arXiv subjects

Esmaeil Rostami

Publications and source records attributed to Esmaeil Rostami.

2 recordsLinked to original sources

arXiv2025

On Semi-simplicity Results in Residuated Lattices

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.

math.LO↗

arXiv2025

On pseudo-irreducibility and Boolean lifting property of filters in residuated lattices

In this paper, we introduce the notion of a pseudo-irreducible filter in a residuated lattice and compare this concept with related notions such as prime and maximal filters. Then, we recall the Boolean lifting property for filters and present useful characterizations for this property using pseudo-irreducible filters and the residuated lattice of fractions. Next, we study the Boolean lifting property of the radical of a filter. Furthermore, we introduce weak MTL-algebras and residuated lattices that have the transitional property of radicals decomposition (TPRD) as generalizations of several algebraic structures, including Boolean algebra, MV-algebra, BL-algebra, MTL-algebra, and Stonean residuated lattice. Moreover, by comparing weak MTL-algebras with other classes of residuated lattices, we address an open question concerning the Boolean lifting property of the radical of a residuated lattice. Finally, we give a topological answer to an open question about the Boolean lifting property of the radical of a residuated lattice. Several additional results are also obtained, further enriching the understanding of the Boolean lifting property in residuated lattices.

math.LO↗