1. INTRODUCTION. In this paper we introduce the reader to two remarkable results in the theory of sets. Both are more than fifty years old, but neither one appears to be well known among nonspecialists. Each one states that a certain proposition implies the Axiom of Choice. First we describe the results, then review definitions, then, finally, present the proofs, most of which are straightforwa...

In this paper, we enlarge the language of triangle algebra by addinga unary operation that describes properties of a state. Thesestructure algebras are called state triangle algebra. The vitalproperties of these algebras are given. The notion of state interval-valued residuated lattice (IVRL)-filters are introduced and givesome examples and properties of them are given. ...

We study the class of functions called monodiffric of the second kind by Rufus Philip Isaacs. They are discrete analogues of holomorphic functions of one or two complex variables. Discrete analogues of the Cauchy–Riemann operator, of domains of holomorphy in one discrete variable, and of the Hartogs phenomenon in two discrete variables are investigated. Two fundamental solutions to the discrete...

Let Z be an analytic subvariety of pure codimension p of a pseudoconvex set X in C. We introduce weighted Koppelman formulas on Z that provides solutions to the ∂̄-equation. If φ is a smooth (0, q + 1)-form on Z with ∂̄φ = 0, we find a smooth (0, q)-form ψ on Zreg with at most polynomial growth at Zsing such that ∂̄ψ = φ. We also present Hartogs theorems for ∂̄-closed forms on Z. As a consequence w...