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The Sandwich Theorems proved in this paper give a new method to show that the partition function a(n) of a partition identity A(x) := ∞ n=0 a(n)x n = ∞ n=1 (1 − x n) −p(n) satisfies the condition RT 1 lim n→∞ a(n − 1) a(n) = 1. This leads to numerous examples of naturally occuring classes of relational structures whose finite members enjoy a logical 0–1 law.
Combinatorial classes T that are recursively defined using combinations of the standard multiset, sequence, directed cycle and cycle constructions , and their restrictions, have generating series T(z) with a positive radius of convergence; for most of these a simple test can be used to quickly show that the form of the asymptotics is the same as that for(More)
The algebra of logic developed by Boole was not Boolean algebra. In this article we give a natural framework that allows one to easily reconstruct his algebra and see the difficulties it created for his successors. It is well known that modern Boolean algebra is connected with the work of George Boole [1815–1864], namely with his two books on a mathematical(More)
Given a finite set A there are only finitely many sequences of the form (Con(An))n>, or (Hom(An, A))n>1; where A is any algebra on A. From this we derive the fact that there are only finitely many primitive positive clones on A, which solves a problem posed by A. F. Danil'ienko in the 1970s. Consequently there are only finitely many model companions for(More)