#5202 ⟨a, b | aab=bb, abba=a⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 12 elements

Element profile

1 element center:
- 1
3 idempotent elements:
- 1, a³b, a⁵

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

a⁶ ⇒ a
ba ⇒ a³
b² ⇒ a²b

# ab:aab=bb,abba=a a/b
aaaaaa=a
ba=aaa
bb=aab

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁵b
1	1	a	b	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁵b
a	a	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a	a⁵b	ab
b	b	a³	a²b	a⁴	a³b	a⁵	a⁴b	a	a⁵b	a²	ab	a²b
a²	a²	a³	a²b	a⁴	a³b	a⁵	a⁴b	a	a⁵b	a²	ab	a²b
ab	ab	a⁴	a³b	a⁵	a⁴b	a	a⁵b	a²	ab	a³	a²b	a³b
a³	a³	a⁴	a³b	a⁵	a⁴b	a	a⁵b	a²	ab	a³	a²b	a³b
a²b	a²b	a⁵	a⁴b	a	a⁵b	a²	ab	a³	a²b	a⁴	a³b	a⁴b
a⁴	a⁴	a⁵	a⁴b	a	a⁵b	a²	ab	a³	a²b	a⁴	a³b	a⁴b
a³b	a³b	a	a⁵b	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁵b
a⁵	a⁵	a	a⁵b	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a⁵b
a⁴b	a⁴b	a²	ab	a³	a²b	a⁴	a³b	a⁵	a⁴b	a	a⁵b	ab
a⁵b	a⁵b	a³	a²b	a⁴	a³b	a⁵	a⁴b	a	a⁵b	a²	ab	a²b

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

37 unique, 618 total

Σ	#	Presentation	Description	Related
7	124	⟨a, b \| aab=a, bbb=1⟩	Finite non-commutative monoid with 12 elements	23 iso, 38 anti-iso
8	458	⟨a, b \| aaaa=1, abbb=1⟩	Isomorphic to ℤ₁₂	325 iso
8	645	⟨a, b \| ab=aa, bbb=b⟩	Finite non-commutative monoid with 12 elements	1 anti-iso
9	1646	⟨a, b \| aab=bb, aba=a⟩	Finite non-commutative monoid with 12 elements	2 iso
9	1963	⟨a, b \| aba=b, aaabb=1⟩	Finite non-Abelian group with 12 elements	30 iso
9	1998	⟨a, b \| aaa=a, bbbb=a⟩	Isomorphic to ℕ_{(12 = 4)}	6 iso
9	2018	⟨a, b \| aaa=b, bbbb=a⟩	Isomorphic to ℕ_{(12 = 1)}	27 iso
9	2019	⟨a, b \| aaa=b, bbbb=b⟩	Isomorphic to ℕ_{(12 = 3)}	23 iso
9	2259	⟨a, b \| ab=aa, bbb=bb⟩	Finite non-commutative monoid with 12 elements
10	5040	⟨a, b \| aaa=aa, bbbb=a⟩	Isomorphic to ℕ_{(12 = 8)}
10	5072	⟨a, b \| aaa=ab, bbbb=a⟩	Isomorphic to ℕ_{(12 = 5)}	7 iso
10	5092	⟨a, b \| aaa=bb, bbbb=a⟩	Isomorphic to ℕ_{(12 = 2)}	43 iso
10	5330	⟨a, b \| aaa=ab, bab=bb⟩	Finite non-commutative monoid with 12 elements	1 iso
10	6597	⟨a, b \| aaa=b, bbbb=bb⟩	Isomorphic to ℕ_{(12 = 6)}	4 iso
10	6660	⟨a, b \| aab=a, bbbb=ba⟩	Finite non-commutative monoid with 12 elements	1 anti-iso
10	6661	⟨a, b \| aab=a, bbbb=bb⟩	Finite non-commutative monoid with 12 elements	1 anti-iso
10	7105	⟨a, b \| ab=aa, bbaa=bb⟩	Finite non-commutative monoid with 12 elements	2 iso
11	12897	⟨a, b \| aab=aaa, baaa=b⟩	Finite non-commutative monoid with 12 elements	5 iso
11	12910	⟨a, b \| aab=aaa, bbbb=a⟩	Isomorphic to ℕ_{(12 = 9)}	2 iso
11	15428	⟨a, b \| aaa=aa, abbbb=b⟩	Finite non-commutative monoid with 12 elements
11	15996	⟨a, b \| aaa=ab, aabb=bb⟩	Finite non-commutative monoid with 12 elements	2 iso
11	16057	⟨a, b \| aaa=bb, aabb=ab⟩	Finite non-commutative monoid with 12 elements	2 iso, 2 anti-iso
11	16104	⟨a, b \| aab=aa, abab=bb⟩	Finite non-commutative monoid with 12 elements
11	16274	⟨a, b \| aab=bb, aaaa=ab⟩	Finite non-commutative monoid with 12 elements	1 iso
11	16275	⟨a, b \| aab=bb, aaaa=ba⟩	Finite non-commutative monoid with 12 elements
11	16448	⟨a, b \| aba=bb, aabb=aa⟩	Finite non-commutative monoid with 12 elements	2 iso
11	19773	⟨a, b \| aba=b, bbbbb=aa⟩	Finite commutative monoid with 12 elements
11	19917	⟨a, b \| aaa=b, bbbb=abb⟩	Isomorphic to ℕ_{(12 = 7)}	2 iso
11	20054	⟨a, b \| aab=b, aaaa=bba⟩	Finite non-commutative monoid with 12 elements
11	20686	⟨a, b \| bb=aa, aaaaab=a⟩	Finite commutative monoid with 12 elements	15 iso
11	20787	⟨a, b \| ab=aa, baaaa=bb⟩	Finite non-commutative monoid with 12 elements	7 iso
11	21086	⟨a, b \| bb=aa, aaaa=aba⟩	Finite non-commutative monoid with 12 elements	4 iso
11	21092	⟨a, b \| bb=aa, aaab=aba⟩	Finite non-commutative monoid with 12 elements	3 iso
11	21112	⟨a, b \| bb=aa, abab=aba⟩	Finite non-commutative monoid with 12 elements
11	24147	⟨a, b \| aa=a, abbbbbb=b⟩	Finite non-commutative monoid with 12 elements
11	24991	⟨a, b \| ab=a, baaaaa=bb⟩	Finite non-commutative monoid with 12 elements
11	25603	⟨a, b \| ab=a, bbaaa=bbb⟩	Finite non-commutative monoid with 12 elements

