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〈a, b | abb=aaa, bbb=ba〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 15 elements

Completion parameters

Reduction order: left-to-right b/a

Complete rewriting system

b⁷ ⇒ b⁵
ab⁶ ⇒ ab⁴
ba ⇒ b³
a²b² ⇒ ab⁴
a³ ⇒ ab²

Idempotents

3 elements

1
b⁶
ab⁴

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	b²	a²b	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶
1	1	a	b	a²	ab	b²	a²b	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶
a	a	a²	ab	ab²	a²b	ab²	ab³	ab⁴	ab³	ab⁵	ab⁴	ab⁴	ab⁵	ab⁵	ab⁴
b	b	b³	b²	b⁵	b⁴	b³	b⁶	b⁵	b⁴	b⁶	b⁵	b⁵	b⁶	b⁶	b⁵
a²	a²	ab²	a²b	ab⁴	ab³	ab⁴	ab⁵	ab⁴	ab⁵	ab⁵	ab⁴	ab⁴	ab⁵	ab⁵	ab⁴
ab	ab	ab³	ab²	ab⁵	ab⁴	ab³	ab⁴	ab⁵	ab⁴	ab⁴	ab⁵	ab⁵	ab⁴	ab⁴	ab⁵
b²	b²	b⁴	b³	b⁶	b⁵	b⁴	b⁵	b⁶	b⁵	b⁵	b⁶	b⁶	b⁵	b⁵	b⁶
a²b	a²b	ab⁵	ab⁴	ab⁵	ab⁴	ab⁵	ab⁴	ab⁵	ab⁴	ab⁴	ab⁵	ab⁵	ab⁴	ab⁴	ab⁵
ab²	ab²	ab⁴	ab³	ab⁴	ab⁵	ab⁴	ab⁵	ab⁴	ab⁵	ab⁵	ab⁴	ab⁴	ab⁵	ab⁵	ab⁴
b³	b³	b⁵	b⁴	b⁵	b⁶	b⁵	b⁶	b⁵	b⁶	b⁶	b⁵	b⁵	b⁶	b⁶	b⁵
ab³	ab³	ab⁵	ab⁴	ab⁵	ab⁴	ab⁵	ab⁴	ab⁵	ab⁴	ab⁴	ab⁵	ab⁵	ab⁴	ab⁴	ab⁵
b⁴	b⁴	b⁶	b⁵	b⁶	b⁵	b⁶	b⁵	b⁶	b⁵	b⁵	b⁶	b⁶	b⁵	b⁵	b⁶
ab⁴	ab⁴	ab⁴	ab⁵	ab⁴	ab⁵	ab⁴	ab⁵	ab⁴	ab⁵	ab⁵	ab⁴	ab⁴	ab⁵	ab⁵	ab⁴
b⁵	b⁵	b⁵	b⁶	b⁵	b⁶	b⁵	b⁶	b⁵	b⁶	b⁶	b⁵	b⁵	b⁶	b⁶	b⁵
ab⁵	ab⁵	ab⁵	ab⁴	ab⁵	ab⁴	ab⁵	ab⁴	ab⁵	ab⁴	ab⁴	ab⁵	ab⁵	ab⁴	ab⁴	ab⁵
b⁶	b⁶	b⁶	b⁵	b⁶	b⁵	b⁶	b⁵	b⁶	b⁵	b⁵	b⁶	b⁶	b⁵	b⁵	b⁶

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

23 unique, 190 total

Length:	Presentation:	Description:	Related:
9	〈a, b \| aaa=bb, aba=a〉	Finite non-commutative monoid with 15 elements	2 isomorphic
9	〈a, b \| aaa=bb, aba=b〉	Finite non-commutative monoid with 15 elements	1 isomorphic
9	〈a, b \| aaab=1, bbbbb=1〉	Isomorphic to ℤ₁₅	138 isomorphic
9	〈a, b \| aaa=a, abbb=b〉	Finite non-commutative monoid with 15 elements	4 isomorphic
10	〈a, b \| aaa=bb, aba=aa〉	Finite non-commutative monoid with 15 elements	1 isomorphic
10	〈a, b \| aaa=bb, aba=bb〉	Finite non-commutative monoid with 15 elements	2 isomorphic
10	〈a, b \| aaa=a, bbbbb=a〉	Finite commutative monoid with 15 elements
10	〈a, b \| aaa=b, bbbbb=a〉	Finite commutative monoid with 15 elements
10	〈a, b \| aaa=b, bbbbb=b〉	Finite commutative monoid with 15 elements
11	〈a, b \| abb=aaa, baaa=b〉	Finite non-commutative monoid with 15 elements	3 isomorphic
11	〈a, b \| abb=aaa, baba=b〉	Finite non-commutative monoid with 15 elements	1 isomorphic
11	〈a, b \| aab=aaa, aba=bb〉	Finite non-commutative monoid with 15 elements
11	〈a, b \| abb=aaa, bbb=ab〉	Finite non-commutative monoid with 15 elements	1 isomorphic
11	〈a, b \| aaa=aa, bbbbb=a〉	Finite commutative monoid with 15 elements
11	〈a, b \| aaa=ab, baabb=b〉	Finite non-commutative monoid with 15 elements	5 isomorphic
11	〈a, b \| aaa=ab, bbbbb=a〉	Finite commutative monoid with 15 elements	1 isomorphic
11	〈a, b \| aaa=bb, bbbbb=a〉	Finite commutative monoid with 15 elements	1 isomorphic
11	〈a, b \| aaa=ab, bbbb=aa〉	Finite non-commutative monoid with 15 elements
11	〈a, b \| aab=aa, bbbb=ab〉	Finite non-commutative monoid with 15 elements
11	〈a, b \| aaa=b, bbbbb=ab〉	Finite commutative monoid with 15 elements
11	〈a, b \| aab=a, bbbbb=ba〉	Finite non-commutative monoid with 15 elements	1 anti-isomorphic
11	〈a, b \| aab=b, bbaa=aaa〉	Finite non-commutative monoid with 15 elements
11	〈a, b \| ab=aa, bbaaa=bb〉	Finite non-commutative monoid with 15 elements	6 isomorphic