On Semi-simplicity Results in Residuated Lattices

TL;DR

This paper advances the understanding of residuated lattices by introducing new filter concepts, providing algebraic and topological characterizations, and establishing a key equivalence between semi-simplicity and hyperarchimedean properties in finite cases.

Contribution

It introduces new types of filters and characterizations, and links semi-simplicity with hyperarchimedean properties in finite residuated lattices.

Findings

Semi-simple filters are characterized algebraically and topologically.

In finite residuated lattices, semi-simplicity is equivalent to being hyperarchimedean.

New insights into the relationship between simple and maximal filters.

Abstract

We develop the theory of residuated lattices by introducing and studying several new types of filters and related concepts, including semi-simple filters, essential filters, the socle of a filter, and independent families of filters. Our primary goal is to understand the inner structure of residuated lattices by analyzing these new objects. First, we establish the key properties of simple and essential filters. Next, we then provide both algebraic and topological characterizations for identifying when a filter is simple or essential. Furthermore, we use the concepts of the socle and independent families to delve deeper into the structure of filters and the residuated lattice itself. We also provide several characterizations for semi-simple filters and residuated lattices. A central result shows that for finite residuated lattices, being semi-simple is equivalent to being hyperarchimedean, highlighting the natural connection between these concepts. Complementary results deepening on the understanding of the relation between simple filters and maximal filters in residuated lattices are also established.