Diamond-free partial orders

This thesis presents initial work in attempting to understand the class of ‘diamond-free’ 3-cs-transitive partial orders. The notion of diamond-freeness, proposed by Gray, says that for any a ≤ b, the set of points between a and b is linearly ordered. A weak transitivity condition called ‘3-cs-transitivity’ is taken from the corresponding notion for cycle-free partial orders, which in that case led to a complete classification [3] of the countable examples. This says that the automorphism group acts transitively on certain isomorphism classes of connected 3-element structures. Classification for diamond-free partial orders seems at present too ambitious, but the strategy is to seek classifications of natural subclasses, and to test conjectures suggested by motivating examples. The body of the thesis is divided into three main inter-related chapters. The first of these, Chapter 3, adopts a topological approach, focussing on an analogue of topological covering maps. It is noted that the class of ‘covering projections’ between diamond-free partial orders can add symmetry or add cycles, and notions such as path connectedness transfer directly. The concept of the ‘nerve’ of a partial order makes this analogy concrete, and leads to useful observations about the fundamental group and the existence of an underlying cycle-free partial order called the universal cover. In Chapter 4, the work of [1] is generalised to show how to decompose ranked diamond- free partial orders. As in the previous chapter, any diamond-free partial order is covered by a specific cycle-free partial order. The paper [1] constructs a diamond-free partial order with cycles of height 1 from a different cycle-free partial order through which the universal covering factors. This is extended to construct a sequence of diamond- free partial orders with cycles of finite height which are not only factors but have the chosen diamond-free partial order as a ‘limit’. This leads to a better understanding of why structures with cycles only of height 1 are special, and the rest divide into structures with cycles of bounded height and a cycle-free backbone, and those for which the cycles have cofinal height. Even these can be expressed as limits of structures with cycles of 6 bounded height, though not directly. A variety of constructions are presented in Chapter 5, based on an underlying cycle- free partial order, and an ‘anomaly’, which in the simplest case given in [5] is a 2-level Dedekind-MacNeille complete 3-cs-transitive partial order, but which here is allowed to be a partial order of greater complexity. A rich class of examples is found, which have very high degrees of homogeneity and help to answer a number of conjectures in the negative.