#1773 ⟨
a
,
b
|
abaabbaba
=
a
⟩
Up:
Monoids with two generators and one relation
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#1772
⟨
a
,
b
|
abaabbaab
=
b
⟩
Next:
#1774
⟨
a
,
b
|
abaabbaba
=
b
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 10
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
b
a
2
b
(
ba
)
2
=
a
and
a
⋅ 1 =
a
, however
b
a
2
b
(
ba
)
2
≠ 1
Not right cancellative, because right multiplication by
a
is not injective:
ab
a
2
b
2
ab
⋅
a
=
a
and 1 ⋅
a
=
a
, however
ab
a
2
b
2
ab
≠ 1
Enveloping group: ⟨
a
,
b
|
aaabab
-1
⟩
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 shortlex with
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ab
a
2
b
2
a
⇒
a
2
b
(
ba
)
2
[2]
2.
a
2
b
(
ba
)
3
⇒
a
[3]
# ab:abaabbaba=a ab abaabba=aabbaba aabbababa=a