#312 ⟨
a
,
b
,
c
|
aab
=
b
,
cac
=1⟩
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Monoid enumeration
Prev:
#311
⟨
a
,
b
,
c
|
aab
=
b
,
cab
=1⟩
Next:
#313
⟨
a
,
b
,
c
|
aab
=
b
,
cba
=1⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not right cancellative, because right multiplication by
b
is not injective:
a
2
⋅
b
=
b
and 1 ⋅
b
=
b
, 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
Reduction order:
Left-to-right shortlex with
c
<
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ac
⇒
ca
[3]
2.
c
2
a
⇒ 1
[4]
3.
c
2
b
⇒
ab
[5]
4.
a
2
b
⇒
b
[1]
# abc:aab=b,cac=1 cab - - ac=ca cca=1 ccb=ab aab=b