#5005 ⟨
a
,
b
,
c
|
ab
=
c
,
aacbc
=1⟩
Up:
Monoid enumeration
Prev:
#4982
⟨
a
,
b
,
c
|
ab
=
a
,
ccccc
=1⟩
Next:
#5007
⟨
a
,
b
,
c
|
ab
=
c
,
aaccb
=1⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
a
2
b
(
ba
)
2
=
a
and
a
⋅ 1 =
a
, however
a
2
b
(
ba
)
2
≠ 1
Not right cancellative, because right multiplication by
a
2
b
2
ab
is not injective:
a
2
b
(
ba
)
2
⋅
a
2
b
2
ab
=
a
2
b
2
ab
and 1 ⋅
a
2
b
2
ab
=
a
2
b
2
ab
, however
a
2
b
(
ba
)
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(
a
) = deg(
b
) = 0,
a
<
b
; deg(
c
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
3
b
2
ab
⇒ 1
[2]
2.
c
⇒
ab
[1]
# abc:ab=c,aacbc=1 ab/c - - aaabbab=1 c=ab