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〈a, b | aaa=a, ababbb=b〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 21 elements

Completion parameters

Reduction order: left-to-right ab

Complete rewriting system

a³ ⇒ a
a²b ⇒ b
bab ⇒ ab²
b⁴ ⇒ b

Idempotents

5 elements

1
a²
b³
ab³a
b³a²

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	b²	aba	ab²	ba²	b²a	b³	aba²	ab²a	ab³	b²a²	b³a	ab²a²	ab³a	b³a²	ab³a²
1	1	a	b	a²	ab	ba	b²	aba	ab²	ba²	b²a	b³	aba²	ab²a	ab³	b²a²	b³a	ab²a²	ab³a	b³a²	ab³a²
a	a	a²	ab	a	b	aba	ab²	ba	b²	aba²	ab²a	ab³	ba²	b²a	b³	ab²a²	ab³a	b²a²	b³a	ab³a²	b³a²
b	b	ba	b²	ba²	ab²	b²a	b³	ab²a	ab³	b²a²	b³a	b	ab²a²	ab³a	ab	b³a²	ba	ab³a²	aba	ba²	aba²
a²	a²	a	b	a²	ab	ba	b²	aba	ab²	ba²	b²a	b³	aba²	ab²a	ab³	b²a²	b³a	ab²a²	ab³a	b³a²	ab³a²
ab	ab	aba	ab²	aba²	b²	ab²a	ab³	b²a	b³	ab²a²	ab³a	ab	b²a²	b³a	b	ab³a²	aba	b³a²	ba	aba²	ba²
ba	ba	ba²	ab²	ba	b²	ab²a	ab³	b²a	b³	ab²a²	ab³a	ab	b²a²	b³a	b	ab³a²	aba	b³a²	ba	aba²	ba²
b²	b²	b²a	b³	b²a²	ab³	b³a	b	ab³a	ab	b³a²	ba	b²	ab³a²	aba	ab²	ba²	b²a	aba²	ab²a	b²a²	ab²a²
aba	aba	aba²	b²	aba	ab²	b²a	b³	ab²a	ab³	b²a²	b³a	b	ab²a²	ab³a	ab	b³a²	ba	ab³a²	aba	ba²	aba²
ab²	ab²	ab²a	ab³	ab²a²	b³	ab³a	ab	b³a	b	ab³a²	aba	ab²	b³a²	ba	b²	aba²	ab²a	ba²	b²a	ab²a²	b²a²
ba²	ba²	ba	b²	ba²	ab²	b²a	b³	ab²a	ab³	b²a²	b³a	b	ab²a²	ab³a	ab	b³a²	ba	ab³a²	aba	ba²	aba²
b²a	b²a	b²a²	ab³	b²a	b³	ab³a	ab	b³a	b	ab³a²	aba	ab²	b³a²	ba	b²	aba²	ab²a	ba²	b²a	ab²a²	b²a²
b³	b³	b³a	b	b³a²	ab	ba	b²	aba	ab²	ba²	b²a	b³	aba²	ab²a	ab³	b²a²	b³a	ab²a²	ab³a	b³a²	ab³a²
aba²	aba²	aba	ab²	aba²	b²	ab²a	ab³	b²a	b³	ab²a²	ab³a	ab	b²a²	b³a	b	ab³a²	aba	b³a²	ba	aba²	ba²
ab²a	ab²a	ab²a²	b³	ab²a	ab³	b³a	b	ab³a	ab	b³a²	ba	b²	ab³a²	aba	ab²	ba²	b²a	aba²	ab²a	b²a²	ab²a²
ab³	ab³	ab³a	ab	ab³a²	b	aba	ab²	ba	b²	aba²	ab²a	ab³	ba²	b²a	b³	ab²a²	ab³a	b²a²	b³a	ab³a²	b³a²
b²a²	b²a²	b²a	b³	b²a²	ab³	b³a	b	ab³a	ab	b³a²	ba	b²	ab³a²	aba	ab²	ba²	b²a	aba²	ab²a	b²a²	ab²a²
b³a	b³a	b³a²	ab	b³a	b	aba	ab²	ba	b²	aba²	ab²a	ab³	ba²	b²a	b³	ab²a²	ab³a	b²a²	b³a	ab³a²	b³a²
ab²a²	ab²a²	ab²a	ab³	ab²a²	b³	ab³a	ab	b³a	b	ab³a²	aba	ab²	b³a²	ba	b²	aba²	ab²a	ba²	b²a	ab²a²	b²a²
ab³a	ab³a	ab³a²	b	ab³a	ab	ba	b²	aba	ab²	ba²	b²a	b³	aba²	ab²a	ab³	b²a²	b³a	ab²a²	ab³a	b³a²	ab³a²
b³a²	b³a²	b³a	b	b³a²	ab	ba	b²	aba	ab²	ba²	b²a	b³	aba²	ab²a	ab³	b²a²	b³a	ab²a²	ab³a	b³a²	ab³a²
ab³a²	ab³a²	ab³a	ab	ab³a²	b	aba	ab²	ba	b²	aba²	ab²a	ab³	ba²	b²a	b³	ab²a²	ab³a	b²a²	b³a	ab³a²	b³a²

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

12 unique, 63 total

Length:	Presentation:	Description:	Related:
8	〈a, b \| aaa=ab, bbb=1〉	Finite non-commutative monoid with 21 elements	6 isomorphic, 9 anti-isomorphic
8	〈a, b \| aaa=1, abbb=b〉	Finite non-commutative monoid with 21 elements	10 isomorphic, 5 anti-isomorphic
9	〈a, b \| aaa=ab, bbb=b〉	Finite non-commutative monoid with 21 elements	1 anti-isomorphic
9	〈a, b \| aaa=1, babb=ab〉	Finite non-commutative monoid with 21 elements	6 isomorphic
10	〈a, b \| aaa=aa, bbb=ab〉	Finite non-commutative monoid with 21 elements
10	〈a, b \| aaa=a, abbbb=b〉	Finite non-commutative monoid with 21 elements
11	〈a, b \| aaaaa=b, abbbb=1〉	Isomorphic to ℤ₂₁	13 isomorphic
11	〈a, b \| bab=aaa, abba=b〉	Finite non-commutative monoid with 21 elements
11	〈a, b \| bab=aaa, bbb=aa〉	Finite non-commutative monoid with 21 elements
11	〈a, b \| aab=ba, ababab=1〉	Finite non-Abelian group with 21 elements	1 isomorphic
11	〈a, b \| aba=b, baaaab=a〉	Finite non-commutative monoid with 21 elements
11	〈a, b \| aaa=1, abbab=aba〉	Finite non-commutative monoid with 21 elements

Other isomorphic instances

1 total

Length:	Presentation:
11	〈a, b \| aaa=a, abbabb=b〉