Clases ∀∃! y funciones algebraicas

CAMPERCHOLI MIGUEL; VAGGIONE DIEGO

Bahía Blanca

Congreso; X Congreso Dr. Antonio Monteiro; 2009

    Given a first order language L, a ∀∃!-sentence in L is a sentence of the form ∀x₁,...,x_{n}∃!y₁,...,y_{m}O(x,y), where O is a quantifier-free L-formula, and ∃! stands for "there exist unique". A ∀∃! class is the class of all models of a set of ∀∃!-sentences. A class of models K has the intersection property (i.p.) if given A∈K and A⊇A_{i}∈K, i∈I, such that ⋂_{i∈I}A_{i}≠∅, then ⋂_{i∈I}A_{i}∈K. It is easy to see that every ∀∃! class has the i.p.. C.C. Chang conjectured in <cite>Chang-IP-Conjecture</cite> that an elementary class K has the i.p. if and only if it is a ∀∃! class. In the paper <cite>Rabin-IP-Charac</cite> M. O. Rabin disproves Chang's conjecture, and also characterizes the elementary classes with i.p. as certain ∀∃ classes. It turns out that ∀∃! classes have a further property not necessarily true of every elementary class with i.p.: a class K is closed under fixed point submodels if for every A∈K and γ an automorphism of A the submodel with universe Fix(γ)={a∈A:γ(a)} is in K (whenever the set Fix(γ) is non-empty). In the case that the elementary class K is formed (up to isomorphism) by a finite number of finite models, we prove that this additional closure condition is enough to ensure that K is a ∀∃! class.<theorem/>Let K be a finite set of finite L-models. Then I(K) is finitely axiomatizable by ∀∃!-sentences if and only if K has the i.p. and K is closed under fixed point submodels.    Given a ∀∃!-sentence ϕ=∀x∃!yO(x,y) and a model A of ϕ we can implicitly define m functions [ϕ]_{i}^{A}:Aⁿ→A, 1≤i≤m, by ([ϕ]₁^{A}(a),...,[ϕ]_{n}^{A}(a))= the unique b such that A⊨O(a,b). The sentence ϕ is called an equational function definition sentence (EFD-sentence) if O is a conjunction of term equalities. A function f is algebraic on A provided there is an EFD-sentence ϕ such that f=[ϕ]_{i}^{A}. An interesting problem is: given a model A characterize all algebraic functions on A. In our talk we show how to apply Theorem 1 to solve this problem for certain kinds of models A.