Finitary ideals of direct products in quantales

Pascal Pankiti,C Nkuimi-Jugnia

doi:10.22481/recic.v3i1.8941

Abstract

The notion of quantale, which designates a complete lattice equipped with an associative binary multiplication distributing over arbitrary joins, appears in various areasof mathematics-in quantaloid theory, in non classical logic as completion of the Lindebaum algebra, and in different representations of the spectrum of a C∗ algebra asmany-valued and non commutative topologies. To put it briefly, its importance is nolonger to be demonstrated. Quantales are ring-like structures in that they share withrings the common fact that while as rings are semi groups in the tensor category ofabelian groups, so quantales are semi groups in the tensor category of sup-lattices.In 2008 Anderson and Kintzinger [1] investigated the ideals, prime ideals, radical ideals, primary ideals, and maximal of a product ring R × S of two commutativenon non necceray unital rings R and S: Something resembling rings are quantales byanalogy with what is studied in ring, we begin an investigation on ideals of a productof two quantales. In this paper, given two quantales Q1 and Q2; not necessarily withidentity, we investigate the ideals, prime ideals, primary ideals, and maximal ideals ofthe quantale Q1 × Q2

Highlights

The notion of quantale, which designates a complete lattice equipped with an associative binary multiplication distributing over arbitrary joins, appears in various areas of mathematics-in quantaloid theory, in non classical logic as completion of the Lindebaum algebra, and in different representations of the spectrum of a ⇤ algebra as many-valued and non com-C mutative topologies
For all quantales Q1 and Q2, non necessary with identity, we investigate the ideals, prime ideals, primary ideals, and maximal ideals of the quantale
We prove that if P2 is a prime ideal of Q2, Q1⇥P2 is a prime ideal of Q1 ⇥ Q2

Summary

Introduction

The notion of quantale, which designates a complete lattice equipped with an associative binary multiplication distributing over arbitrary joins, appears in various areas of mathematics-in quantaloid theory, in non classical logic as completion of the Lindebaum algebra, and in different representations of the spectrum of a ⇤ algebra as many-valued and non com-. R is a ring, let () Id R the collection of all left ideals of R; Id(RW) has a structure of quantale where the operWations and multiplication are defined as follows. Let M = (M, ⇤, e) be a monoid; the power set P(M ) has the structure of quantale where the suprema is given by the union and the multiplication is defined as follows: for each subsets A and B of M , A B is the set given by A B = {a ⇤ b : a 2 A, b 2 B}. The identity in this quantale is the subset {e}. It is proved in reference [4] that each quantale is isomorphic to a relational quantale

Ideals of product of two quantales

For each element a of Q1 and each element b of

Primeness

Conclusion