⟨a, b, c | aa=b, acc=1⟩

Up:

Monoid enumeration

#89 ⟨a, b, c | aa=b, abc=1⟩

#94 ⟨a, b, c | aa=b, bbc=1⟩

Properties

Sum of relation sides is 6

Infinite non-cancellative non-commutative monoid

Element a has infinite order

Not left cancellative, because left multiplication by a is not injective:

a ⋅ c²a = a and a ⋅ 1 = a, however c²a ≠ 1

Not right cancellative, because right multiplication by c² is not injective:

c²a ⋅ c² = c² and 1 ⋅ c² = c², however c²a ≠ 1

Rewriting system

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	1 1

Reduction order:
- Left-to-right recursive path with deg(a) = deg(c) = 0, a < c; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	ac² ⇒ 1	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,acc=1 ac/b - - acc=1 b=aa

Isomorphic instances

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

Σ	#	Presentation	Mapping
6	110	⟨a, b, c \| ab=a, caa=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
6	127	⟨a, b, c \| ab=c, baa=1⟩	φ(a) = c, φ(b) = a, φ(c) = ca
7	273	⟨a, b, c \| aaa=b, acc=1⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
7	321	⟨a, b, c \| aab=c, abb=1⟩	φ(a) = a, φ(b) = c, φ(c) = aac
7	326	⟨a, b, c \| aab=c, baa=1⟩	φ(a) = c, φ(b) = a, φ(c) = cca
7	368	⟨a, b, c \| aba=c, abb=1⟩	φ(a) = a, φ(b) = c, φ(c) = aca
7	373	⟨a, b, c \| aba=c, baa=1⟩	φ(a) = c, φ(b) = a, φ(c) = cac
7	450	⟨a, b, c \| ac=ab, baa=1⟩	φ(a) = c, φ(b) = a, φ(c) = a
7	730	⟨a, b, c \| aa=a, abcc=1⟩	φ(a) = 1, φ(b) = a, φ(c) = c
7	804	⟨a, b, c \| ab=a, bcaa=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
7	814	⟨a, b, c \| ab=a, caab=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
7	816	⟨a, b, c \| ab=a, caba=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
7	822	⟨a, b, c \| ab=a, cbaa=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
7	1306	⟨a, b, c \| ab=1, abcb=a⟩	φ(a) = ac, φ(b) = c, φ(c) = a
7	1310	⟨a, b, c \| ab=1, abcc=b⟩	φ(a) = a, φ(b) = cc, φ(c) = c
7	1443	⟨a, b, c \| ab=1, aab=cb⟩	φ(a) = ac, φ(b) = c, φ(c) = a
7	1457	⟨a, b, c \| ab=1, aba=cb⟩	φ(a) = ac, φ(b) = c, φ(c) = a
7	1525	⟨a, b, c \| ab=1, bcc=bb⟩	φ(a) = a, φ(b) = cc, φ(c) = c
8	1559	⟨a, b, c \| aaa=ab, caa=1⟩	φ(a) = c, φ(b) = cc, φ(c) = a
8	1604	⟨a, b, c \| aab=aa, caa=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	1639	⟨a, b, c \| aab=ac, baa=1⟩	φ(a) = c, φ(b) = a, φ(c) = ca
8	1752	⟨a, b, c \| aab=cb, baa=1⟩	φ(a) = c, φ(b) = a, φ(c) = cc
8	1830	⟨a, b, c \| aba=ac, baa=1⟩	φ(a) = c, φ(b) = a, φ(c) = ac
8	1961	⟨a, b, c \| abc=ac, caa=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	2002	⟨a, b, c \| abc=bc, bcc=1⟩	φ(a) = 1, φ(b) = a, φ(c) = c
8	4191	⟨a, b, c \| aab=1, abcc=b⟩	φ(a) = a, φ(b) = cccc, φ(c) = c
8	4612	⟨a, b, c \| aa=a, aabcc=1⟩	φ(a) = 1, φ(b) = a, φ(c) = c
8	4617	⟨a, b, c \| aa=a, abacc=1⟩	φ(a) = 1, φ(b) = a, φ(c) = c
8	4624	⟨a, b, c \| aa=a, abcac=1⟩	φ(a) = 1, φ(b) = a, φ(c) = c
8	4866	⟨a, b, c \| ab=a, bbcaa=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4876	⟨a, b, c \| ab=a, bcaab=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4878	⟨a, b, c \| ab=a, bcaba=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4884	⟨a, b, c \| ab=a, bcbaa=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4906	⟨a, b, c \| ab=a, caabb=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4912	⟨a, b, c \| ab=a, cabab=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4914	⟨a, b, c \| ab=a, cabba=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4930	⟨a, b, c \| ab=a, cbaab=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4932	⟨a, b, c \| ab=a, cbaba=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4938	⟨a, b, c \| ab=a, cbbaa=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a

