bab=b⟩	φ(a) = a, φ(b) = b
7	272	⟨a, b \| ab=a, bba=b⟩	φ(a) = a, φ(b) = b
8	943	⟨a, b \| ab=a, babb=b⟩	φ(a) = a, φ(b) = b
8	947	⟨a, b \| ab=a, bbab=b⟩	φ(a) = a, φ(b) = b
8	949	⟨a, b \| ab=a, bbba=b⟩	φ(a) = a, φ(b) = b
9	2039	⟨a, b \| aab=a, baab=b⟩	φ(a) = a, φ(b) = b
9	2041	⟨a, b \| aab=a, baba=b⟩	φ(a) = a, φ(b) = b
9	2043	⟨a, b \| aab=a, babb=b⟩	φ(a) = a, φ(b) = b
9	2045	⟨a, b \| aab=a, bbaa=b⟩	φ(a) = a, φ(b) = b
9	2047	⟨a, b \| aab=a, bbab=b⟩	φ(a) = a, φ(b) = b
9	2049	⟨a, b \| aab=a, bbba=b⟩	φ(a) = a, φ(b) = b
9	2983	⟨a, b \| ab=a, babbb=b⟩	φ(a) = a, φ(b) = b
9	2991	⟨a, b \| ab=a, bbabb=b⟩	φ(a) = a, φ(b) = b
9	2995	⟨a, b \| ab=a, bbbab=b⟩	φ(a) = a, φ(b) = b
9	2997	⟨a, b \| ab=a, bbbba=b⟩	φ(a) = a, φ(b) = b
10	8807	⟨a, b \| ab=a, babbbb=b⟩	φ(a) = a, φ(b) = b
10	8823	⟨a, b \| ab=a, bbabbb=b⟩	φ(a) = a, φ(b) = b
10	8831	⟨a, b \| ab=a, bbbabb=b⟩	φ(a) = a, φ(b) = b
10	8835	⟨a, b \| ab=a, bbbbab=b⟩	φ(a) = a, φ(b) = b
10	8837	⟨a, b \| ab=a, bbbbba=b⟩	φ(a) = a, φ(b) = b
11	14444	⟨a, b \| aaab=a, baaab=b⟩	φ(a) = a, φ(b) = b
11	14446	⟨a, b \| aaab=a, baaba=b⟩	φ(a) = a, φ(b) = b
11	14450	⟨a, b \| aaab=a, babaa=b⟩	φ(a) = a, φ(b) = b
11	14458	⟨a, b \| aaab=a, bbaaa=b⟩	φ(a) = a, φ(b) = b
11	14574	⟨a, b \| aaba=a, baaba=b⟩	φ(a) = a, φ(b) = b
11	14578	⟨a, b \| aaba=a, babaa=b⟩	φ(a) = a, φ(b) = b
11	14586	⟨a, b \| aaba=a, bbaaa=b⟩	φ(a) = a, φ(b) = b
11	14712	⟨a, b \| aabb=a, babbb=b⟩	φ(a) = a, φ(b) = b
11	14720	⟨a, b \| aabb=a, bbabb=b⟩	φ(a) = a, φ(b) = b
11	14724	⟨a, b \| aabb=a, bbbab=b⟩	φ(a) = a, φ(b) = b
11	14726	⟨a, b \| aabb=a, bbbba=b⟩	φ(a) = a, φ(b) = b
11	14776	⟨a, b \| abab=a, babbb=b⟩	φ(a) = a, φ(b) = b
11	14784	⟨a, b \| abab=a, bbabb=b⟩	φ(a) = a, φ(b) = b
11	14788	⟨a, b \| abab=a, bbbab=b⟩	φ(a) = a, φ(b) = b
11	14790	⟨a, b \| abab=a, bbbba=b⟩	φ(a) = a, φ(b) = b
11	18903	⟨a, b \| aab=a, baaabb=b⟩	φ(a) = a, φ(b) = b
11	18907	⟨a, b \| aab=a, baabab=b⟩	φ(a) = a, φ(b) = b
11	18909	⟨a, b \| aab=a, baabba=b⟩	φ(a) = a, φ(b) = b
11	18911	⟨a, b \| aab=a, baabbb=b⟩	φ(a) = a, φ(b) = b
11	18915	⟨a, b \| aab=a, babaab=b⟩	φ(a) = a, φ(b) = b
11	18917	⟨a, b \| aab=a, bababa=b⟩	φ(a) = a, φ(b) = b
11	18919	⟨a, b \| aab=a, bababb=b⟩	φ(a) = a, φ(b) = b
11	18921	⟨a, b \| aab=a, babbaa=b⟩	φ(a) = a, φ(b) = b
11	18923	⟨a, b \| aab=a, babbab=b⟩	φ(a) = a, φ(b) = b
11	18925	⟨a, b \| aab=a, babbba=b⟩	φ(a) = a, φ(b) = b
11	18931	⟨a, b \| aab=a, bbaaab=b⟩	φ(a) = a, φ(b) = b
11	18933	⟨a, b \| aab=a, bbaaba=b⟩	φ(a) = a, φ(b) = b
11	18935	⟨a, b \| aab=a, bbaabb=b⟩	φ(a) = a, φ(b) = b
11	18937	⟨a, b \| aab=a, bbabaa=b⟩	φ(a) = a, φ(b) = b
11	18939	⟨a, b \| aab=a, bbabab=b⟩	φ(a) = a, φ(b) = b
11	18941	⟨a, b \| aab=a, bbabba=b⟩	φ(a) = a, φ(b) = b
11	18945	⟨a, b \| aab=a, bbbaaa=b⟩	φ(a) = a, φ(b) = b
11	18947	⟨a, b \| aab=a, bbbaab=b⟩	φ(a) = a, φ(b) = b
11	18949	⟨a, b \| aab=a, bbbaba=b⟩	φ(a) = a, φ(b) = b
11	18953	⟨a, b \| aab=a, bbbbaa=b⟩	φ(a) = a, φ(b) = b
11	24523	⟨a, b \| ab=a, babbbbb=b⟩	φ(a) = a, φ(b) = b
11	24555	⟨a, b \| ab=a, bbabbbb=b⟩	φ(a) = a, φ(b) = b
11	24571	⟨a, b \| ab=a, bbbabbb=b⟩	φ(a) = a, φ(b) = b
11	24579	⟨a, b \| ab=a, bbbbabb=b⟩	φ(a) = a, φ(b) = b
11	24583	⟨a, b \| ab=a, bbbbbab=b⟩	φ(a) = a, φ(b) = b
11	24585	⟨a, b \| ab=a, bbbbbba=b⟩	φ(a) = a, φ(b) = b

