#19809 ⟨
a
,
b
|
aaa
=
a
,
abab
=
abb
⟩
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Monoid enumeration
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#19808
⟨
a
,
b
|
aaa
=
a
,
abab
=
aba
⟩
Next:
#19810
⟨
a
,
b
|
aaa
=
a
,
abab
=
baa
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element
ca
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
a
2
=
a
and
a
⋅ 1 =
a
, however
a
2
≠ 1
Not right cancellative, because right multiplication by
a
is not injective:
a
2
⋅
a
=
a
and 1 ⋅
a
=
a
, however
a
2
≠ 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
Auxiliary generators:
c
=
ab
Reduction order:
Left-to-right shortlex with
a
<
c
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
Morphocompletion
: how the auxiliary generators were found.
#
Rule
Proof
1.
ab
⇒
c
[3]
2.
cb
⇒
c
2
[5]
3.
a
3
⇒
a
[1]
4.
a
2
c
⇒
c
[6]
# ab:aaa=a,abab=abb acb ab=c morph:2/0 ab=c cb=cc aaa=a aac=c