An extension of the ordering based on nullnorms

summary:In this paper, we generally study an order induced by nullnorms on bounded lattices. We investigate monotonicity property of nullnorms on bounded lattices with respect to the $F$-partial order. Also, we introduce the set of incomparable elements with respect to the F-partial order for any nullnorm on a bounded lattice. Finally, we investigate the relationship between the order induced by a nullnorm and the distributivity property for nullnorms

On the constructions of t-norms and t-conorms on some special classes of bounded lattices

summary:Recently, the topic related to the construction of triangular norms and triangular conorms on bounded lattices using ordinal sums has been extensively studied. In this paper, we introduce a new ordinal sum construction of triangular norms and triangular conorms on an appropriate bounded lattice. Also, we give some illustrative examples for clarity. Then, we show that a new construction method can be generalized by induction to a modified ordinal sum for triangular norms and triangular conorms on an appropriate bounded lattice, respectively. And we provide some illustrative examples

A principal topology obtained from uninorms

summary:We obtain a principal topology and some related results. We also give some hints of possible applications. Some mathematical systems are both lattice and topological space. We show that a topology defined on the any bounded lattice is definable in terms of uninorms. Also, we see that these topologies satisfy the condition of the principal topology. These topologies can not be metrizable except for the discrete metric case. We show an equivalence relation on the class of uninorms on a bounded lattice based on equality of the topologies induced by uninorms

About the equivalence of nullnorms on bounded lattice

summary:In this paper, an equivalence on the class of nullnorms on a bounded lattice based on the equality of the orders induced by nullnorms is introduced. The set of all incomparable elements w.r.t. the order induced by nullnorms is investigated. Finally, the recently posed open problems have been solved

A characterization of uninorms on bounded lattices via closure and interior operators

summary:Uninorms on bounded lattices have been recently a remarkable field of inquiry. In the present study, we introduce two novel construction approaches for uninorms on bounded lattices with a neutral element, where some necessary and sufficient conditions are required. These constructions exploit a t-norm and a closure operator, or a t-conorm and an interior operator on a bounded lattice. Some illustrative examples are also included to help comprehend the newly added classes of uninorms

Nullnorms on bounded lattices

INCE, MEHMET AKIF/0000-0002-8146-5509WOS: 000362380600014Nullnorms are generalization of triangular norms and triangular conorms with a zero element in the interior of the unit interval. in this paper, we generalize nullnorms to an arbitrary bounded lattices. We show the existence of nullnorms with zero element a for arbitrary element a is an element of L\{0, 1} using the fact that some t-norms and t-conorms on an arbitrary fixed bounded lattice L always exist. As a by-product we give a method of constructing nullnorms. Using this method, we obtain also the smallest nullnorm and the greatest nullnorm with given zero element a. Some other construction methods for nullnorms are also added. (C) 2015 Elsevier Inc. All rights reserved.European Regional Development Fund in IT4 Innovations Centre of Excellence Project [CZ.1.05/1.1.00/02.0070]; [APVV-0073-10]The last author was supported by grants APVV-0073-10 and by the European Regional Development Fund in IT4 Innovations Centre of Excellence Project (CZ.1.05/1.1.00/02.0070)

Medians and nullnorms on bounded lattices

INCE, MEHMET AKIF/0000-0002-8146-5509WOS: 000368997800006Nullnorms are generalizations of triangular norms and triangular conorms with a zero element in the interior of the unit interval. in this paper, we work on nullnorms which are defined on an arbitrary bounded lattice. We introduce a general median-based method for constructing nullnorms by means of triangular norms and triangular conorms. Furthermore, we highlight a significant difference between the (existence and representation of) (idempotent) nullnorms on chains, distributive bounded lattices and general bounded lattices. (C) 2015 Elsevier B.V. All rights reserved.Research Fund of Karadeniz Technical University [12802]; Research and Development for Innovations Operational Programme - Structural Funds of Europe Union [CZ.1.05/1.1.00/02.0070]; state budget of the Czech Republic [CZ.1.05/1.1.00/02.0070]; [VEGA 1/0420/15]The work on this paper was supported by the Research Fund of Karadeniz Technical University (Project no. 12802) and grant VEGA 1/0420/15. A significant part of the work of the third author has been done also in connection with project IT4Innovations Centre of Excellence, reg. no. CZ.1.05/1.1.00/02.0070 supported by Research and Development for Innovations Operational Programme financed by Structural Funds of Europe Union and from the means of state budget of the Czech Republic