Factorization from an order-theoretic view 1&2

TL;DR

This paper introduces an order-theoretic framework for ideal theory and factorization, connecting algebra and topology through B-ideals and lattice structures, inspired by Noether and Ehresmann.

Contribution

It develops a novel order-theoretic approach to ideal decomposition, replacing algebraic operations with lattice-theoretic constructs, applicable to general posets.

Findings

B-ideals form a complete lattice enabling decomposition theorems.

Decomposition is equivalent to complete distributivity in the lattice.

The cotopology based on multiplication differs from Zariski topology by using prime-powers.

Abstract

Drawing inspiration from Emmy Noether'set-theoretic foundations for algebra and Charles Ehresmann's topology without points, we adopt a new order-theoretic approach to ideal theory. For this we emphasize the order of divisibility in factorization and use it as a medium for relating algebra to topology 1. Replacing principal ideals and their intersections by equivalence classes and their collections respectively, we transform integral divisorial ideals into B-ideals in order to provide an order-theoretic frame for treating decomposition dispensing with addition. The idea of a B-ideal is connected closely with generalized-algebraicty originated from semantics for programme languages. 2. Since B-ideals constitute a complete lattice, we can utilize the fact that decomposition, which means that each element can be decomposed into the join of all elements way-below it, is equivalent to complete distributivity. B-ideals with decomposition theorems in themselves do not depend on algebraic structures and can be applied to any poset 3. Closed-set lattice is cotopology based on multiplication and independent of a partioular prime in the sense of pointless topology by Ehresmann. It differs from Zariski topology in using prime-powers rather than primes so that multiplicity in algebra acquires geometric meaning. 4. Factorial group is also a free module with multiplication instead of addition. Hence poset-theoretic constructions have corresponding algebraic analoques. They are introduced based on Noether's set-theoretic approach but quotient is within like a subset rather than outside.