5.1 Ensayos de campo electromagnético
5.1.1 Ensayos de campo de impulso
In this section we show there exists a first-order L(J )-theory that axiomatises the class J of partialt-• algebras with∪-representations. Hence J, viewed as a class of L(J)-structures, is elementary. We do• the same for the class K of partial
•
−-algebras with
•
\-representations (as sets) and the class L of partial (t,• −)-algebras with (• ∪,•
•
\)-representations.
Definition 7.3.1. If A1 ⊆ A2 are similar partial algebras and the inclusion map is a partial-algebra
embedding then we say that A1 is a partial-subalgebra of A2. Let Ai = (Ai, Ω0, . . . ) be partial
algebras, for i ∈ I, and let U be an ultrafilter over I. The ultraproduct Πi∈IAi/U is defined in the
normal way, noting that, for example, [(ai)i∈I]
•
t [(bi)i∈I] (where ai, bi∈ Aifor all i ∈ I) is defined in
the ultraproduct if and only if {i ∈ I | ai
•
t biis defined in Ai} ∈ U . Ultrapowers and ultraroots also
have their normal definitions: an ultrapower is an ultraproduct of identical partial algebras and A is an ultraroot of B if B is an ultrapower of A.
It is clear that a partial-subalgebra of A is always a substructure of A, as relational structures, and also that any substructure of A is a partial algebra, that is, validates (7.1). However, in order for a relational substructure of A to be a partial-subalgebra it is necessary that it be closed under the partial operations, wherever they are defined in A.
It is almost trivial that the class of ∪-representable partial algebras is closed under partial-• subalgebras. This class is not however closed under substructures. Indeed it is easy to construct a partialt-algebra A with a disjoint-union representation but where an L(J)-substructure of A has no• disjoint-union representation. We give an example now.
Example 7.3.2. The collection ℘{1, 2, 3} of sets forms a disjoint-union partial algebra of sets and so is trivially a∪-representable partial• t-algebra, if we identify• t with• ∪.•
The substructure with domain ℘{1, 2, 3} \ {1, 2, 3} is not ∪-representable, because {1}• t• {2}, {2} t {3}, and {3}• t {1} all exist, so {1}, {2}, {3} would have to be represented by pairwise-• disjoint sets. But then {1, 2}t {3} would have to exist, which is not the case.•
We obtain the following corollary.
Corollary 7.3.3. The isomorphic closure of the class of disjoint-union partial algebras of sets is not axiomatisable by a universal first-orderL(J )-theory.
We now return to our objective of proving that the classes J, K, and L are elementary. Theorem 7.3.4. Let σ be any one of the signatures (∪), (•
•
\), or (∪,•
•
\). The class of partial algebras σ-representable as sets, viewed as a class of relational structures, is elementary.
Proof. The classes in question are J, K, and L. We are going to show that each of these classes is closed under isomorphisms, ultraproducts, and ultraroots. This is a well-known algebraic characterisation of elementarity (for example see [18, Theorem 6.1.16]).
We start with J. By definition, J is closed under isomorphism. Next we show that J is pseudoele- mentary, hence also closed under ultraproducts (Theorem 2.3.22).
Consider a two-sorted language, with an algebra sort and a base sort. The signature consists of a ternary operation J on the algebra sort, and a binary predicate ∈, written infix, of type base × algebra. Consider the formulas
a 6= b → ∃x((x ∈ a ∧ x 6∈ b) ∨ (x 6∈ a ∧ x ∈ b)) ∃cJ abc ↔ ¬∃x(x ∈ a ∧ x ∈ b)
J abc → ((x ∈ c) ↔ (x ∈ a ∨ x ∈ b))
where a, b, c are algebra-sorted variables and x is a base-sorted variable.
These formulas merely state that the base-sorted elements form the base of a representation of the algebra-sorted elements. Hence J is the class of J -reducts of restrictions of models of the formulas to algebra-sorted elements, that is, J is pseudoelementary. Hence J is closed under ultraproducts.
To show that J is closed under ultraroots, we show that ultraroots are (isomorphic to) partial subal- gebras. As we remarked earlier, J is closed under partial subalgebras.
Let A be an ultraroot of U ∈ J. Then A is isomorphic to its image A0under the diagonal embedding of A into U (by Corollary 2.3.20). To show A0 is a partial subalgebra of U, we need to show that for all a1, a2 ∈ A0, it holds that a1
•
tA0 a2is defined if and only a1
•
tU a2is defined, and that when they are
defined they are equal. The fact that whenever a1
•
tA0 a2 is defined, a1
•
tU a2 is defined and equals
a1
•
tA0 a2, follows from the fact that, viewed as J -structures, A0 is a substructure of U (since diagonal
embeddings are embeddings). Now suppose a1
•
tA0 a2is undefined. Then A0, (a1, a2) |= ¬∃yJ x1x2y.
As diagonal embeddings are elementary embeddings, it follows that U, (a1, a2) |= ¬∃yJ x1x2y, and
hence a1
•
tUa2is undefined. We conclude that J is closed under ultraroots.
We now know that J is closed under isomorphism, ultraproducts, and ultraroots. Then as J is elementary and closed under substructures it is universally axiomatisable, by the Ło´s–Tarski preservation theorem.
For K and L the same line of reasoning applies. Each is by definition closed under isomorphism. For K we show closure under ultraproducts via pseudoelementarity, using the formulas
a 6= b → ∃x((x ∈ a ∧ x 6∈ b) ∨ (x 6∈ a ∧ x ∈ b)) ∃cKabc ↔ (x ∈ b → x ∈ a)
Kabc → ((x ∈ c) ↔ (x ∈ a ∨ x 6∈ b))
and for L we do the same using the union of the formulas for J and the formulas for K. The proofs of closure under ultraroots are the same as for J.
We can now easily establish elementarity in all cases without composition. Corollary 7.3.5. Let σ be any signature whose symbols are a subset of {∪,•
•
\, ∅}. The class of partial algebras that areσ-representable by sets is elementary.
Proof. The previous theorem gives us the result for the three signatures (∪), (•
•
\), and (∪,•
•
\). Then as we noted in Remark 7.2.10, axiomatisations for these signatures yield axiomatisations for the signatures (∪, ∅), (•
•
\, ∅), and (∪,•
•
\, ∅) with the addition of a single extra axiom, either J(0, 0, 0) or K(0, 0, 0). The remaining cases, the empty signature and the signature (∅), trivially are axiomatised by the empty theory.
Corollary 7.3.6. Let σ be any signature whose symbols are a subset of {•
^,
•
\, ∅}. The class of partial algebras that areσ-representable by partial functions is elementary.
Proof. By Proposition 7.2.9(1) these representation classes are the same as those in Corollary 7.3.5.