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In-Depth Information
3. A map
σ
:
O(r
1
,...,r
k
,r)
→
O(r
σ(
1
)
,...,r
σ(k)
,r)
,
f
−→
fσ
for any permu-
tation
σ
∈R
k
such that:
(a) (associativity) Whenever both sides make sense,
f
·
(g
1
·
(h
11
,...,h
1
i
1
),...,
g
k
·
(h
k
1
,...,h
ki
k
))
=
(f
·
(g
1
,...,g
k
))
·
(h
11
,...,h
1
i
1
,...,h
k
1
,...,h
ki
k
)
;
(b) For any
f
∈
O(r
1
,...,r
k
,r)
,
f
=
1
r
·
f
=
f
·
(
1
r
1
,...,
1
r
k
)
;
(c) For any
f
∈
O(r
1
,...,r
k
,r)
and
σ,σ
1
∈R
k
,
f(σσ
1
)
=
(f σ)σ
1
;
(d) For any
f
∈
O(r
1
,...,r
k
,r)
,
σ
∈R
k
and
g
1
∈
O(r
11
,...,r
1
i
1
,r
1
),...,g
k
∈
O(r
k
1
,...,r
ki
k
,r
k
)
,
(f σ)
·
(g
σ(
1
)
,...,g
σ(k)
)
=
(f
·
(g
1
,...,g
k
))ρ(σ)
where
:R
k
−→ R
i
1
+···+
i
k
is the obvious homomorphism;
(e) For any
f
ρ
∈
O(r
1
,...,r
k
,r)
,
g
1
∈
O(r
11
,...,r
1
i
1
,r
1
),...,g
k
∈
O(r
k
1
,...,
r
ki
k
,r
k
)
and
σ
1
∈R
i
1
,...,σ
k
∈R
i
k
,
f
(g
1
σ
1
,...,g
k
σ
k
)
=
f
(g
1
,...,g
k
)
1
(σ
1
,...,σ
k
),
·
·
where
1
:R
i
1
×···×R
i
k
−→ R
i
1
+···+
i
k
is the obvious homomorphism.
Thus, an R-operad
O
can be considered as an algebra where the variables (a car-
rier set) are the typed operators
f
O(r
1
,...,r
k
,r)
and the signature is composed
of the set of basic operations used for composition of the elements of this carrier
set:
1. The binary associative composition operation (a partial function) '
∈
·
' such that
·
f,(g
1
,...,g
k
)
=
f
·
(g
1
,...,g
k
),
if
f
is a
k
-ary typed operator.
2. The left and right identity (partial) operations 1
r
·
_, _
·
(
1
r
1
,...,
1
r
1
)
such that
for a
k
-ary typed operator
f
,
1
r
·
_
(f )
=
1
r
·
f
=
f,
_
·
(
1
r
1
,...,
1
r
1
)(f )
=
f
·
(
1
r
1
,...,
1
r
1
)
=
f.
3. The permutation (partial) operations,
σ
:
O(r
1
,...,r
k
,r)
→
O(r
σ(
1
)
,...,r
σ(k)
,r),
fσ
.
Thus, all operations in the signature of this R-operads algebra are the
partial
func-
tions (defined only for the subsets of elements of the carrier set of this algebra).
It is clear that in this formal syntax of R-operads algebra, the relational symbols
are left out of its syntax; thus, the terms of this algebra do not contain the rela-
tional symbols. Consequently, it appears fundamentally different from the relational
algebras (as, for example, the basic SPRJU Codd's algebra and its extensions, ex-
amined in Chap.
5
). However, in Chap.
5
, where we will consider the extensions
of the Codd's relational algebra, we will take a different point of view of the R-
algebras of operads by considering its carrier set (the variables) equal to the set
such that for each
f
∈
O(r
1
,...,r
k
,r)
,
σ(f)
=
R
of relational symbols (considered only as implicit “types” for operads). Hence, each
f
∈
O(r
1
,...,r
k
,r)
will be a composed operation of such a relational
Σ
α
algebra,
