#5242 ⟨
a
,
b
|
aba
=
ab
,
aaaa
=
a
⟩
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Monoid enumeration
Prev:
#5241
⟨
a
,
b
|
aba
=
aa
,
bbbb
=
b
⟩
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#5259
⟨
a
,
b
|
aba
=
ab
,
baaa
=
b
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 10
Infinite non-cancellative non-commutative monoid
Element
c
has infinite order
Not left cancellative, because left multiplication by
c
is not injective:
c
⋅
a
=
c
and
c
⋅ 1 =
c
, however
a
≠ 1
Not right cancellative, because right multiplication by
a
is not injective:
a
3
⋅
a
=
a
and 1 ⋅
a
=
a
, however
a
3
≠ 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.
ca
⇒
c
[5]
3.
cb
⇒
c
2
[6]
4.
a
4
⇒
a
[2]
5.
a
3
c
⇒
c
[7]
# ab:aba=ab,aaaa=a acb ab=c morph:2/1 ab=c ca=c cb=cc aaaa=a aaac=c