On Semi-simplicity Results in Residuated Lattices
Esmaeil Rostami
TL;DR
This paper develops an order-theoretic framework for semisimplicity in residuated lattices by introducing simple and essential filters, the socle of a filter, and independent families of filters. It provides algebraic and topological characterizations of simple and essential filters, linking semi-simple filters to hyperarchimedean and semi-local conditions, with a key result showing that in finite residuated lattices, being semi-simple is equivalent to being hyperarchimedean. The work also analyzes decompositions of the lattice via direct sums determined by Boolean elements and investigates the radical, maximal spectrum, and socle to reveal deeper inner structure, including broad equivalences and density criteria for special filter families. Together, these results extend classical notions of semisimplicity into the residuated-lattice setting, offering new tools for structural analysis and potential applications to logic and algebraic semantics.
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.