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. 1. Introduction. Let P be a partially ordered set (poset) and K a field of characteristic 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