#360 ⟨
a
,
b
,
c
|
aba
=
b
,
bac
=1⟩
Up:
Monoid enumeration
Prev:
#359
⟨
a
,
b
,
c
|
aba
=
b
,
acc
=1⟩
Next:
#361
⟨
a
,
b
,
c
|
aba
=
b
,
bbc
=1⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element
b
has infinite order
Not left cancellative, because left multiplication by
b
is not injective:
b
⋅
b
c
2
b
=
b
and
b
⋅ 1 =
b
, however
b
c
2
b
≠ 1
Not right cancellative, because right multiplication by
b
c
2
is not injective:
b
c
2
b
⋅
b
c
2
=
b
c
2
and 1 ⋅
b
c
2
=
b
c
2
, however
b
c
2
b
≠ 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(
c
) = deg(
b
) = 0,
c
<
b
; deg(
a
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
b
2
c
2
⇒ 1
[6]
2.
b
3
c
⇒
bc
b
2
[5]
3.
bc
b
2
c
⇒
b
[4]
4.
a
⇒
bc
[3]
# abc:aba=b,bac=1 cb/a - - bbcc=1 bbbc=bcbb bcbbc=b a=bc