#3003 ⟨
a
,
b
,
c
|
aab
=
c
,
cbc
=
b
⟩
Up:
Monoid enumeration
Prev:
#3002
⟨
a
,
b
,
c
|
aab
=
c
,
cbc
=
a
⟩
Next:
#3004
⟨
a
,
b
,
c
|
aab
=
c
,
cbc
=
c
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 8
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
2
a
2
⋅
b
=
b
and 1 ⋅
b
=
b
, however
a
2
b
2
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 recursive path with deg(
b
) = deg(
a
) = 0,
b
<
a
; deg(
c
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
2
b
3
⇒
b
2
a
2
b
[3]
2.
a
2
b
2
a
2
b
⇒
b
[2]
3.
c
⇒
a
2
b
[1]
# abc:aab=c,cbc=b ba/c - - aabbb=bbaab aabbaab=b c=aab