#1243 ⟨
a
,
b
|
abaab
=
abaa
⟩
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Monoids with two generators and one relation
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#1242
⟨
a
,
b
|
abaab
=
aabb
⟩
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#1244
⟨
a
,
b
|
abaab
=
abab
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 9
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
b
a
2
b
=
ab
a
2
and
a
⋅
b
a
2
=
ab
a
2
, however
b
a
2
b
≠
b
a
2
Enveloping group is isomorphic to ℤ
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
Auxiliary generators:
c
=
abaa
Reduction order:
Left-to-right recursive path with deg(
b
) = deg(
c
) = 0,
b
<
c
; deg(
a
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
cb
⇒
c
[4]
2.
c
a
2
⇒
abac
[5]
3.
ab
a
2
⇒
c
[2]
# ab:abaab=abaa bc/a abaa=c magic:0 cb=c caa=abac abaa=c