Replication via Invalidating the Applicability of the Fixed Point Theorem

TL;DR

This paper constructs various infinite complete partial orders (CPOs), demonstrating how invalidating the fixed point theorem's applicability can lead to the concept of replication in denotational semantics.

Contribution

It introduces new CPO constructions and shows how invalidating the fixed point theorem's applicability relates to the concept of replication.

Findings

Some CPOs allow fixed points for all continuous functions

Other CPOs do not support the fixed point theorem

Invalidation of the fixed point theorem relates to replication concept

Abstract

We present a construction of a certain infinite complete partial order (CPO) that differs from the standard construction used in Scott's denotational semantics. In addition, we construct several other infinite CPO's. For some of those, we apply the usual Fixed Point Theorem (FPT) to yield a fixed point for every continuous function $\mu:2\to 2$ (where 2 denotes the set $\{0,1\}$), while for the other CPO's we cannot invoke that theorem to yield such fixed points. Every element of each of these CPO's is a binary string in the monotypic form and we show that invalidation of the applicability of the FPT to the CPO that Scott's constructed yields the concept of replication.