#2040 ⟨
a
,
b
|
abaababa
=
ba
⟩
Up:
Monoids with two generators and one relation
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#2039
⟨
a
,
b
|
abaababa
=
ab
⟩
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#2041
⟨
a
,
b
|
abaababa
=
bb
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 10
Infinite non-cancellative non-commutative monoid
Not right cancellative, because right multiplication by
a
is not injective:
(
aba
)
2
b
⋅
a
=
ba
and
b
⋅
a
=
ba
, however (
aba
)
2
b
≠
b
Enveloping group is isomorphic to ℤ
2
∗ 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
=
ba
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.
aca
c
2
⇒
c
[4]
2.
ba
⇒
c
[2]
3.
bc
⇒
c
2
a
c
2
[5]
# ab:abaababa=ba ac/b ba=c magic:0 acacc=c ba=c bc=ccacc