Invariant Definability (Extended Abstract)

To prove the conjecture for this case we observe that if K 0 is FOL{deenable with equality only, then L = INV K0 (L) by proposition 5, and hence EV EN is not K 0-invariantly LFP-deenable. Hence we have Corollary 26. If K 0 is f-categorical and closed under substructures and L 6 = INV K0 (L) then a linear order is deenable with parameters on K 0. In particular , this is the case if EV EN 2 INV K0 (LFP), L INV K0 (LFP) or P INV K0 (LFP). The general case of the conjecture seems diicult, as K can be rather complicated and the assumption that there is no parametrically deenable order is hard to handle. The latter reminds us of Shelah's characterization of stability, cf. She90], although the situation is not quite the same: Our models are nite and the order, if present, has to be total. On the other side, we do not deal with complete theories. E. Rosen has announced that the conjecture is true if K consists of a class of pre-orders, i.e linear orders of equivalence classes. More precisely: Theorem 27 (E. Rosen). If K is a class of pre-orders such that every K 1 2 P is K-invariantly LFP-deenable then ORD is parametrically LFP-deenable (actually even FOL-deenable) in K. The proof will be published in RM97]. References AF90] M. Ajtai and R. Fagin. Reachability is harder for directed than for undirected nite graphs. 1 {formulae on nite structures. Example 5. Let = fExyg, and let K 2 = fA j A is a chaing. Here, A is a chain ii it is a connected simple undirected graph such that every vertex has degree exactly 2. Observe that, up to isomorphism, there is a unique chain of each nite cardinality, and that it is non-rigid with automorphism group of size = 2 card(A). Observe that a linear order is FOL{deenable with parameters over K 1 and LFP{deenable with parameters over K 2. We conjecture that this deenability property provides both a necessary and suucient condition for INV K (LFP) to contain P, (the suuciency having just been established). More precisely, as stated in the introduction Conjecture: For any class K, P INV K (LFP) ii a linear order is parametrically deenable on K. This is the major conjecture of this note, though one can also ask many similar questions concerning invariant deenability …