By Trofimchuk E. P.
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Additional info for A. M. Samoilenkos method for the determination of the periodic solutions of quasilinear differential equations
9 This just refers to the fact that quantifiers range over elements, not subsets. 1. Suppose that a is from M and that M is a direct summand of N. Then ppM (a) = ppN (a). 7 to the injection of M into N and also to a projection from N onto M to conclude that a satisfies exactly the same pp conditions in M as in N. 1. 2. 12 Then ppM (a) is the intersection of the ppMi (a i ). Proof. If a satisfies a pp condition φ, say φ(x) is ∃y (x y)H = 0, choose b = (bi )i from M such that (a b)H = 0. Then (a i bi )H = 0 for each i and so each a i satisfies φ in Mi .
If M is any D-module, then End(M) acts transitively on the non-zero elements of M, so the only pp-definable subgroups of M are 0 and M. Sometimes we will use the following notation from model theory: if χ is a condition with free variables x, we write χ (x) if we wish to display these variables, and if a is a tuple of elements from the module M, then the notation M |= χ (a), read as “M satisfies χ (a)” or “a satisfies the condition χ in M”, means a ∈ χ (M), where χ (M) denotes the solution set of χ in M.
Also, the implication preordering, ψ ≤ φ, on pp conditions induces an ordering on equivalence classes of pp conditions. It is then immediate that the poset of equivalence classes of pp conditions (in, say, n free variables) is a lattice, with + and ∧ being the join and meet operations respectively. We denote this lattice of pp conditions by ppnR , often dropping the superscript when n = 1. We usually identify a point in this lattice – an equivalence class of pp conditions – with any pp condition which represents it.
A. M. Samoilenkos method for the determination of the periodic solutions of quasilinear differential equations by Trofimchuk E. P.