Characterization of uninorms with continuous underlying t-norm and t-conorm by their set of discontinuity points

Uninorms with continuous underlying t-norm and t-conorm are discussed and properties of the set of discontinuity points of such a uninorm are shown. This set is proved to be a subset of the graph of a special symmetric, surjective, non-increasing multi-function. A sufficient condition for a uninorm to have continuous underlying operations is also given. Several examples are included.Comment: Submitted to IEEE TFS (on October 27, 2014) as a part of the longer paper. This part remained in IEEE (resubmission on April 27, 2015). arXiv admin note: text overlap with arXiv:1506.07820, arXiv:1506.0695

Group-like Uninorms

Uninorms play a prominent role both in the theory and the applications of Aggregations and Fuzzy Logic. In this paper the class of group-like uninorms is introduced and characterized. First, two variants of a general construction -- called partial-lexicographic product -- will be recalled from \cite{Jenei_Hahn}; these construct odd involutive FL$_e$-algebras. Then two particular ways of applying the partial-lexicographic product construction will be specified. The first method constructs, starting from $\mathbb R$ (the additive group of the reals) and modifying it in some way by $\mathbb Z$'s (the additive group of the integers), what we call basic group-like uninorms, whereas with the second method one can modify any group-like uninorm by a basic group-like uninorm to obtain another group-like uninorm. All group-like uninorms obtained this way have finitely many idempotent elements. On the other hand, we prove that given any group-like uninorm which has finitely many idempotent elements, it can be constructed by consecutive applications of the second construction (finitely many times) using only basic group-like uninorms as building blocks. Hence any basic group-like uninorm can be built using the first method, and any group-like uninorm which has finitely many idempotent elements can be built using the second method from only basic group-like uninorms. In this way a complete characterization for group-like uninorms which possess finitely many idempotent elements is given: ultimately, all such uninorms can be built from $\mathbb R$ and $\mathbb Z$. This characterization provides, for potential applications in several fields of fuzzy theory or aggregation theory, the whole spectrum of choice of those group-like uninorms which possess finitely many idempotent elements

Characterization of uninorms with continuous underlying t-norm and t-conorm by means of an extended ordinal sum

The uninorms with continuous underlying t-norm and t-conorm are characterized via an extended ordinal sum construction. Using the results of [18], where each uninorm with continuous underlying operations was characterized by properties of its set of discontinuity points, it is shown that each such a uninorm can be decomposed into an extended ordinal sum of representable uninorms, continuous Archimedean t-norms, continuous Archimedean t-conorms and internal uninorms.Comment: Originally submitted to IEEE TFS (on October 27, 2014) as a part of the longer paper which was after recommendation divided into two papers. This part was submitted to International Journal of Approximate Reasoning on February 04, 2015. arXiv admin note: text overlap with arXiv:1506.0695