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〈a, b | aaa=bb, aba=a〉

Monoid presentation of length 9

Properties

Finite non-commutative monoid with 15 elements

Completion parameters

Reduction order: left-to-right a/b

Complete rewriting system

a⁸ ⇒ a³
ba³ ⇒ a⁷
aba ⇒ a
a³b ⇒ a⁷
b² ⇒ a³

Idempotents

4 elements

1
a⁵
ba
ab

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	a³	a²b	ba²	bab	a⁴	ba²b	a⁵	a⁶	a⁷
1	1	a	b	a²	ab	ba	a³	a²b	ba²	bab	a⁴	ba²b	a⁵	a⁶	a⁷
a	a	a²	ab	a³	a²b	a	a⁴	a⁷	a²	ab	a⁵	a²b	a⁶	a⁷	a³
b	b	ba	a³	ba²	bab	a⁴	a⁷	ba²b	a⁵	a³	a³	a⁴	a⁴	a⁵	a⁶
a²	a²	a³	a²b	a⁴	a⁷	a²	a⁵	a³	a³	a²b	a⁶	a⁷	a⁷	a³	a⁴
ab	ab	a	a⁴	a²	ab	a⁵	a³	a²b	a⁶	a⁴	a⁴	a⁵	a⁵	a⁶	a⁷
ba	ba	ba²	bab	a⁷	ba²b	ba	a³	a⁶	ba²	bab	a⁴	ba²b	a⁵	a⁶	a⁷
a³	a³	a⁴	a⁷	a⁵	a³	a³	a⁶	a⁴	a⁴	a⁷	a⁷	a³	a³	a⁴	a⁵
a²b	a²b	a²	a⁵	a³	a²b	a⁶	a⁴	a⁷	a⁷	a⁵	a⁵	a⁶	a⁶	a⁷	a³
ba²	ba²	a⁷	ba²b	a³	a⁶	ba²	a⁴	a⁷	a⁷	ba²b	a⁵	a⁶	a⁶	a⁷	a³
bab	bab	ba	a³	ba²	bab	a⁴	a⁷	ba²b	a⁵	a³	a³	a⁴	a⁴	a⁵	a⁶
a⁴	a⁴	a⁵	a³	a⁶	a⁴	a⁴	a⁷	a⁵	a⁵	a³	a³	a⁴	a⁴	a⁵	a⁶
ba²b	ba²b	ba²	a⁴	a⁷	ba²b	a⁵	a³	a⁶	a⁶	a⁴	a⁴	a⁵	a⁵	a⁶	a⁷
a⁵	a⁵	a⁶	a⁴	a⁷	a⁵	a⁵	a³	a⁶	a⁶	a⁴	a⁴	a⁵	a⁵	a⁶	a⁷
a⁶	a⁶	a⁷	a⁵	a³	a⁶	a⁶	a⁴	a⁷	a⁷	a⁵	a⁵	a⁶	a⁶	a⁷	a³
a⁷	a⁷	a³	a⁶	a⁴	a⁷	a⁷	a⁵	a³	a³	a⁶	a⁶	a⁷	a⁷	a³	a⁴

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

23 unique, 188 total

Length:	Presentation:	Description:	Related:
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 \| abb=aaa, bbb=ba〉	Finite non-commutative monoid with 15 elements
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

Other isomorphic instances

2 total

Length:	Presentation:
11	〈a, b \| aba=a, aaaba=bb〉
11	〈a, b \| aba=a, aabaa=bb〉