#5589 ⟨
a
,
b
,
c
|
ab
=
c
,
bcbc
=
b
⟩
Up:
Monoid enumeration
Prev:
#5587
⟨
a
,
b
,
c
|
ab
=
c
,
bcac
=
c
⟩
Next:
#5590
⟨
a
,
b
,
c
|
ab
=
c
,
bcbc
=
c
⟩
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
b
is not injective:
b
⋅
aba
b
2
=
b
and
b
⋅ 1 =
b
, however
aba
b
2
≠ 1
Not right cancellative, because right multiplication by
b
is not injective:
b
(
ab
)
2
⋅
b
=
b
and 1 ⋅
b
=
b
, however
b
(
ab
)
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.
b
2
ab
⇒
ba
b
2
[3]
2.
(
ba
)
2
b
2
⇒
b
[4]
3.
c
⇒
ab
[1]
# abc:ab=c,bcbc=b ab/c - - bbab=babb bababb=b c=ab