By the means of lower and upper fuzzy approximations we define quasiorders. Their properties are used to prove our main results. First, we characterize the pairs of fuzzy sets that form fuzzy rough sets w.r.t. a t-similarity relation θ on U, for certain t-norms and implicators. If U is finite or the range of θ and of the fuzzy sets is a fixed finite chain, we establish conditions under which fuzzy rough sets form lattices. We show that this is the case for the min t-norm and any S-implicator defined by an involutive negator and the max co-norm.

We study the pseudo-Kleene algebras of the Dedekind–MacNeille completion of the ordered set of rough set determined by a reflexive relation. We characterize the cases when PBZ and PBZ*-lattices can be defined on these pseudo-Kleene algebras.

One way to view the multigranular rough set model is that the assessment of more experts are considered when determining the approximations of a set, instead of a single equivalence relation expressing the indiscernibility of the objects. There are two approaches for modeling this: the optimistic and the pessimistic multigranular rough set models. In this paper, we analyze both approaches from a lattice-theoretic point of view. We generalize existing results for two equivalence relations (two experts) to n equivalence relations, completing them with additional findings. We also characterize the order structures of optimistic and pessimistic rough sets and determine when they form complete, or even completely distributive lattices. Additionally, we examine the properties of the Dedekind-MacNeille completion of the poset of optimistic multigranular rough sets. Some applications for information tables and for recommendation theory are also presented.

The combination of fuzzy set theory and rough set theory has been discussed in a lot of research papers over the years. In this paper, we examine one such combination, namely the notion of rough L-fuzzy sets. We provide a representation theorem that determines when a pair of L-fuzzy sets is a rough L-fuzzy set, and we establish a connection between the lattice of rough fuzzy sets and the lattice of rough relations. Furthermore, we investigate the properties of the lattice of rough L-fuzzy sets and characterize the case when a three-valued Łukasiewicz-algebra can be defined on it.

We answer the question what properties a collection F of three-valued functions on a set U must fulfil so that there exists a quasiorder ≤ on U such that the rough sets determined by ≤ coincide with the core–support pairs of the functions in F. Applying this characterization, we give a new representation of rough sets determined by equivalences in terms of three-valued Łukasiewicz algebras of three-valued functions.

We study multigranulation spaces of two equivalences. The lattice-theoretical properties of so-called “optimistic” and “pessimistic” multigranular approximation systems are given. We also consider the ordered sets of rough sets determined by these approximation pairs.

Since the theory of rough sets was introduced by Zdzislaw Pawlak, several approaches have been proposed to combine rough set theory with fuzzy set theory. In this paper, we examine one of these approaches, namely fuzzy rough sets with crisp reference sets, from a lattice-theoretic point of view. We connect the lower and upper approximations of a fuzzy relation R to the approximations of the core and support of R. We also show that the lattice of fuzzy rough sets corresponding to a fuzzy equivalence relation R and the crisp subsets of its universe is isomorphic to the lattice of rough sets for the (crisp) equivalence relation E, where E is the core of R. We establish a connection between the exact (fuzzy) sets of R and the exact (crisp) sets of the support of R.

An alternative proof is given of the existence of greatest lower bounds in the imbalance order of binary maximal instantaneous codes of a given size. These codes are viewed as maximal antichains of a given size in the infinite binary tree of 0–1 words. The proof proposed makes use of a single balancing operation within the same imbalance poset of codes of the same fixed size, instead of moving back and forth between posets of codes corresponding to two different code sizes using expansion and contraction, as in the previous proofs of the existence of glb. It also makes use of a new combinatorial characterization of the imbalance order.

In this paper, we consider tolerances induced by irredundant coverings. Each tolerance R on U determines a quasiorder ≤ R by setting x ≤ R y if and only if R(x) ⊆ R(y). We prove that for a tolerance R induced by a covering H of U, the covering H is irredundant if and only if the quasiordered set (U,≤ R ) is bounded by minimal elements and the tolerance R coincides with the product ≥ R ° ≤ R . We also show that in such a case H = {↑m | m is minimal in (U,≤ R )}, and for each minimal m, we have R(m) = ↑m. Additionally, this irredundant covering H inducing R consists of some blocks of the tolerance R. We give necessary and sufficient conditions under which H and the set of R-blocks coincide. These results are established by applying the notion of Helly numbers of quasiordered sets.

The theory of rough sets provides a widely used modern tool, and in particular, rough sets induced by quasiorders are in the focus of the current interest, because they are strongly interrelated with the applications of preference relations and intuitionistic logic. In this paper, a structural characterisation of rough sets induced by quasiorders is given. These rough sets form Nelson algebras defined on algebraic lattices. We prove that any Nelson algebra can be represented as a subalgebra of an algebra defined on rough sets induced by a suitable quasiorder. We also show that Monteiro spaces, rough sets induced by quasiorders and Nelson algebras defined on T 0 -spaces that are Alexandrov topologies can be considered as equivalent structures, because they determine each other up to isomorphism.