Other isomorphic instances

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

9 total

Σ	#	Presentation	Mapping
10	5210	⟨a, b \| aab=bb, baba=a⟩	φ(a) = a, φ(b) = b
10	5214	⟨a, b \| aab=bb, bbaa=a⟩	φ(a) = a, φ(b) = b
11	12298	⟨a, b \| aaab=bb, aaba=a⟩	φ(a) = aaaa, φ(b) = b
11	12302	⟨a, b \| aaab=bb, abaa=a⟩	φ(a) = aaaa, φ(b) = b
11	12310	⟨a, b \| aaab=bb, baaa=a⟩	φ(a) = aaaa, φ(b) = b
11	15741	⟨a, b \| aab=bb, aaaba=a⟩	φ(a) = a, φ(b) = b
11	15745	⟨a, b \| aab=bb, aabaa=a⟩	φ(a) = a, φ(b) = b
11	15753	⟨a, b \| aab=bb, abaaa=a⟩	φ(a) = a, φ(b) = b
11	15769	⟨a, b \| aab=bb, baaaa=a⟩	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.

32 total

Σ	#	Presentation	Mapping
11	19443	⟨a, b \| aab=a, baaaa=bb⟩	φ(a) = aaa, φ(b) = b
11	20585	⟨a, b \| ab=aa, baaaaa=b⟩	φ(a) = b, φ(b) = aa
11	20587	⟨a, b \| ab=aa, baaaab=b⟩	φ(a) = b, φ(b) = aa
11	20589	⟨a, b \| ab=aa, baaaba=b⟩	φ(a) = b, φ(b) = aa
11	20591	⟨a, b \| ab=aa, baaabb=b⟩	φ(a) = b, φ(b) = aa
11	20593	⟨a, b \| ab=aa, baabaa=b⟩	φ(a) = b, φ(b) = aa
11	20595	⟨a, b \| ab=aa, baabab=b⟩	φ(a) = b, φ(b) = aa
11	20597	⟨a, b \| ab=aa, baabba=b⟩	φ(a) = b, φ(b) = aa
11	20599	⟨a, b \| ab=aa, baabbb=b⟩	φ(a) = b, φ(b) = aa
11	20601	⟨a, b \| ab=aa, babaaa=b⟩	φ(a) = b, φ(b) = aa
11	20603	⟨a, b \| ab=aa, babaab=b⟩	φ(a) = b, φ(b) = aa
11	20605	⟨a, b \| ab=aa, bababa=b⟩	φ(a) = b, φ(b) = aa
11	20607	⟨a, b \| ab=aa, bababb=b⟩	φ(a) = b, φ(b) = aa
11	20609	⟨a, b \| ab=aa, babbaa=b⟩	φ(a) = b, φ(b) = aa
11	20611	⟨a, b \| ab=aa, babbab=b⟩	φ(a) = b, φ(b) = aa
11	20613	⟨a, b \| ab=aa, babbba=b⟩	φ(a) = b, φ(b) = aa
11	20615	⟨a, b \| ab=aa, babbbb=b⟩	φ(a) = b, φ(b) = aa
11	20617	⟨a, b \| ab=aa, bbaaaa=b⟩	φ(a) = b, φ(b) = aa
11	20619	⟨a, b \| ab=aa, bbaaab=b⟩	φ(a) = b, φ(b) = aa
11	20621	⟨a, b \| ab=aa, bbaaba=b⟩	φ(a) = b, φ(b) = aa
11	20623	⟨a, b \| ab=aa, bbaabb=b⟩	φ(a) = b, φ(b) = aa
11	20625	⟨a, b \| ab=aa, bbabaa=b⟩	φ(a) = b, φ(b) = aa
11	20627	⟨a, b \| ab=aa, bbabab=b⟩	φ(a) = b, φ(b) = aa
11	20629	⟨a, b \| ab=aa, bbabba=b⟩	φ(a) = b, φ(b) = aa
11	20631	⟨a, b \| ab=aa, bbabbb=b⟩	φ(a) = b, φ(b) = aa
11	20633	⟨a, b \| ab=aa, bbbaaa=b⟩	φ(a) = b, φ(b) = aa
11	20635	⟨a, b \| ab=aa, bbbaab=b⟩	φ(a) = b, φ(b) = aa
11	20637	⟨a, b \| ab=aa, bbbaba=b⟩	φ(a) = b, φ(b) = aa
11	20639	⟨a, b \| ab=aa, bbbabb=b⟩	φ(a) = b, φ(b) = aa
11	20641	⟨a, b \| ab=aa, bbbbaa=b⟩	φ(a) = b, φ(b) = aa
11	20643	⟨a, b \| ab=aa, bbbbab=b⟩	φ(a) = b, φ(b) = aa
11	20645	⟨a, b \| ab=aa, bbbbba=b⟩	φ(a) = b, φ(b) = aa

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