#3095 ⟨
a
,
b
,
c
|
ab
=
aa
,
bacb
=1⟩
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Monoid enumeration
Prev:
#3092
⟨
a
,
b
,
c
|
ab
=
aa
,
baac
=1⟩
Next:
#3096
⟨
a
,
b
,
c
|
ab
=
aa
,
bacc
=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
⋅
b
=
a
2
and
a
⋅
a
=
a
2
, however
b
≠
a
Not right cancellative, because right multiplication by
c
is not injective:
c
b
2
a
⋅
c
=
c
and 1 ⋅
c
=
c
, however
c
b
2
a
≠ 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(
c
) = 0,
a
<
c
; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
4
c
⇒
a
[5]
2.
ab
⇒
a
2
[1]
3.
acb
⇒
bac
[4]
4.
b
2
ac
⇒ 1
[6]
# abc:ab=aa,bacb=1 ac/b - - aaaac=a ab=aa acb=bac bbac=1