#1123 ⟨a, b | aa=1, abbbb=b⟩

Properties

Presentation has sum-of-sides 8
Finite non-commutative monoid with 14 elements

Element profile

1 element center:
- 1
3 idempotent elements:
- 1, b³a, b⁶

Complete rewriting system

Format:	Pretty Plain
Word to reduce:
	Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Reduction strategy:	Leftmost Rightmost
Path to normal form:	1 1

Reduction order:
- Left-to-right recursive path with deg(b) = 0; deg(a) = 1

b⁷ ⇒ b
ab ⇒ b⁴
a² ⇒ 1

# ab:aa=1,abbbb=b b/a
bbbbbbb=b
ab=bbbb
aa=1

Cayley table

Idempotents are shown in bold.

	1	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a
1	1	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a
a	a	1	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b	ba	b²	b²a	b³	b³a
b	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b	ba
ba	ba	b	b⁵	b⁵a	b⁶	b⁶a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a
b²	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b	ba	b²	b²a
b²a	b²a	b²	b⁶	b⁶a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a
b³	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b	ba	b²	b²a	b³	b³a
b³a	b³a	b³	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a
b⁴	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a
b⁴a	b⁴a	b⁴	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b	ba
b⁵	b⁵	b⁵a	b⁶	b⁶a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a
b⁵a	b⁵a	b⁵	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b	ba	b²	b²a
b⁶	b⁶	b⁶a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a
b⁶a	b⁶a	b⁶	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b	ba	b²	b²a	b³	b³a

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

30 unique, 257 total

Σ	#	Presentation	Description	Related
9	1682	⟨a, b \| aba=bb, bab=a⟩	Finite non-commutative monoid with 14 elements	1 iso
9	3034	⟨a, b \| aa=a, bbbb=ab⟩	Finite non-commutative monoid with 14 elements	2 iso
10	3773	⟨a, b \| aaaa=bb, abbb=1⟩	Isomorphic to ℤ₁₄	165 iso
10	5218	⟨a, b \| aab=bb, bbba=a⟩	Finite non-commutative monoid with 14 elements	5 iso, 1 anti-iso
10	5404	⟨a, b \| aab=bb, aba=aa⟩	Finite non-commutative monoid with 14 elements	1 iso
10	6718	⟨a, b \| aab=b, bbba=aa⟩	Finite non-commutative monoid with 14 elements	2 iso
11	12183	⟨a, b \| aaaa=ab, baab=b⟩	Finite non-commutative monoid with 14 elements	3 iso
11	12206	⟨a, b \| aaaa=bb, abbb=a⟩	Finite commutative monoid with 14 elements	1 iso
11	12207	⟨a, b \| aaaa=bb, abbb=b⟩	Finite commutative monoid with 14 elements	1 iso
11	12441	⟨a, b \| aabb=aa, baab=b⟩	Finite non-commutative monoid with 14 elements	3 iso
11	12499	⟨a, b \| abab=aa, abba=b⟩	Finite non-commutative monoid with 14 elements
11	14383	⟨a, b \| aaaa=b, aabbb=a⟩	Isomorphic to ℕ_{(14 = 1)}	15 iso
11	14384	⟨a, b \| aaaa=b, aabbb=b⟩	Isomorphic to ℕ_{(14 = 4)}	5 iso
11	15532	⟨a, b \| aaa=bb, abbbb=b⟩	Isomorphic to ℕ_{(14 = 3)}	11 iso
11	15539	⟨a, b \| aaa=bb, babbb=a⟩	Finite commutative monoid with 14 elements
11	16020	⟨a, b \| aaa=ab, baab=bb⟩	Finite non-commutative monoid with 14 elements	1 iso
11	16079	⟨a, b \| aaa=bb, bbbb=ab⟩	Finite non-commutative monoid with 14 elements
11	16293	⟨a, b \| aab=bb, abab=aa⟩	Finite non-commutative monoid with 14 elements
11	16470	⟨a, b \| aba=bb, bbbb=aa⟩	Finite non-commutative monoid with 14 elements
11	19552	⟨a, b \| aab=b, abbaa=aa⟩	Finite non-commutative monoid with 14 elements	2 iso
11	20844	⟨a, b \| ab=aa, bbbbb=aa⟩	Finite non-commutative monoid with 14 elements	1 iso
11	20846	⟨a, b \| ab=aa, bbbbb=ba⟩	Finite non-commutative monoid with 14 elements
11	21023	⟨a, b \| ab=aa, bbaa=bbb⟩	Finite non-commutative monoid with 14 elements	1 iso
11	21040	⟨a, b \| ab=aa, bbbb=aaa⟩	Finite non-commutative monoid with 14 elements	3 iso
11	21044	⟨a, b \| ab=aa, bbbb=baa⟩	Finite non-commutative monoid with 14 elements	1 iso
11	21046	⟨a, b \| ab=aa, bbbb=bba⟩	Finite non-commutative monoid with 14 elements
11	21110	⟨a, b \| bb=aa, abab=aaa⟩	Finite non-commutative monoid with 14 elements	2 anti-iso
11	24186	⟨a, b \| aa=a, bbbbbbb=a⟩	Isomorphic to ℕ_{(14 = 7)}
11	24331	⟨a, b \| aa=b, bbbbbbb=b⟩	Isomorphic to ℕ_{(14 = 2)}
11	25055	⟨a, b \| ab=a, bbaaaa=bb⟩	Finite non-commutative monoid with 14 elements

