#25200 ⟨
a
,
b
|
aa
=
a
,
abbab
=
aba
⟩
Up:
Monoid enumeration
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#25197
⟨
a
,
b
|
aa
=
a
,
ababb
=
bbb
⟩
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#25201
⟨
a
,
b
|
aa
=
a
,
abbab
=
abb
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element
a
b
3
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
a
=
a
and
a
⋅ 1 =
a
, however
a
≠ 1
Not right cancellative, because right multiplication by
a
is not injective:
a
⋅
a
=
a
and 1 ⋅
a
=
a
, however
a
≠ 1
Rewriting system
Format:
Pretty
Plain
Word:
Enter a word above to compute its normal form. Tips:
Lowercase letters stand for generators.
Spaces are ignored.
Numbers repeat the previous letter, e.g.
b90
.
Strategy:
Leftmost
Rightmost
Result:
1
1
Reduction order:
Left-to-right shortlex with
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
2
⇒
a
[1]
2.
a
(
ba
)
2
⇒ (
ab
)
2
[4]
3.
aba
b
2
⇒
aba
[5]
4.
a
b
2
ab
⇒
aba
[2]
# ab:aa=a,abbab=aba ab - - aa=a ababa=abab ababb=aba abbab=aba