© Hindawi Publishing Corp. A REMARK ON THE EXTENSION OF THE CONCEPT OF INCIDENCE ALGEBRAS TO NONLOCALLY FINITE PARTIALLY ORDERED SETS

An incidence algebra of a nonlocally finite partially ordered set Q is a very rare concept, perhaps nonexistent. In this note, we will attempt to construct such an algebra. 2000 Mathematics Subject Classification: 06A11. 1. Introduction. Let P be a partially ordered set (poset) and K a field of character-istic 0. The functions f: P ×P → K, such that x y implies f(x,y) = 0, are called the incidence functions of P over K. The set of such functions is denoted by ϑ(K,P). P is called locally finite if for every x,y ∈ P the interval [x,y] = {t ∈ P | x ≤ t ≤ y} is finite. When P is locally finite, ϑ(K,P) becomes a K-algebra under a multiplication (∗