#7118 ⟨
a
,
b
|
ba
=
ab
,
aaaa
=
aa
⟩
Up:
Monoid enumeration
Prev:
#7117
⟨
a
,
b
|
ab
=
aa
,
bbbb
=
bb
⟩
Next:
#7119
⟨
a
,
b
|
ba
=
ab
,
aaaa
=
ab
⟩
Contents
Properties
Commutative structure
Staircase diagram
Rewriting system
Properties
Sum of relation sides is 10
Isomorphic to ℕ
(4 = 2)
⊕ ℕ
Infinite non-cancellative commutative monoid
Element
b
has infinite order
Not cancellative, because multiplication by
a
is not injective:
a
⋅
a
3
=
a
2
and
a
⋅
a
=
a
2
, however
a
3
≠
a
Commutative structure
Gröbner basis: ⟨
a
,
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
4 Archimedian components:
1,
a
,
b
,
ab
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
[1]
2.
a
4
⇒
a
2
[2]
# ab:ba=ab,aaaa=aa ab - - ba=ab aaaa=aa