Other anti-isomorphic instances

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

17 total

Σ	#	Presentation	Mapping
9	2091	⟨a, b \| aba=a, aabb=b⟩	φ(a) = b, φ(b) = a
9	2093	⟨a, b \| aba=a, abab=b⟩	φ(a) = b, φ(b) = a
9	2097	⟨a, b \| aba=a, abbb=b⟩	φ(a) = b, φ(b) = a
11	14544	⟨a, b \| aaba=a, aaabb=b⟩	φ(a) = b, φ(b) = a
11	14548	⟨a, b \| aaba=a, aabab=b⟩	φ(a) = b, φ(b) = a
11	14556	⟨a, b \| aaba=a, abaab=b⟩	φ(a) = b, φ(b) = a
11	14820	⟨a, b \| abba=a, abbbb=b⟩	φ(a) = b, φ(b) = a
11	19103	⟨a, b \| aba=a, aaabbb=b⟩	φ(a) = b, φ(b) = a
11	19109	⟨a, b \| aba=a, aababb=b⟩	φ(a) = b, φ(b) = a
11	19113	⟨a, b \| aba=a, aabbab=b⟩	φ(a) = b, φ(b) = a
11	19117	⟨a, b \| aba=a, aabbbb=b⟩	φ(a) = b, φ(b) = a
11	19123	⟨a, b \| aba=a, abaabb=b⟩	φ(a) = b, φ(b) = a
11	19125	⟨a, b \| aba=a, ababab=b⟩	φ(a) = b, φ(b) = a
11	19129	⟨a, b \| aba=a, ababbb=b⟩	φ(a) = b, φ(b) = a
11	19131	⟨a, b \| aba=a, abbaab=b⟩	φ(a) = b, φ(b) = a
11	19133	⟨a, b \| aba=a, abbabb=b⟩	φ(a) = b, φ(b) = a
11	19135	⟨a, b \| aba=a, abbbab=b⟩	φ(a) = b, φ(b) = a

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