A partially ordered extension of the integers (Q1893122)

This paper consists of two parts. In Part I, the authors present a monotonic system of Post algebras of order \(\omega + \omega^*\) whose chain of Post constants is isomorphic with \(0 \leq 1 \leq 2 \leq \cdots \leq -3 \leq -2 \leq -1\). The basic development of the theory is presented and the study of filters and representability is discussed. Besides monotonic operations, other unary operations are considered; namely, disjoint operations, the quasi-complement, successor and predecessor operations. The successor operation is a modification of Peano's successor function for the natural numbers. Finally, the authors mention applications of Post algebras of order \(\omega + \omega^*\) to approximate reasoning. In Part II, a stronger version of Post algebras of order \(\omega + \omega^*\), called Post algebras of order \(\omega + \omega^*\) in the strict sense, is presented. These algebras are defined by adding an axiom, called axiom of pivot elimination, that guarantees that a pivot element \(e\) in the chain of constants does not exist. Fundamental properties, a filter theory and set-theoretic representability of these algebras are developed.