Discrete t-norms and operations on extended multisets

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,…} rather than of concrete tokens{x,x,x,y,y,…} of them. The set of multisets on a universe is a partially ordered set for a functionally defined relationship of order. Moreover, it is a product of chains.Several pointwise defined operations as the addition, the union and the intersection between multisets have been defined and their properties investigated in several papers. The union and the intersection of two multisets is defined by means of the maximum and the minimum of the respective functions in 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⊎B and of course is not idempotent. The addition and the union satisfy the properties of t-conorms in the same way that 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 the divisibility. We distinguish the cases where the multisets are bounded or not.