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.
A partial order on a bounded lattice L is called t-order if it is defined by means of the t-norm on L. It is obtained that for a t-norm on a bounded lattice L the relation a⪯Tb iff a=T(x,b) for some x∈L is a partial order. The goal of the paper is to determine some conditions such that the new partial order induces a bounded lattice on the subset of all idempotent elements of L and a complete lattice on the subset A of all elements of L which are the supremum of a subset of atoms.
In this paper, some generating methods for principal topology are introduced by means of some logical operators such as uninorms and triangular norms and their properties are investigated. Defining a pre-order obtained from the closure operator, the properties of the pre-order are studied.
In this paper, we define the set of incomparable elements with respect to the triangular order for any t-norm on a bounded lattice. By means of the triangular order, an equivalence relation on the class of t-norms on a bounded lattice is defined and this equivalence is deeply investigated. Finally, we discuss some properties of this equivalence.