Presentation

Mapping

110

⟨a, b, c | ab=a, caa=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

127

⟨a, b, c | ab=c, baa=1⟩

φ(a) = c, φ(b) = a, φ(c) = ca

273

⟨a, b, c | aaa=b, acc=1⟩

φ(a) = a, φ(b) = aaa, φ(c) = c

321

⟨a, b, c | aab=c, abb=1⟩

φ(a) = a, φ(b) = c, φ(c) = aac

326

⟨a, b, c | aab=c, baa=1⟩

φ(a) = c, φ(b) = a, φ(c) = cca

368

⟨a, b, c | aba=c, abb=1⟩

φ(a) = a, φ(b) = c, φ(c) = aca

373

⟨a, b, c | aba=c, baa=1⟩

φ(a) = c, φ(b) = a, φ(c) = cac

450

⟨a, b, c | ac=ab, baa=1⟩

φ(a) = c, φ(b) = a, φ(c) = a

730

⟨a, b, c | aa=a, abcc=1⟩

φ(a) = 1, φ(b) = a, φ(c) = c

804

⟨a, b, c | ab=a, bcaa=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

814

⟨a, b, c | ab=a, caab=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

816

⟨a, b, c | ab=a, caba=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

822

⟨a, b, c | ab=a, cbaa=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

1306

⟨a, b, c | ab=1, abcb=a⟩

φ(a) = ac, φ(b) = c, φ(c) = a

1310

⟨a, b, c | ab=1, abcc=b⟩

φ(a) = a, φ(b) = cc, φ(c) = c

1443

⟨a, b, c | ab=1, aab=cb⟩

φ(a) = ac, φ(b) = c, φ(c) = a

1457

⟨a, b, c | ab=1, aba=cb⟩

φ(a) = ac, φ(b) = c, φ(c) = a

1525

⟨a, b, c | ab=1, bcc=bb⟩

φ(a) = a, φ(b) = cc, φ(c) = c

1559

⟨a, b, c | aaa=ab, caa=1⟩

φ(a) = c, φ(b) = cc, φ(c) = a

1604

⟨a, b, c | aab=aa, caa=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

1639

⟨a, b, c | aab=ac, baa=1⟩

φ(a) = c, φ(b) = a, φ(c) = ca

1752

⟨a, b, c | aab=cb, baa=1⟩

φ(a) = c, φ(b) = a, φ(c) = cc

1830

⟨a, b, c | aba=ac, baa=1⟩

φ(a) = c, φ(b) = a, φ(c) = ac

1961

⟨a, b, c | abc=ac, caa=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

2002

⟨a, b, c | abc=bc, bcc=1⟩

φ(a) = 1, φ(b) = a, φ(c) = c

4191

⟨a, b, c | aab=1, abcc=b⟩

φ(a) = a, φ(b) = cccc, φ(c) = c

4612

⟨a, b, c | aa=a, aabcc=1⟩

φ(a) = 1, φ(b) = a, φ(c) = c

4617

⟨a, b, c | aa=a, abacc=1⟩

φ(a) = 1, φ(b) = a, φ(c) = c

4624

⟨a, b, c | aa=a, abcac=1⟩

φ(a) = 1, φ(b) = a, φ(c) = c

4866

⟨a, b, c | ab=a, bbcaa=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4876

⟨a, b, c | ab=a, bcaab=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4878

⟨a, b, c | ab=a, bcaba=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4884

⟨a, b, c | ab=a, bcbaa=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4906

⟨a, b, c | ab=a, caabb=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4912

⟨a, b, c | ab=a, cabab=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4914

⟨a, b, c | ab=a, cabba=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4930

⟨a, b, c | ab=a, cbaab=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4932

⟨a, b, c | ab=a, cbaba=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4938

⟨a, b, c | ab=a, cbbaa=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

Anti-isomorphic instances

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

Σ	#	Presentation	Mapping
7	442	⟨a, b, c \| ab=aa, cca=1⟩	φ(a) = a, φ(b) = a, φ(c) = c
8	4161	⟨a, b, c \| aab=1, aabb=c⟩	φ(a) = c, φ(b) = a, φ(c) = a
8	4163	⟨a, b, c \| aab=1, aabc=b⟩	φ(a) = c, φ(b) = a, φ(c) = a
8	4221	⟨a, b, c \| aab=1, baab=c⟩	φ(a) = c, φ(b) = a, φ(c) = a
8	4289	⟨a, b, c \| aab=1, caab=b⟩	φ(a) = c, φ(b) = a, φ(c) = a

Presentation

Mapping

442

⟨a, b, c | ab=aa, cca=1⟩

φ(a) = a, φ(b) = a, φ(c) = c

4161

⟨a, b, c | aab=1, aabb=c⟩

φ(a) = c, φ(b) = a, φ(c) = a

4163

⟨a, b, c | aab=1, aabc=b⟩

φ(a) = c, φ(b) = a, φ(c) = a

4221

⟨a, b, c | aab=1, baab=c⟩

φ(a) = c, φ(b) = a, φ(c) = a

4289

⟨a, b, c | aab=1, caab=b⟩

φ(a) = c, φ(b) = a, φ(c) = a

#92 ⟨a, b, c | aa=b, acc=1⟩

Contents

Properties

Rewriting system

Isomorphic instances

Anti-isomorphic instances

Up:	Monoid enumeration
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