#261 ⟨
a
,
b
|
abab
=
aba
⟩
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Monoids with two generators and one relation
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#260
⟨
a
,
b
|
abab
=
aab
⟩
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#262
⟨
a
,
b
|
abab
=
baa
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 7
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
bab
=
aba
and
a
⋅
ba
=
aba
, however
bab
≠
ba
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
=
aba
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.
ca
⇒
abc
[5]
3.
aba
⇒
c
[2]
# ab:abab=aba bc/a aba=c magic:0 cb=c ca=abc aba=c