⟨a, b, c | aa=b, bc=1⟩

Up:

Sum of relation sides is 5

Infinite non-cancellative non-commutative monoid

Element a has infinite order

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

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

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

aca ⋅ ac = ac and 1 ⋅ ac = ac, however aca ≠ 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.	a²c ⇒ 1	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,bc=1 ac/b - - aac=1 b=aa

Isomorphic instances

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

Σ	#	Presentation	Mapping
5	13	⟨a, b, c \| ab=c, ac=1⟩	φ(a) = a, φ(b) = c, φ(c) = ac
6	88	⟨a, b, c \| aa=b, aac=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
6	116	⟨a, b, c \| ab=a, cca=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
6	120	⟨a, b, c \| ab=c, aab=1⟩	φ(a) = a, φ(b) = c, φ(c) = ac
7	269	⟨a, b, c \| aaa=b, aac=1⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
7	329	⟨a, b, c \| aab=c, bba=1⟩	φ(a) = c, φ(b) = a, φ(c) = cca
7	453	⟨a, b, c \| ac=ab, bba=1⟩	φ(a) = c, φ(b) = a, φ(c) = a
7	456	⟨a, b, c \| ac=ab, bca=1⟩	φ(a) = c, φ(b) = a, φ(c) = a
7	581	⟨a, b, c \| aab=1, aabc=1⟩	φ(a) = a, φ(b) = c, φ(c) = 1
7	619	⟨a, b, c \| aab=1, caab=1⟩	φ(a) = a, φ(b) = c, φ(c) = 1
7	727	⟨a, b, c \| aa=a, abbc=1⟩	φ(a) = 1, φ(b) = a, φ(c) = c
7	810	⟨a, b, c \| ab=a, bcca=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
7	828	⟨a, b, c \| ab=a, cbca=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
7	832	⟨a, b, c \| ab=a, ccab=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
7	834	⟨a, b, c \| ab=a, ccba=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
7	1283	⟨a, b, c \| ab=1, aabc=b⟩	φ(a) = a, φ(b) = ac, φ(c) = c
7	1297	⟨a, b, c \| ab=1, abac=b⟩	φ(a) = a, φ(b) = ac, φ(c) = c
7	1309	⟨a, b, c \| ab=1, abcc=a⟩	φ(a) = aa, φ(b) = c, φ(c) = a
7	1373	⟨a, b, c \| ab=1, cabc=a⟩	φ(a) = aa, φ(b) = c, φ(c) = a
7	1444	⟨a, b, c \| ab=1, aab=cc⟩	φ(a) = aa, φ(b) = c, φ(c) = a
7	1458	⟨a, b, c \| ab=1, aba=cc⟩	φ(a) = aa, φ(b) = c, φ(c) = a
7	1471	⟨a, b, c \| ab=1, aca=ba⟩	φ(a) = a, φ(b) = ac, φ(c) = c
7	1501	⟨a, b, c \| ab=1, bac=bb⟩	φ(a) = a, φ(b) = ac, φ(c) = c
7	1524	⟨a, b, c \| ab=1, bcc=ba⟩	φ(a) = aa, φ(b) = c, φ(c) = a
8	1565	⟨a, b, c \| aaa=ab, cca=1⟩	φ(a) = c, φ(b) = cc, φ(c) = a
8	1610	⟨a, b, c \| aab=aa, cca=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	1642	⟨a, b, c \| aab=ac, bba=1⟩	φ(a) = c, φ(b) = a, φ(c) = ca
