#5870 ⟨
a
,
b
|
ababab
=
ababa
⟩
Up:
Monoids with two generators and one relation
Prev:
#5869
⟨
a
,
b
|
ababab
=
abaab
⟩
Next:
#5871
⟨
a
,
b
|
ababab
=
abbaa
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 11
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
b
(
ab
)
2
=
a
(
ba
)
2
and
a
⋅ (
ba
)
2
=
a
(
ba
)
2
, however
b
(
ab
)
2
≠ (
ba
)
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
=
ababa
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.
a
(
ba
)
2
⇒
c
[2]
# ab:ababab=ababa bc/a ababa=c magic:0 cb=c ca=abc ababa=c