Interactive Computation of Homology of Finite Partially Ordered Sets

We outline a method for practical use of an interactive system (APL) to compute the homology of finite partially ordered sets. 1. Prerequisites. All partially ordered sets (posets) are assumed finite. Given a poset , we say that b covers a if b > a and a g c ^ h implies a = c or b = c. Since we deal with finite posets, the order relation can be obtained as the reflexive, transitive closure of the cover relation. Our programs allow the user to describe posets by the cover relation considered as a list of ordered pairs (more precisely, as an N x 2 matrix). For its own convenience, the program only accepts cover relations which are subsets of the usual order on the natural numbers. There is no difficulty representing any poset in this fashion, e.g., the cover relation of a "labeled Hasse diagram" [1]. In order to calculate the homology of a poset, we define a functor, C: .^0-> .V/77, from the category of finite posets to the category of finite chain complexes of abelian groups. If P is a poset, then the group of n-chains, Cn(P), is the free abelian group generated by symbols a0 < a l < • • • < an in P. The boundary operator r is defined on each generator a0 < a, < < an by the formula d(a0 < • • • <an)= £ (-l)'flo < ' <«,-< <«« O&i;&n; where a0 < • • • < a t < • • • < an is the generator of Cn _,(P) obtained from a 0 < • • • < a, < • • • < an by deleting the element a,. The n-th homology group of P, Hn(P), is defined to be the nth homology group of the complex C(P). For the category of small categories, #at, which includes ^0 , homology is usually defined as the homology of the simplicial set nerve of P, N(P). It is well known [3], [6], that these homology theories are isomorphic. 2. Method. We begin by describing some of the functions in our APL-workspace: PO: PO computes the graph of the < relation in poset P and represents it as an N x N matrix called POMAT. CHAIN: CHAIN computes the list of K-1-chains in the poset P from the list of K-chains and POMAT. BD: BD computes …