#112 ⟨
a
,
b
|
abba
=
ab
⟩
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Monoids with two generators and one relation
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#111
⟨
a
,
b
|
abba
=
aa
⟩
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#113
⟨
a
,
b
|
abba
=
bb
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 6
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
b
2
a
=
ab
and
a
⋅
b
=
ab
, however
b
2
a
≠
b
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
=
abb
Reduction order:
Left-to-right shortlex with
c
<
b
<
a
Certificate
: derivations of all rewriting rules from the defining relations.
Morphocompletion
: how the auxiliary generators were found.
#
Rule
Proof
1.
cb
⇒
c
2
[5]
2.
ab
⇒
ca
[3]
3.
c
2
a
⇒
c
[4]
# ab:abba=ab cba abb=c morph:3/1 cb=cc ab=ca cca=c