A Fibonacci Sequence of Distributive Lattices

In this paper we describe an order-theoretic realization of the Fibonacci numbers 1, 2, 3, 5, 8, 13, . .. and of the Bisection Lucas numbers 3, 7, 18, 47, 123, .. . . The Bisection Lucas numbers are part of the Lucas sequence and are obtained from the Lucas numbers 2, 1, 3, 4, 7, 11, ... by deleting 2, 1, 4, and then every second number after that. We represent the Fibonacci numbers and the Bisection Lucas numbers as the cardinalities of sequences of distributive lattices that we glue together from simple building blocks. The gluing process is described in Section 2, and the main results are formulated in Section 3 as Theorem 3.1, Theorem 3.4, and their corollaries. In Section 1, we introduce some essential terminology and necessary facts about function lattices. For a more complete treatment of these topics, we refer the reader to the standard textbooks [1], [2], [5], and to [3]. For a related recursive construction of a sequence of modular lattices whose cardinalities are the polygonal numbers, we refer the reader to [6]. It should be noted that the construction discussed in [6] is very different from the construction discussed here in Section 2.