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3. The permutation in a given operad's operation
f
∈
O(r
1
,...,r
k
,r)
of its rela-
tional symbols in
{
r
1
,...,r
k
}
is possible on the left-hand side of implication of
⇒
the expression
e
(
_
)
because the atoms of these relational symbols are con-
nected by the logical conjunction
∧
which is a commutative operation. Thus,
all properties for the permutations in point 4 of Definition
8
are satisfied for the
mapping-operads as well.
Corollary 3
The database schemas and the composition of their operad-mappings
can be represented by a sketch category
.
Proof
It is a direct result of Corollary
2
, the equality of these two representations,
and from Proposition
1
. From the fact that for each schema
A
=
(S
A
,Σ
A
)
there
is its identity mapping-operad
Id
A
:
A
→
A
,
and that the composition of mapping-operads is associative, we conclude that the
set of schemas (the objects) and the set of mapping-operads (the morphisms) can
define a sketch category. The associativity of composition is a consequence of the
associativity of the operads-composition '
such that
Id
A
={
1
r
|
r
∈
S
A
}∪{
1
r
∅
}
·
' defined in point 4(a) of Definition
8
.
Thus,
M
CD
◦
(
M
BC
◦
M
AB
)
=
(
M
CD
◦
M
BC
)
◦
M
AB
.
2.4.1 R-Algebras, Tarski's Interpretations and Instance-Database
Mappings
The theory of typed operads represents the syntax of the database mapping, trans-
lated from the schema-database logic into the algebraic framework. The semantic
part of the operad's algebra theory, corresponding to the semantics of FOL based
on Tarski's interpretations, is represented by the R-algebras which are particular
interpretations of the operads.
Let us now define the R-algebra of a database mapping-operad based on homo-
topy theory [
1
,
3
,
7
], where its abstract operations are represented by actual func-
tions:
U
Definition 10
For a given universe of values
and R-operad
O
, an R-algebra
α
\
consists of (here '
' denotes the set-difference):
1. A set
α(r)
for any
r
∈ R
, which is a set of tuples (relation), with the empty
relation
α(r
∅
)
=⊥={}
, unary universe-relation
α(r
∞
)
={
d
|
d
∈
U
}
, and
.The
α
∗
is the extension
of
α
to a list of symbols
α
∗
(
{
r
1
,...,r
k
}
)
{
α(r
1
),...,α(r
k
),α(r
∅
)
}
binary relation
α(r
)
=
R
=
for the “equality” type
r
.
2. A mapping function
α(q
i
)
:
R
1
×···×
R
k
−→
α(r)
for any
q
i
∈
O(r
1
,...,r
k
,r)
,
where for each 1
ar(r
i
)
≤
i
≤
k
,
R
i
=
U
\
α(r
i
)
if the place symbol
(
_
)
i
∈
q
i
is
preceded by negation operator
;
α(r
i
)
otherwise.
Consequently,
α
∗
(
{
q
1
,...,q
k
}
)
{
α(q
1
),...,α(q
k
)
}
¬
, and
(a) For any
r
∈R
,
α(
1
r
)
acts as an identity function on relation
α(r)
.The
q
⊥
=
α(
1
r
∅
)
:⊥→⊥
is the empty function (with
q
(
)
=
for the empty tuple
⊥
) with the empty graph).
