The dual space of a finite simple Ockham algebra

Abstract

Let (L; f) be a finite simple Ockham algebra and let (X;g) be its dual space. We first prove that every connected component of X is either a singleton or a generalised crown (i.e. an ordered set that is connected, has length 1, and all vertices of which have the same degree). The representation of a generalised crown by a square (0,1)-matrix in which all line sums are equal is used throughout, and a complete description of X, including the number of connected components and the degree of the vertices, is given. We then examine the converse problem of when a generalised crown can be made into a connected component of (X; g). We also determine the number of non-isomorphic finite simple Ockham algebras that belong properly to a given subvariety P _2n,0. Finally, we show that the number of fixed points of (L; f) is 0,1, or 2 according to the nature of X.