#310 ⟨
a
,
b
,
c
|
aab
=
b
,
caa
=1⟩
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Monoid enumeration
Prev:
#309
⟨
a
,
b
,
c
|
aab
=
b
,
bcc
=1⟩
Next:
#311
⟨
a
,
b
,
c
|
aab
=
b
,
cab
=1⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
c
is not injective:
c
⋅
a
2
c
=
c
and
c
⋅ 1 =
c
, however
a
2
c
≠ 1
Not right cancellative, because right multiplication by
b
is not injective:
c
⋅
b
=
b
and 1 ⋅
b
=
b
, however
c
≠ 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
<
c
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
cb
⇒
b
[3]
2.
a
2
b
⇒
b
[1]
3.
c
a
2
⇒ 1
[2]
4.
cab
⇒
ab
[4]
# abc:aab=b,caa=1 abc - - cb=b aab=b caa=1 cab=ab