#5333 ⟨
a
,
b
|
aaa
=
ab
,
bba
=
ba
⟩
Up:
Monoid enumeration
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#5332
⟨
a
,
b
|
aaa
=
ab
,
bba
=
ab
⟩
Next:
#5334
⟨
a
,
b
|
aaa
=
ab
,
bba
=
bb
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 10
Infinite non-cancellative non-commutative monoid
Element
b
has infinite order
Not left cancellative, because left multiplication by
b
is not injective:
b
⋅
ba
=
ba
and
b
⋅
a
=
ba
, however
ba
≠
a
Not right cancellative, because right multiplication by
a
is not injective:
b
2
⋅
a
=
ba
and
b
⋅
a
=
ba
, however
b
2
≠
b
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 recursive path with deg(
a
) = 0; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
6
⇒
a
4
[3]
2.
ab
⇒
a
3
[1]
3.
b
2
a
⇒
ba
[2]
# ab:aaa=ab,bba=ba a/b - - aaaaaa=aaaa ab=aaa bba=ba