Other isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

42 total

Σ	#	Presentation	Mapping
8	1202	⟨a, b \| aa=1, bbbb=ab⟩	φ(a) = a, φ(b) = b
9	3377	⟨a, b \| aa=1, ababbb=b⟩	φ(a) = a, φ(b) = ba
9	3381	⟨a, b \| aa=1, abbabb=b⟩	φ(a) = a, φ(b) = ba
9	3535	⟨a, b \| aa=1, babbb=ab⟩	φ(a) = a, φ(b) = ba
9	3539	⟨a, b \| aa=1, bbabb=ab⟩	φ(a) = a, φ(b) = ba
10	9687	⟨a, b \| aa=1, aaabbbb=b⟩	φ(a) = a, φ(b) = b
10	9727	⟨a, b \| aa=1, abaabbb=b⟩	φ(a) = a, φ(b) = b
10	9735	⟨a, b \| aa=1, ababbab=b⟩	φ(a) = a, φ(b) = b
10	9743	⟨a, b \| aa=1, abbaabb=b⟩	φ(a) = a, φ(b) = b
10	9745	⟨a, b \| aa=1, abbabab=b⟩	φ(a) = a, φ(b) = b
10	9751	⟨a, b \| aa=1, abbbaab=b⟩	φ(a) = a, φ(b) = b
10	9983	⟨a, b \| aa=1, aabbbb=ab⟩	φ(a) = a, φ(b) = b
10	10040	⟨a, b \| aa=1, baabbb=ab⟩	φ(a) = a, φ(b) = b
10	10048	⟨a, b \| aa=1, babbab=ab⟩	φ(a) = a, φ(b) = b
10	10055	⟨a, b \| aa=1, bbaabb=ab⟩	φ(a) = a, φ(b) = b
10	10297	⟨a, b \| aa=1, abbbb=aab⟩	φ(a) = a, φ(b) = b
10	10300	⟨a, b \| aa=1, abbbb=baa⟩	φ(a) = a, φ(b) = b
10	10530	⟨a, b \| aa=1, bbbb=aaab⟩	φ(a) = a, φ(b) = b
11	26485	⟨a, b \| aa=1, aaababbb=b⟩	φ(a) = a, φ(b) = ba
11	26493	⟨a, b \| aa=1, aaabbabb=b⟩	φ(a) = a, φ(b) = ba
11	26567	⟨a, b \| aa=1, abaaabbb=b⟩	φ(a) = a, φ(b) = ba
11	26573	⟨a, b \| aa=1, abaababb=b⟩	φ(a) = a, φ(b) = ba
11	26583	⟨a, b \| aa=1, ababaabb=b⟩	φ(a) = a, φ(b) = ba
11	26585	⟨a, b \| aa=1, abababab=b⟩	φ(a) = a, φ(b) = ba
11	26591	⟨a, b \| aa=1, ababbaab=b⟩	φ(a) = a, φ(b) = ba
11	26605	⟨a, b \| aa=1, abbaaabb=b⟩	φ(a) = a, φ(b) = ba
11	26613	⟨a, b \| aa=1, abbabaab=b⟩	φ(a) = a, φ(b) = ba
11	27054	⟨a, b \| aa=1, aababbb=ab⟩	φ(a) = a, φ(b) = ba
11	27066	⟨a, b \| aa=1, aabbabb=ab⟩	φ(a) = a, φ(b) = ba
11	27124	⟨a, b \| aa=1, ababbba=ba⟩	φ(a) = a, φ(b) = ba
11	27180	⟨a, b \| aa=1, baaabbb=ab⟩	φ(a) = a, φ(b) = ba
11	27187	⟨a, b \| aa=1, baababb=ab⟩	φ(a) = a, φ(b) = ba
11	27199	⟨a, b \| aa=1, babaabb=ab⟩	φ(a) = a, φ(b) = ba
11	27203	⟨a, b \| aa=1, bababab=ab⟩	φ(a) = a, φ(b) = ba
11	27221	⟨a, b \| aa=1, bbaaabb=ab⟩	φ(a) = a, φ(b) = ba
11	27651	⟨a, b \| aa=1, ababbb=aab⟩	φ(a) = a, φ(b) = ba
11	27654	⟨a, b \| aa=1, ababbb=baa⟩	φ(a) = a, φ(b) = ba
11	27667	⟨a, b \| aa=1, abbabb=aab⟩	φ(a) = a, φ(b) = ba
11	27670	⟨a, b \| aa=1, abbabb=baa⟩	φ(a) = a, φ(b) = ba
11	28261	⟨a, b \| aa=1, babbb=aaab⟩	φ(a) = a, φ(b) = ba
11	28264	⟨a, b \| aa=1, babbb=abaa⟩	φ(a) = a, φ(b) = ba
11	28277	⟨a, b \| aa=1, bbabb=aaab⟩	φ(a) = a, φ(b) = ba

