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Discrete t-norms and operations on extended multisets. (English) Zbl 1176.03023
Summary: Multisets are set-like structures where an element can appear more than once. They are also called bags. A set means a collection of types of objects $$\{x,y,\dots \}$$ rather than of concrete tokens $$\{x,x,x,y,y,\dots \}$$ of them. The set of multisets on a universe is a partially ordered set for a functionally defined relation of order. Moreover, it is a product of chains. Several pointwise defined operations, such as addition, union and intersection, on multisets have been defined and their properties investigated in several papers. The union and the intersection of two multisets are defined by means of the maximum respectively the minimum of the respective functions in $$\overline {\mathbb N} = \mathbb N\cup \{\infty\}$$, and a lattice structure is obtained for the previously defined poset. But, for multisets, the addition is an important operation because it corresponds to the simultaneous consideration of two multisets on a universe (or the consideration on a multiset twice). The addition can be defined as the disjoint union, i.e. $$A+B=A\uplus B$$ and of course is not idempotent. The addition and the union satisfy the properties of t-conorms in the same way as the intersection is a t-norm on the poset of the multisets and they are functionally, even pointwise, defined. In this paper, we are concerned with some representation of all the possible functionally defined t-norms and t-conorms over the poset of the multisets that satisfy some interesting property, like divisibility. We distinguish the cases where the multisets are bounded or not.

##### MSC:
 3e+70 Nonclassical and second-order set theories 3e+72 Theory of fuzzy sets, etc.
##### Keywords:
discrete t-norms; fuzzy sets; extended multisets
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##### References:
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