#52 ⟨
a
,
b
|
aba
=
ab
⟩
Up:
Monoids with two generators and one relation
Prev:
#51
⟨
a
,
b
|
aba
=
aa
⟩
Next:
#53
⟨
a
,
b
|
aba
=
bb
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 5
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
ba
=
ab
and
a
⋅
b
=
ab
, however
ba
≠
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
=
ab
Reduction order:
Left-to-right shortlex with
a
<
c
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ab
⇒
c
[2]
2.
ca
⇒
c
[4]
3.
cb
⇒
c
2
[5]
# ab:aba=ab acb ab=c magic:0 ab=c ca=c cb=cc