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〈a, b | aaaa=a, aabab=b〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 16 elements

Completion parameters

Reduction order: left-to-right ab

Complete rewriting system

b² ⇒ b
bab ⇒ ab
a⁴ ⇒ a
a³b ⇒ b
ba²b ⇒ a²b

Idempotents

6 elements

1
b
a³
a²ba
aba²
ba³

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	a³	a²b	aba	ba²	a²ba	aba²	ba³	a²ba²	aba³	a²ba³
1	1	a	b	a²	ab	ba	a³	a²b	aba	ba²	a²ba	aba²	ba³	a²ba²	aba³	a²ba³
a	a	a²	ab	a³	a²b	aba	a	b	a²ba	aba²	ba	a²ba²	aba³	ba²	a²ba³	ba³
b	b	ba	b	ba²	ab	ba	ba³	a²b	aba	ba²	a²ba	aba²	ba³	a²ba²	aba³	a²ba³
a²	a²	a³	a²b	a	b	a²ba	a²	ab	ba	a²ba²	aba	ba²	a²ba³	aba²	ba³	aba³
ab	ab	aba	ab	aba²	a²b	aba	aba³	b	a²ba	aba²	ba	a²ba²	aba³	ba²	a²ba³	ba³
ba	ba	ba²	ab	ba³	a²b	aba	ba	b	a²ba	aba²	ba	a²ba²	aba³	ba²	a²ba³	ba³
a³	a³	a	b	a²	ab	ba	a³	a²b	aba	ba²	a²ba	aba²	ba³	a²ba²	aba³	a²ba³
a²b	a²b	a²ba	a²b	a²ba²	b	a²ba	a²ba³	ab	ba	a²ba²	aba	ba²	a²ba³	aba²	ba³	aba³
aba	aba	aba²	a²b	aba³	b	a²ba	aba	ab	ba	a²ba²	aba	ba²	a²ba³	aba²	ba³	aba³
ba²	ba²	ba³	a²b	ba	b	a²ba	ba²	ab	ba	a²ba²	aba	ba²	a²ba³	aba²	ba³	aba³
a²ba	a²ba	a²ba²	b	a²ba³	ab	ba	a²ba	a²b	aba	ba²	a²ba	aba²	ba³	a²ba²	aba³	a²ba³
aba²	aba²	aba³	b	aba	ab	ba	aba²	a²b	aba	ba²	a²ba	aba²	ba³	a²ba²	aba³	a²ba³
ba³	ba³	ba	b	ba²	ab	ba	ba³	a²b	aba	ba²	a²ba	aba²	ba³	a²ba²	aba³	a²ba³
a²ba²	a²ba²	a²ba³	ab	a²ba	a²b	aba	a²ba²	b	a²ba	aba²	ba	a²ba²	aba³	ba²	a²ba³	ba³
aba³	aba³	aba	ab	aba²	a²b	aba	aba³	b	a²ba	aba²	ba	a²ba²	aba³	ba²	a²ba³	ba³
a²ba³	a²ba³	a²ba	a²b	a²ba²	b	a²ba	a²ba³	ab	ba	a²ba²	aba	ba²	a²ba³	aba²	ba³	aba³

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

26 unique, 174 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
9	〈a, b \| aab=a, bbbb=b〉	Finite non-commutative monoid with 16 elements
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 \| 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

Other isomorphic instances

1 total

Length:	Presentation:
11	〈a, b \| aaaa=a, abaab=b〉