Other anti-isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

17 total

Σ	#	Presentation	Mapping
10	9713	⟨a, b \| aa=1, aabbbba=b⟩	φ(a) = a, φ(b) = b
10	9984	⟨a, b \| aa=1, aabbbb=ba⟩	φ(a) = a, φ(b) = b
10	10022	⟨a, b \| aa=1, abbbba=ab⟩	φ(a) = a, φ(b) = b
10	10041	⟨a, b \| aa=1, baabbb=ba⟩	φ(a) = a, φ(b) = b
10	10045	⟨a, b \| aa=1, bababb=ba⟩	φ(a) = a, φ(b) = b
10	10298	⟨a, b \| aa=1, abbbb=aba⟩	φ(a) = a, φ(b) = b
10	10531	⟨a, b \| aa=1, bbbb=aaba⟩	φ(a) = a, φ(b) = b
11	26539	⟨a, b \| aa=1, aabbabba=b⟩	φ(a) = a, φ(b) = ba
11	26545	⟨a, b \| aa=1, aabbbaba=b⟩	φ(a) = a, φ(b) = ba
11	27067	⟨a, b \| aa=1, aabbabb=ba⟩	φ(a) = a, φ(b) = ba
11	27074	⟨a, b \| aa=1, aabbbab=ba⟩	φ(a) = a, φ(b) = ba
11	27143	⟨a, b \| aa=1, abbabba=ab⟩	φ(a) = a, φ(b) = ba
11	27188	⟨a, b \| aa=1, baababb=ba⟩	φ(a) = a, φ(b) = ba
11	27192	⟨a, b \| aa=1, baabbab=ba⟩	φ(a) = a, φ(b) = ba
11	27668	⟨a, b \| aa=1, abbabb=aba⟩	φ(a) = a, φ(b) = ba
11	27676	⟨a, b \| aa=1, abbbab=aba⟩	φ(a) = a, φ(b) = ba
11	28278	⟨a, b \| aa=1, bbabb=aaba⟩	φ(a) = a, φ(b) = ba

Up:	Monoids with two generators and two relations
Prev:	#1121 ⟨a, b \| aa=1, abbba=b⟩
Next:	#1129 ⟨a, b \| aa=1, babab=b⟩