Name
Title | ||
|---|---|---|
Generalized continuous and left-continuous t-norms arising from algebraic semantics for fuzzy logics |
Abstract | ||
|---|---|---|
This paper focuses on the issue of how generalizations of continuous and left-continuous t-norms over linearly ordered sets should be from a logical point of view. Taking into account recent results in the scope of algebraic semantics for fuzzy logics over chains with a monoidal residuated operation, we advocate linearly ordered BL-algebras and MTL-algebras as adequate generalizations of continuous and left-continuous t-norms respectively. In both cases, the underlying basic structure is that of linearly ordered residuated lattices. Although the residuation property is equivalent to left-continuity in t-norms, continuous t-norms have received much more attention due to their simpler structure. We review their complete description in terms of ordinal sums and discuss the problem of describing the structure of their generalization to BL-chains. In particular we show the good behavior of BL-algebras over a finite or complete chain, and discuss the partial knowledge of rational BL-chains. Then we move to the general non-continuous case corresponding to left-continuous t-norms and MTL-chains. The unsolved problem of describing the structure of left-continuous t-norms is presented together with a fistful of construction-decomposition techniques that apply to some distinguished families of t-norms and, finally, we discuss the situation in the general study of MTL-chains as a natural generalization of left-continuous t-norms. |
Year | DOI | Venue |
|---|---|---|
2010 | 10.1016/j.ins.2009.12.011 | Inf. Sci. |
Keywords | Field | DocType |
mv-algebras,monoidal t-norm based logic mtl,mtl-chains,mathematical fuzzy logic,continuous t-norms,complete chain,complete description,adequate generalization,simpler structure,general non-continuous case,fuzzy logic,left-continuous t-norms,monoidal residuated operation,underlying basic structure,algebraic semantics,general study,bl-chains,rational chains. | Ordered set,T-norm fuzzy logics,Discrete mathematics,Lattice (order),Generalization,Ordinal number,Fuzzy logic,Monoidal t-norm logic,Mathematics,Algebraic semantics | Journal |
Volume | Issue | ISSN |
180 | 8 | 0020-0255 |
Citations | PageRank | References |
12 | 0.69 | 41 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Carles Noguera | 1 | 462 | 33.93 |
| Francesc Esteva | 2 | 1885 | 200.14 |
| Lluís Godo | 3 | 888 | 56.28 |