#2042 ⟨
a
,
b
,
c
|
aaaa
=1,
bbbc
=1⟩
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Monoid enumeration
Prev:
#2041
⟨
a
,
b
,
c
|
aaaa
=1,
bacc
=1⟩
Next:
#2044
⟨
a
,
b
,
c
|
aaaa
=1,
bbcc
=1⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element
a
2
ba
has infinite order
Not left cancellative, because left multiplication by
b
is not injective:
b
⋅
b
2
cb
=
b
and
b
⋅ 1 =
b
, however
b
2
cb
≠ 1
Not right cancellative, because right multiplication by
b
2
c
is not injective:
b
2
cb
⋅
b
2
c
=
b
2
c
and 1 ⋅
b
2
c
=
b
2
c
, however
b
2
cb
≠ 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 shortlex with
a
<
b
<
c
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
4
⇒ 1
[1]
2.
b
3
c
⇒ 1
[2]
# abc:aaaa=1,bbbc=1 abc - - aaaa=1 bbbc=1