# Associative Polynomial Functions over Bounded Distributive Lattices

@article{Couceiro2011AssociativePF, title={Associative Polynomial Functions over Bounded Distributive Lattices}, author={Miguel Couceiro and Jean-Luc Marichal}, journal={Order}, year={2011}, volume={28}, pages={1-8} }

The associativity property, usually defined for binary functions, can be generalized to functions of a given fixed arity n ⩾ 1 as well as to functions of multiple arities. In this paper, we investigate these two generalizations in the case of polynomial functions over bounded distributive lattices and present explicit descriptions of the corresponding associative functions. We also show that, in this case, both generalizations of associativity are essentially the same.

## 19 Citations

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## References

SHOWING 1-10 OF 30 REFERENCES

Polynomial Functions Over Bounded Distributive Lattices

- MathematicsJ. Multiple Valued Log. Soft Comput.
- 2012

Several characterizations of those Ln → L mappings that are polynomial functions, i.e., functions which can be obtained from projections and constant functions using binary joins and meets are given.

Characterizations of discrete Sugeno integrals as polynomial functions over distributive lattices

- Mathematics, Computer ScienceFuzzy Sets Syst.
- 2010

The Arity Gap of Polynomial Functions over Bounded Distributive Lattices

- Mathematics, Computer Science2010 40th IEEE International Symposium on Multiple-Valued Logic
- 2010

This paper presents a characterization of the essential arguments of polynomial functions, which is used to show that almost all lattice polynometric functions have arity gap 1, with the exception of truncated median functions, whoseArity gap is 2.

Representations and Characterizations of Polynomial Functions on Chains

- Mathematics, Computer ScienceJ. Multiple Valued Log. Soft Comput.
- 2010

This paper discusses representations of lattice polynomial functions given in terms of standard simplices and presents new axiomatizations of these functions by relaxing some of the conditions given in [arXiv 0901.4888, arXiv 0808.2619] and by considering further conditions, namely comonotonic minitivity and maxitivity.

On the Lattice of Equational Classes of Boolean Functions and Its Closed Intervals

- MathematicsJ. Multiple Valued Log. Soft Comput.
- 2008

For | A |=2, the aim of this paper is to provide a better understanding of this uncountable lattice of equational classes of Boolean functions, by analyzing its “closed" intervals, for idempotent classes C 1 and C 2.

An Extension of Fung-Fu's Theorem

- MathematicsInt. J. Uncertain. Fuzziness Knowl. Based Syst.
- 1996

We give the general form of an idempotent, associative, nondecreasing and continuous binary aggregation operation in a connected order topological space. The particular case of the unit interval is…

Aggregation Functions (Encyclopedia of Mathematics and its Applications)

- Computer Science
- 2009

This is a comprehensive, rigorous and self-contained exposition of aggregation functions, which include triangular norms and conorms, copulas, means and averages, and those based on nonadditive integrals.

On some old and new problems in n-ary groups

- Mathematics
- 2001

In this paper some old unsolved problems connected with skew elements in n-ary groups are discussed.