#24680 ⟨
a
,
b
|
aa
=
a
,
abbbab
=
bb
⟩
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Monoid enumeration
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#24679
⟨
a
,
b
|
aa
=
a
,
abbbab
=
ba
⟩
Next:
#24682
⟨
a
,
b
|
aa
=
a
,
abbbba
=
ab
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element
ba
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
b
2
⇒
b
2
[3]
3.
b
2
ab
⇒
b
3
[7]
4.
b
4
⇒
b
2
[6]
# ab:aa=a,abbbab=bb ab - - aa=a abb=bb bbab=bbb bbbb=bb