We construct a completely regular ordered space such that is an -space, the topology of is metrizable and the bitopological space is pairwise regular, but not pairwise completely regular. (Here denotes the upper topology and the lower topology of .)
In this paper, on a bounded lattice , we give a new approach to construct uninorms via a given uninorm on the subinterval (or ) of under additional constraint conditions on and . This approach makes our methods generalize some known construction methods for uninorms in the literature. Meanwhile, some illustrative examples for the construction of uninorms on bounded lattices are provided.
We give a representation of an observable on a fuzzy quantum poset of type II by a pointwise defined real-valued function. This method is inspired by that of Kolesárová [6] and Mesiar [7], and our results extend representations given by the author and Dvurečenskij [4]. Moreover, we show that in this model, the converse representation fails, in general.
We present a unified treatment of pointfree metrization theorems based on an analysis of special properties of bases. It essentially covers all the facts concerning metrization from Engelking [1] which make pointfree sense. With one exception, where the generalization is shown to be false, all the theorems extend to the general pointfree context.
In the context of the atomic poset, we propose several new methods of constructing the complete lattice and the algebraic lattice, and the mutual decision of relationship between atomic posets and complete lattices (algebraic lattices) is studied.
Pseudo -autonomous lattices are non-commutative generalizations of -autonomous lattices. It is proved that the class of pseudo -autonomous lattices is a variety of algebras which is term equivalent to the class of dualizing residuated lattices. It is shown that the kernels of congruences of pseudo -autonomous lattices can be described as their normal ideals.
We show that splitting of elements of an independent family of infinite regular size will produce a full size independent set.
We characterize ideals of ortholattices which are congruence kernels. We show that every congruence class determines a kernel.