A class of partially ordered sets

This paper is concerned with a theory of discrete partially ordered sets which satisfy a number of additional axioms. In particular, the sets will be assumed to be ‘stable’. That is, if a and b are two elements of such a set, with the property that all elements above b are also above a, and all elements below a are below b then—necessarily—we must have (as expected) a b. It is found that such sets have certain geometrical structures which are analogous to the usual Euclidean geometry. A probability theory for a class of such sets is proposed, and it is found that sets which contain particle-like chains of elements are most likely.