8	1755	⟨a, b, c \| aab=cb, bba=1⟩	φ(a) = c, φ(b) = a, φ(c) = cc
8	1833	⟨a, b, c \| aba=ac, bba=1⟩	φ(a) = c, φ(b) = a, φ(c) = ac
8	4155	⟨a, b, c \| aab=1, aaaa=c⟩	φ(a) = a, φ(b) = c, φ(c) = aaaa
8	4156	⟨a, b, c \| aab=1, aaab=c⟩	φ(a) = a, φ(b) = c, φ(c) = a
8	4160	⟨a, b, c \| aab=1, aaba=c⟩	φ(a) = a, φ(b) = c, φ(c) = a
8	4162	⟨a, b, c \| aab=1, aabc=a⟩	φ(a) = a, φ(b) = c, φ(c) = a
8	4174	⟨a, b, c \| aab=1, abaa=c⟩	φ(a) = a, φ(b) = c, φ(c) = acaa
8	4175	⟨a, b, c \| aab=1, abab=c⟩	φ(a) = a, φ(b) = c, φ(c) = acac
8	4177	⟨a, b, c \| aab=1, abac=b⟩	φ(a) = a, φ(b) = acac, φ(c) = c
8	4179	⟨a, b, c \| aab=1, abba=c⟩	φ(a) = a, φ(b) = c, φ(c) = acca
8	4180	⟨a, b, c \| aab=1, abbb=c⟩	φ(a) = a, φ(b) = c, φ(c) = accc
8	4220	⟨a, b, c \| aab=1, baaa=c⟩	φ(a) = a, φ(b) = c, φ(c) = caaa
8	4225	⟨a, b, c \| aab=1, baba=c⟩	φ(a) = a, φ(b) = c, φ(c) = caca
8	4226	⟨a, b, c \| aab=1, babb=c⟩	φ(a) = a, φ(b) = c, φ(c) = cacc
8	4239	⟨a, b, c \| aab=1, bbaa=c⟩	φ(a) = a, φ(b) = c, φ(c) = ccaa
8	4240	⟨a, b, c \| aab=1, bbab=c⟩	φ(a) = a, φ(b) = c, φ(c) = ccac
8	4244	⟨a, b, c \| aab=1, bbba=c⟩	φ(a) = a, φ(b) = c, φ(c) = ccca
8	4245	⟨a, b, c \| aab=1, bbbb=c⟩	φ(a) = a, φ(b) = c, φ(c) = cccc
8	4288	⟨a, b, c \| aab=1, caab=a⟩	φ(a) = a, φ(b) = c, φ(c) = a
8	4497	⟨a, b, c \| abc=1, aabc=b⟩	φ(a) = a, φ(b) = a, φ(c) = c
8	4525	⟨a, b, c \| abc=1, abca=b⟩	φ(a) = a, φ(b) = a, φ(c) = c
8	4527	⟨a, b, c \| abc=1, abcb=a⟩	φ(a) = a, φ(b) = a, φ(c) = c
8	4609	⟨a, b, c \| aa=a, aabbc=1⟩	φ(a) = 1, φ(b) = a, φ(c) = c
8	4614	⟨a, b, c \| aa=a, ababc=1⟩	φ(a) = 1, φ(b) = a, φ(c) = c
8	4618	⟨a, b, c \| aa=a, abbac=1⟩	φ(a) = 1, φ(b) = a, φ(c) = c
8	4620	⟨a, b, c \| aa=a, abbca=1⟩	φ(a) = 1, φ(b) = a, φ(c) = c
8	4872	⟨a, b, c \| ab=a, bbcca=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4890	⟨a, b, c \| ab=a, bcbca=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4894	⟨a, b, c \| ab=a, bccab=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4896	⟨a, b, c \| ab=a, bccba=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4944	⟨a, b, c \| ab=a, cbbca=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4948	⟨a, b, c \| ab=a, cbcab=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4950	⟨a, b, c \| ab=a, cbcba=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4960	⟨a, b, c \| ab=a, ccabb=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4966	⟨a, b, c \| ab=a, ccbab=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4968	⟨a, b, c \| ab=a, ccbba=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a

