Nowhere simplicity in matroids

R. Downey
1983 Journal of the Australian Mathematical Society  
We examine the concepts of nowhere simplicity in a wide class of abstract dependence systems. Initially we examine how many of the existing results valid for L(o>), the lattice of r.e. sets, have analogues valid for more general lattices. For example, we show that any r. e. subspace of V x can be decomposed into a pair of nowhere simple subspaces. Later we begin an analysis of effective nowhere simplicity by giving a number of results which are new for both L(oi) and more general systems. In
more » ... eral systems. In particular we examine and define the concept of being a member of a maximal pair. We prove: THEOREM. An r.e. (closed) set A may be decomposed into an re. maximal pair if and only if A is simple. COROLLARY. There exist effectively nowhere simple (closed) sets in each r. e. degree (¥= 0).
doi:10.1017/s1446788700024757 fatcat:addjwp64wbbpjekxh7jptgwuha