#1259 ⟨
a
,
b
|
ababa
=
abab
⟩
Up:
Monoids with two generators and one relation
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#1258
⟨
a
,
b
|
ababa
=
aabb
⟩
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#1260
⟨
a
,
b
|
ababa
=
abba
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 9
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅ (
ba
)
2
= (
ab
)
2
and
a
⋅
bab
= (
ab
)
2
, however (
ba
)
2
≠
bab
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
=
abab
Reduction order:
Left-to-right recursive path with deg(
a
) = deg(
c
) = 0,
a
<
c
; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ca
⇒
c
[4]
2.
cb
⇒
abc
[5]
3.
(
ab
)
2
⇒
c
[2]
# ab:ababa=abab ac/b abab=c magic:0 ca=c cb=abc abab=c