#6762 ⟨
a
,
b
|
aba
=
b
,
aaaa
=
aa
⟩
Up:
Monoid enumeration
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#6761
⟨
a
,
b
|
aba
=
a
,
bbbb
=
bb
⟩
Next:
#6764
⟨
a
,
b
|
aba
=
b
,
aaaa
=
bb
⟩
Contents
Properties
Commutative structure
Staircase diagram
Rewriting system
Properties
Sum of relation sides is 10
Infinite non-cancellative commutative monoid
Element
b
has infinite order
Not cancellative, because multiplication by
ab
is not injective:
ab
⋅
a
2
=
ab
and
ab
⋅ 1 =
ab
, however
a
2
≠ 1
Commutative structure
Gröbner basis: ⟨
a
,
b
|
a
2
b
=
b
,
a
4
=
a
2
⟩
Cancellative quotient is isomorphic to ℤ
2
⊕ ℕ
Enveloping group is isomorphic to ℤ
2
⊕ ℤ
Group of units is isomorphic to ℤ
1
3 Archimedian components:
1,
a
,
b
Staircase diagram
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
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ba
⇒
ab
[5]
2.
a
2
b
⇒
b
[4]
3.
a
4
⇒
a
2
[2]
# ab:aba=b,aaaa=aa ab - - ba=ab aab=b aaaa=aa
