On the representation of cardinalities of interval-valued fuzzy sets: the valuation property (Q691717)

In this paper the authors investigate the representation of the cardinalities of interval-valued fuzzy sets which they introduced in a previous paper [ibid. 158, No. 15, 1728--1750 (2007; Zbl 1120.03034)] by following Wygralak's axiomatic theory of the scalar cardinality of fuzzy sets. Considering the lattice \(\mathcal{L}^{1}=(\mathcal{L}^{1},\leq _{ \mathcal{L}^{1}}),\) where \[ \mathcal{L}^{1}=\{\left[ x_{1},x_{2}]:(x_{1},x_{2})\in [ 0,1]^{2},x_{1}\leq x_{2}\right] \] and \[ [ x_{1},x_{2}]\leq _{\mathcal{L}^{1}}[y_{1},y_{2}]\Leftrightarrow \left( x_{1}\leq y_{1}\text{ \& }x_{2}\leq y_{2}\right) , \] and the class of all finite interval-valued fuzzy sets over a set \(U\), denoted by \(\mathcal{F}_{\mathcal{L}^{1}}^{F}(U)\) (that is, the class of all mappings \(A:U\rightarrow \mathcal{L}^{1}\) with finite support), a mapping \(\mathrm{card}_{I}:\mathcal{F}_{\mathcal{L}^{1}}^{F}(U)\rightarrow \overline{L }_{+}^{I}\), where \[ \overline{L}_{+}^{I}=\{[x_{1},x_{2}]\in \mathcal{L}^{1}:x_{1}\geq 0\}, \] is called a scalar cardinality of interval-valued fuzzy sets if this mapping is coincident, monotonous and additive (see Definition 2.7 in the paper). The main results of the paper consist in numerous characterizations of the valuation property of cardinalities of finite interval-valued fuzzy sets. The mapping \(\mathrm{card}_{I}:\mathcal{F}_{\mathcal{L}^{1}}^{F}(U)\rightarrow \overline{L}_{+}^{I}\) has the valuation property for a t-norm \(\mathcal{T}\) and a t-conorm \(\mathcal{S}\) on \(\mathcal{L}^{1}\) if for all \(A,B\in \mathcal{F}_{\mathcal{L}^{1}}^{F}(U)\), \[ \mathrm{card}_{I}\left( A\cap _{\mathcal{T}}B\right) +\mathrm{card}_{I}\left( A\cup _{\mathcal{ S}}B\right) =\mathrm{card}_{I}\left( A\right) +\mathrm{card}_{I}\left( B\right) \text{.} \] Among the most important results I mention Theorems 3.8, 4.3, 4.10, 4.11, 5.4, 5.7, 5.8 and 6.1 (there are also many important corollaries derived from theses theorems), where the valuation property is characterized for the case when the cardinalities are generated by so-called cardinality patterns which can be represented by special classes of t-norms and t-conorms, respectively, such as generalized pseudo-t-representable t-norms and t-conorms, type-1 and type-2 pseudo-t-representable t-norms and t-conorms or t-representable t-norms and t-conorms, respectively.