∀s ∈L, s_{G} | ={((x_{0},y_{0}),..., (x_{ns-1}, y_{ns-1}))| s_{E}(x_{0},..., x_{ns-1}) ∧s_{F}(y_{0},..., y_{ns-1})} |
={(x×y) | x∈s_{E} ∧ y∈s_{F}} | |
={z∈G^{ns} | π_{0} ০ z ∈s_{E} ∧ π_{1} ০ z ∈s_{F}} |
f ∈ Mor_{L}(F,P) | ⇔∀s∈L, ∀x∈s_{F}, f০x ∈ s_{P} |
⇔∀s∈L, ∀x∈s_{F}, ∀i∈I, π_{i}০f ০x ∈ s_{i} | |
⇔ ∀i∈I, ∀s∈L, ∀x∈s_{F}, f_{i} ০x ∈ s_{i} | |
⇔ ∀i∈I, f_{i} ∈ Mor_{L}(F,E_{i}) |
Now let us define the product of algebras, taking as L an
algebraic language (set of operation symbols): for any family of L-algebras
(E_{i})_{i}_{∈}_{I}
, their product P=∏_{i}_{∈}_{I}
E_{i} is an L-algebra interpreting each
symbol s∈L on each tuple x=∏_{i∈I
}x_{i} ∈ P^{ns},
by
This comes as a particular case of the product of relational
systems, obtained by replacing each n-ary operation s
by its graph, the (n+1)-ary relation (y=s(x)):
An operation defined by a term in a product of algebras, coincides with the product of operations defined by the same term in all component algebras.
For any L-algebras E, F, and any f : E → F, f ∈ Mor_{L}(E,F) ⇔ Gr f ∈ Sub_{L} (E×F).
Proof 1: ∀s∈L, | (∀x∈ E^{ns}, f(s_{E}(x)) = s_{F}(f০x)) | ⇔∀x∈E^{ns},∀y∈F^{ns},(y=f০x⇒ f(s_{E}(x))=s_{F}(y)) |
⇔ ∀(x×y)∈(Gr f)^{ns}, (s_{E}(x), s_{F}(y))∈ Gr f | ||
⇔ ∀z∈ (Gr f)^{ns}, s_{E×F}(z) ∈ Gr f |
By seeing morphisms as subalgebras, we can write another
construction of recursive sequences u∈Mor_{(0,S)}(ℕ,(E,a,f)),
as follows.
Let M be the minimal subalgebra of ℕ×E_{a,f}, and let A={n∈ℕ | ∃!x∈E, (n,x)∈M}.
As M is a minimal (0,S)-algebra, M = {(0,a)}∪ Im S_{M}.
Substituting this into the definition of A we get
∀p∈ℕ, p∈A ⇔ (∃!y∈E, (p=0 ∧ y=a)∨∃(n,x)∈M, (p=Sn ∧ y=f(x))).From 0 ∉ Im S we get 0∈A, and
∀n∈ℕ, Sn∈A ⇔ ∃!y∈E, ∃(n',x)∈M, (Sn=Sn' ∧ y=f(x)).From the injectivity of S we get
∀n∈ℕ, Sn∈A ⇔ ∃!y∈E, ∃x∈E, (n,x)∈M∧ y=f(x).Thus (∀n∈A, Sn∈A), so that A = ℕ, i.e. M is the graph of a function u ∈ E^{ℕ}. As M ∈ Sub(ℕ×E_{a,f}), we conclude u ∈ Mor_{(0,S)}(ℕ,E_{a,f}).