Database Reference
In-Depth Information
B
(as in the mapping (
3.1
)). Consequently, we define the complex mapping-operad
M
(A
†
B)C
by:
M
(A
†
B)C
[
M
AC
,
M
BC
]:
A
†
B
→
C
(3.3)
where
M
AC
MakeOperads(
{
Φ
A
}
)
=
(q
1
,...,q
k
,
1
r
∅
}:
A
→
C
and
M
BC
(q
1
,...,q
m
,
1
r
∅
}
MakeOperads(
{
Φ
B
}=
)
:
B
→
C
are two simple mapping-operads
by introducing a structural-operator
. Information flux of complex mapping-
operads is different from the information flux of ordinary (simple) mapping-
operads:
Flux
α,
[
_
,
_
]
M
AC
,
M
BC
]
[
Flux(α,
M
AC
)
Flux(α,
M
BC
)
=
(
1
,a)
|
a
∈
Flux(α,
M
AC
)
∪
(
2
,b)
|
b
∈
Flux(α,
M
BC
)
.
(3.4)
Analogously, each complex mapping-operad
M
D(A
†
B)
from a simple (non-separa-
ted) schema
D
into separated schema
A
†
B
is denoted as
M
DA
,
M
DB
:
D
→
A
†
B
where
M
DA
:
D
→
A
and
M
DB
:
D
→
B
are ordinary (simple) mapping-
operads (equal to
{
1
r
∅
}
if they are not defined), with
Flux
α,
M
DA
,
M
DB
Flux(α,
M
DA
)
Flux(α,
M
DB
).
(3.5)
Any pair of simple mapping-operads
M
AC
,
M
BD
can be expressed by single com-
plex mapping-operad:
M
(A
†
B)(C
†
D)
=
M
AC
†
M
BD
M
AC
M
BD
:
A
†
B
→
C
†
D
,
(3.6)
with
Flux(α,
M
AC
Flux(α,
M
DB
)
.
Now we introduce formally these types of complex mapping-operads, as follows:
M
BD
)
Flux(α,
M
DA
)
Definition 15
The
n
-ary composition of separated simple schemas define the fol-
lowing four types of complex morphisms:
1. A mapping-operad
(
M
A
1
C
1
,...,
M
A
n
C
n
)
:
A
1
†
···
†
A
n
→
C
1
†
···
†
C
n
with
Flux
α,
(
M
A
1
C
1
,...,
M
A
n
C
n
)
Flux(α,
M
A
1
C
1
),...,Flux(α,
M
A
n
C
n
)
;
=
2. A mapping-operad
[
M
A
1
C
,...,
M
A
n
C
]:
A
1
†
···
†
A
n
→
C
with
Flux(α,
M
A
1
C
),...,Flux(α,
M
A
n
C
)
;
Flux
α,
M
A
1
C
,...,
M
A
n
C
]
=
[
M
DA
1
,...,
M
DA
n
:
→
A
1
†
···
A
n
with
3. A mapping-operad
D
†
Flux(α,
M
DA
1
),...,Flux(α,
M
DA
n
)
.
Flux
α,
M
DA
1
,...,
M
DA
n
=
