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〈a, b | aab=a, bbbb=b〉

Monoid presentation of length 9

Properties

Finite non-commutative monoid with 16 elements

Completion parameters

Reduction order: left-to-right a/b

Complete rewriting system

a⁴ ⇒ a
ab ⇒ a³
b⁴ ⇒ b

Idempotents

6 elements

1
a³
ba
b²a²
b³
b³a³

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	b²	a³	ba²	b²a	b³	ba³	b²a²	b³a	b²a³	b³a²	b³a³
1	1	a	b	a²	ba	b²	a³	ba²	b²a	b³	ba³	b²a²	b³a	b²a³	b³a²	b³a³
a	a	a²	a³	a³	a	a²	a	a²	a³	a	a³	a	a²	a²	a³	a
b	b	ba	b²	ba²	b²a	b³	ba³	b²a²	b³a	b	b²a³	b³a²	ba	b³a³	ba²	ba³
a²	a²	a³	a	a	a²	a³	a²	a³	a	a²	a	a²	a³	a³	a	a²
ba	ba	ba²	ba³	ba³	ba	ba²	ba	ba²	ba³	ba	ba³	ba	ba²	ba²	ba³	ba
b²	b²	b²a	b³	b²a²	b³a	b	b²a³	b³a²	ba	b²	b³a³	ba²	b²a	ba³	b²a²	b²a³
a³	a³	a	a²	a²	a³	a	a³	a	a²	a³	a²	a³	a	a	a²	a³
ba²	ba²	ba³	ba	ba	ba²	ba³	ba²	ba³	ba	ba²	ba	ba²	ba³	ba³	ba	ba²
b²a	b²a	b²a²	b²a³	b²a³	b²a	b²a²	b²a	b²a²	b²a³	b²a	b²a³	b²a	b²a²	b²a²	b²a³	b²a
b³	b³	b³a	b	b³a²	ba	b²	b³a³	ba²	b²a	b³	ba³	b²a²	b³a	b²a³	b³a²	b³a³
ba³	ba³	ba	ba²	ba²	ba³	ba	ba³	ba	ba²	ba³	ba²	ba³	ba	ba	ba²	ba³
b²a²	b²a²	b²a³	b²a	b²a	b²a²	b²a³	b²a²	b²a³	b²a	b²a²	b²a	b²a²	b²a³	b²a³	b²a	b²a²
b³a	b³a	b³a²	b³a³	b³a³	b³a	b³a²	b³a	b³a²	b³a³	b³a	b³a³	b³a	b³a²	b³a²	b³a³	b³a
b²a³	b²a³	b²a	b²a²	b²a²	b²a³	b²a	b²a³	b²a	b²a²	b²a³	b²a²	b²a³	b²a	b²a	b²a²	b²a³
b³a²	b³a²	b³a³	b³a	b³a	b³a²	b³a³	b³a²	b³a³	b³a	b³a²	b³a	b³a²	b³a³	b³a³	b³a	b³a²
b³a³	b³a³	b³a	b³a²	b³a²	b³a³	b³a	b³a³	b³a	b³a²	b³a³	b³a²	b³a³	b³a	b³a	b³a²	b³a³

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

26 unique, 175 total

Length:	Presentation:	Description:	Related:
8	〈a, b \| bb=aa, abab=1〉	Finite non-Abelian group with 16 elements	4 isomorphic
8	〈a, b \| aa=1, ababbb=1〉	Finite non-Abelian group with 16 elements	40 isomorphic
9	〈a, b \| aaaa=b, bbbb=1〉	Isomorphic to ℤ₁₆	67 isomorphic
10	〈a, b \| aaab=ba, abab=1〉	Finite non-Abelian group with 16 elements	7 isomorphic
10	〈a, b \| bab=aaa, aabb=1〉	Finite non-Abelian group with 16 elements	6 isomorphic
10	〈a, b \| aaaa=a, aabb=b〉	Finite non-commutative monoid with 16 elements	1 isomorphic
10	〈a, b \| aaaa=a, bbbb=a〉	Finite commutative monoid with 16 elements	1 isomorphic
10	〈a, b \| aaaa=b, bbbb=a〉	Finite commutative monoid with 16 elements	5 isomorphic
10	〈a, b \| aba=b, aaaabb=1〉	Finite non-Abelian group with 16 elements	3 isomorphic
10	〈a, b \| aba=b, aabbbb=1〉	Finite non-Abelian group with 16 elements	5 isomorphic
11	〈a, b \| aaaa=aa, bbbb=a〉	Finite commutative monoid with 16 elements
11	〈a, b \| aaaa=ab, bbbb=a〉	Finite commutative monoid with 16 elements	2 isomorphic
11	〈a, b \| aaaa=bb, bbbb=a〉	Finite commutative monoid with 16 elements
11	〈a, b \| aaab=bb, abba=a〉	Finite non-commutative monoid with 16 elements	2 isomorphic
11	〈a, b \| bab=aab, bbb=aa〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| bab=aba, bbb=aa〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| aaaa=a, aabab=b〉	Finite non-commutative monoid with 16 elements	1 isomorphic
11	〈a, b \| aab=bb, abbba=a〉	Finite non-commutative monoid with 16 elements	3 isomorphic
11	〈a, b \| aaa=ab, babb=bb〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| aaa=ab, bbaa=bb〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| aaa=bb, abab=aa〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| aba=aa, bbbb=ab〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| aaa=b, abbbbb=b〉	Finite commutative monoid with 16 elements	2 isomorphic
11	〈a, b \| aba=b, aaaa=abb〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| ab=aa, bbba=bbb〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| aa=a, ababab=bb〉	Finite non-commutative monoid with 16 elements