#1564 ⟨
a
,
b
,
c
|
aaa
=
ab
,
cbc
=1⟩
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Monoid enumeration
Prev:
#1563
⟨
a
,
b
,
c
|
aaa
=
ab
,
cbb
=1⟩
Next:
#1566
⟨
a
,
b
,
c
|
aaa
=
ab
,
ccb
=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
c
2
=
a
and
a
⋅ 1 =
a
, however
a
2
c
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(
c
) = 0,
a
<
c
; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
3
c
2
⇒
a
[5]
2.
b
c
2
⇒ 1
[4]
3.
ab
⇒
a
3
[1]
4.
cb
⇒
bc
[3]
# abc:aaa=ab,cbc=1 ac/b - - aaacc=a bcc=1 ab=aaa cb=bc