Let K := (K; +, •, D, 0, 1) be a differentially closed field of characteristic 0 with field of constants C.In the first part of the paper we explore the connection between Ax-Schanuel type theorems (predimension inequalities) for a differential equation E(x, y) and the geometry of the fibres U s := {y : E(s, y) ∧ y / ∈ C} where s is a non-constant element. We show that certain types of predimension inequalities imply strong minimality and geometric triviality of U s . Moreover, the induced structure on the Cartesian powers of U s is given by special subvarieties. In particular, since the j-function satisfies an Ax-Schanuel inequality of the required form (due to Pila and Tsimerman), applying our results to the j-function we recover a theorem of Freitag and Scanlon stating that the differential equation of j defines a strongly minimal set with trivial geometry.In the second part of the paper we study strongly minimal sets in the j-reducts of differentially closed fields. Let E j (x, y) be the (two-variable) differential equation of the j-function. We prove a Zilber style classification result for strongly minimal sets in the reduct K := (K; +, •, E j ). More precisely, we show that in K all strongly minimal sets are geometrically trivial or non-orthogonal to C. Our proof is based on the Ax-Schanuel theorem and a matching Existential Closedness statement which asserts that systems of equations in terms of E j have solutions in K unless having a solution contradicts Ax-Schanuel.
It is commonly known that the free Boolean algebra on n free generators is isomorphic to the Boolean algebra of Boolean functions of n variables. The free bounded distributive lattice on n free generators is isomorphic to the bounded lattice of monotone Boolean functions of n variables. In this paper, we introduce the concept of De Morgan function and prove that the free De Morgan algebra on n free generators is isomorphic to the De Morgan algebra of De Morgan functions of n variables. This is a solution of the problem suggested by B. I. Plotkin.
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