Some geometrical methods, the so called Triangular Schemes and Principles, are introduced and investigated for weak congruences of algebras. They are analogues of the corresponding notions for congruences. Particular versions of Triangular Schemes are equivalent to weak congruence modularity and to weak congruence distributivity. For algebras in congruence permutable varieties, stronger properties—Triangular Principles—are equivalent to weak congruence modularity and distributivity.
Eighteen open problems posed during FSTA 2010 (Liptovský Ján, Slovakia) are presented. These problems concern copulas, triangular norms and related aggregation functions. Some open problems concerning effect algebras are also included.
In decision processes some objects may not be comparable with respect to a preference relation, especially if several criteria are considered. To provide a model for such cases a poset valued preference relation is introduced as a fuzzy relation on a set of alternatives with membership values in a partially ordered set. We analyze its properties and prove the representation theorem in terms of particular order reversing involution on the co-domain poset. We prove that for every set of alternatives there is a poset valued preference whose cut relations are all relations on this domain. We also deal with particular transitivity of such preferences.