Presentation

Mapping

⟨a, b, c | ab=c, ac=1⟩

φ(a) = a, φ(b) = c, φ(c) = ac

⟨a, b, c | aa=b, aac=1⟩

φ(a) = a, φ(b) = aa, φ(c) = c

116

⟨a, b, c | ab=a, cca=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

120

⟨a, b, c | ab=c, aab=1⟩

φ(a) = a, φ(b) = c, φ(c) = ac

269

⟨a, b, c | aaa=b, aac=1⟩

φ(a) = a, φ(b) = aaa, φ(c) = c

329

⟨a, b, c | aab=c, bba=1⟩

φ(a) = c, φ(b) = a, φ(c) = cca

453

⟨a, b, c | ac=ab, bba=1⟩

φ(a) = c, φ(b) = a, φ(c) = a

456

⟨a, b, c | ac=ab, bca=1⟩

φ(a) = c, φ(b) = a, φ(c) = a

581

⟨a, b, c | aab=1, aabc=1⟩

φ(a) = a, φ(b) = c, φ(c) = 1

619

⟨a, b, c | aab=1, caab=1⟩

φ(a) = a, φ(b) = c, φ(c) = 1

727

⟨a, b, c | aa=a, abbc=1⟩

φ(a) = 1, φ(b) = a, φ(c) = c

810

⟨a, b, c | ab=a, bcca=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

828

⟨a, b, c | ab=a, cbca=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

832

⟨a, b, c | ab=a, ccab=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

834

⟨a, b, c | ab=a, ccba=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

1283

⟨a, b, c | ab=1, aabc=b⟩

φ(a) = a, φ(b) = ac, φ(c) = c

1297

⟨a, b, c | ab=1, abac=b⟩

φ(a) = a, φ(b) = ac, φ(c) = c

1309

⟨a, b, c | ab=1, abcc=a⟩

φ(a) = aa, φ(b) = c, φ(c) = a

1373

⟨a, b, c | ab=1, cabc=a⟩

φ(a) = aa, φ(b) = c, φ(c) = a

1444

⟨a, b, c | ab=1, aab=cc⟩

φ(a) = aa, φ(b) = c, φ(c) = a

1458

⟨a, b, c | ab=1, aba=cc⟩

φ(a) = aa, φ(b) = c, φ(c) = a

1471

⟨a, b, c | ab=1, aca=ba⟩

φ(a) = a, φ(b) = ac, φ(c) = c

1501

⟨a, b, c | ab=1, bac=bb⟩

φ(a) = a, φ(b) = ac, φ(c) = c

1524

⟨a, b, c | ab=1, bcc=ba⟩

φ(a) = aa, φ(b) = c, φ(c) = a

1565

⟨a, b, c | aaa=ab, cca=1⟩

φ(a) = c, φ(b) = cc, φ(c) = a

1610

⟨a, b, c | aab=aa, cca=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

1642

⟨a, b, c | aab=ac, bba=1⟩

φ(a) = c, φ(b) = a, φ(c) = ca

1755

⟨a, b, c | aab=cb, bba=1⟩

φ(a) = c, φ(b) = a, φ(c) = cc

1833

⟨a, b, c | aba=ac, bba=1⟩

φ(a) = c, φ(b) = a, φ(c) = ac

4155

⟨a, b, c | aab=1, aaaa=c⟩

φ(a) = a, φ(b) = c, φ(c) = aaaa

4156

⟨a, b, c | aab=1, aaab=c⟩

φ(a) = a, φ(b) = c, φ(c) = a

4160

⟨a, b, c | aab=1, aaba=c⟩

φ(a) = a, φ(b) = c, φ(c) = a

4162

⟨a, b, c | aab=1, aabc=a⟩

φ(a) = a, φ(b) = c, φ(c) = a

4174

⟨a, b, c | aab=1, abaa=c⟩

φ(a) = a, φ(b) = c, φ(c) = acaa

4175

⟨a, b, c | aab=1, abab=c⟩

φ(a) = a, φ(b) = c, φ(c) = acac

4177

⟨a, b, c | aab=1, abac=b⟩

φ(a) = a, φ(b) = acac, φ(c) = c

4179

⟨a, b, c | aab=1, abba=c⟩

φ(a) = a, φ(b) = c, φ(c) = acca

4180

⟨a, b, c | aab=1, abbb=c⟩

φ(a) = a, φ(b) = c, φ(c) = accc

