#2344 ⟨
a
,
b
|
abbaaab
=
aab
⟩
Up:
Monoids with two generators and one relation
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#2343
⟨
a
,
b
|
abbaaab
=
aaa
⟩
Next:
#2345
⟨
a
,
b
|
abbaaab
=
aba
⟩
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
2
a
3
b
=
a
2
b
and
a
⋅
ab
=
a
2
b
, however
b
2
a
3
b
≠
ab
Not right cancellative, because right multiplication by
b
is not injective:
a
b
2
a
3
⋅
b
=
a
2
b
and
a
2
⋅
b
=
a
2
b
, however
a
b
2
a
3
≠
a
2
Enveloping group: ⟨
a
,
b
|
aabb
⟩
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
=
aaab
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.
Morphocompletion
: how the auxiliary generators were found.
#
Rule
Proof
1.
ac
⇒
cbc
[5]
2.
ab
(
bc
)
2
⇒
c
[4]
3.
a
2
b
⇒
a
b
2
c
[3]
# ab:abbaaab=aab bc/a aaab=c morph:4/1 ac=cbc abbcbc=c aab=abbc