Overview
This article investigates the upper and lower bounds of triangular subnorms, specifically those generated by continuous, strictly decreasing additive generators. The work extends to explore the existence of strict and nilpotent bounds for finite families of these subnorms. A dual analysis is conducted for triangular superconorms, which are generated by continuous, strictly increasing additive generators, yielding analogous results. The study also provides algorithms for computing these bounds, illustrating them with examples.
Research Context
The research focuses on triangular subnorms and superconorms within a mathematical framework. Specifically, it addresses subnorms created by continuous and strictly decreasing additive generators, and superconorms generated by continuous and strictly increasing additive generators. The core problems explored are the conditions under which these subnorms are comparable and the existence of specific types of bounds (strict and nilpotent) for collections of these mathematical structures.
Approach
The research approach involves a two-pronged strategy. Firstly, it establishes necessary and sufficient conditions for the comparability of triangular subnorms. This involves a direct analysis of their properties when derived from continuous, strictly decreasing additive generators. Secondly, it investigates the existence of strict and nilpotent bounds. This is conducted for a finite family of strict or nilpotent triangular subnorms, also generated by continuous, strictly decreasing additive generators. A dual approach is then employed: completely analogous results are derived for triangular superconorms, which are generated by continuous, strictly increasing additive generators. For both subnorms and superconorms, the methodology includes supplying corresponding algorithms for computing the identified bounds. These algorithms are then illustrated through various examples.
Findings
- Necessary and sufficient conditions for the comparability of triangular subnorms generated by continuous, strictly decreasing additive generators have been established.
- The existence of strict bounds for a finite family of strict triangular subnorms generated by continuous, strictly decreasing additive generators has been explored.
- The existence of nilpotent bounds for a finite family of nilpotent triangular subnorms generated by continuous, strictly decreasing additive generators has been explored.
- By duality, completely analogous results for triangular superconorms generated by continuous, strictly increasing additive generators have been derived.
- Algorithms for computing the bounds of triangular subnorms and superconorms have been supplied.
Why This Matters
The provision of algorithms for computing bounds, alongside illustrative examples, offers practical tools for analyzing these mathematical structures. This framework may be relevant in fields requiring the characterization or comparison of operations defined by specific types of additive generators.