4220

⟨a, b, c | aab=1, baaa=c⟩

φ(a) = a, φ(b) = c, φ(c) = caaa

4225

⟨a, b, c | aab=1, baba=c⟩

φ(a) = a, φ(b) = c, φ(c) = caca

4226

⟨a, b, c | aab=1, babb=c⟩

φ(a) = a, φ(b) = c, φ(c) = cacc

4239

⟨a, b, c | aab=1, bbaa=c⟩

φ(a) = a, φ(b) = c, φ(c) = ccaa

4240

⟨a, b, c | aab=1, bbab=c⟩

φ(a) = a, φ(b) = c, φ(c) = ccac

4244

⟨a, b, c | aab=1, bbba=c⟩

φ(a) = a, φ(b) = c, φ(c) = ccca

4245

⟨a, b, c | aab=1, bbbb=c⟩

φ(a) = a, φ(b) = c, φ(c) = cccc

4288

⟨a, b, c | aab=1, caab=a⟩

φ(a) = a, φ(b) = c, φ(c) = a

4497

⟨a, b, c | abc=1, aabc=b⟩

φ(a) = a, φ(b) = a, φ(c) = c

4525

⟨a, b, c | abc=1, abca=b⟩

φ(a) = a, φ(b) = a, φ(c) = c

4527

⟨a, b, c | abc=1, abcb=a⟩

φ(a) = a, φ(b) = a, φ(c) = c

4609

⟨a, b, c | aa=a, aabbc=1⟩

φ(a) = 1, φ(b) = a, φ(c) = c

4614

⟨a, b, c | aa=a, ababc=1⟩

φ(a) = 1, φ(b) = a, φ(c) = c

4618

⟨a, b, c | aa=a, abbac=1⟩

φ(a) = 1, φ(b) = a, φ(c) = c

4620

⟨a, b, c | aa=a, abbca=1⟩

φ(a) = 1, φ(b) = a, φ(c) = c

4872

⟨a, b, c | ab=a, bbcca=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4890

⟨a, b, c | ab=a, bcbca=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4894

⟨a, b, c | ab=a, bccab=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4896

⟨a, b, c | ab=a, bccba=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4944

⟨a, b, c | ab=a, cbbca=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4948

⟨a, b, c | ab=a, cbcab=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4950

⟨a, b, c | ab=a, cbcba=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4960

⟨a, b, c | ab=a, ccabb=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4966

⟨a, b, c | ab=a, ccbab=1⟩

φ(a) = c, φ(b) = 1, φ(c) = a

4968

⟨a, b, c | ab=a, ccbba=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	436	⟨a, b, c \| ab=aa, caa=1⟩	φ(a) = a, φ(b) = a, φ(c) = c
7	437	⟨a, b, c \| ab=aa, cab=1⟩	φ(a) = a, φ(b) = a, φ(c) = c
7	439	⟨a, b, c \| ab=aa, cba=1⟩	φ(a) = a, φ(b) = a, φ(c) = c
8	4529	⟨a, b, c \| abc=1, abcb=c⟩	φ(a) = c, φ(b) = a, φ(c) = a

Presentation

Mapping

436

⟨a, b, c | ab=aa, caa=1⟩

φ(a) = a, φ(b) = a, φ(c) = c

437

⟨a, b, c | ab=aa, cab=1⟩

φ(a) = a, φ(b) = a, φ(c) = c

439

⟨a, b, c | ab=aa, cba=1⟩

φ(a) = a, φ(b) = a, φ(c) = c

4529

⟨a, b, c | abc=1, abcb=c⟩

φ(a) = c, φ(b) = a, φ(c) = a

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Prev:	#5 ⟨a, b, c \| aa=b, ac=1⟩
Next:	#7 ⟨a, b, c \| aa=b, cc=1⟩

#6 ⟨a, b, c | aa=b, bc=1⟩

Contents

Properties

Rewriting system

Isomorphic instances

Anti-isomorphic instances