Abstract. In this paper, we show that every quasiorder R induces a Nelson algebra RS such that the underlying rough set lattice RS is algebraic. We note that RS is a three-valued Lukasiewicz algebra if and only if R is an equivalence. Our main result says that if A is a Nelson algebra defined on an algebraic lattice, then there exists a set U and a quasiorder R on U such that A ∼ = RS.

We show that for any tolerance $R$ on $U$, the ordered sets of lower and upper rough approximations determined by $R$ form ortholattices. These ortholattices are completely distributive, thus forming atomistic Boolean lattices, if and only if $R$ is induced by an irredundant covering of $U$, and in such a case, the atoms of these Boolean lattices are described. We prove that the ordered set $\mathit{RS}$ of rough sets determined by a tolerance $R$ on $U$ is a complete lattice if and only if it is a complete subdirect product of the complete lattices of lower and upper rough approximations. We show that $R$ is a tolerance induced by an irredundant covering of $U$ if and only if $\mathit{RS}$ is an algebraic completely distributive lattice, and in such a situation a quasi-Nelson algebra can be defined on $\mathit{RS}$. We present necessary and sufficient conditions which guarantee that for a tolerance $R$ on $U$, the ordered set $\mathit{RS}_X$ is a lattice for all $X \subseteq U$, where $R_X$ denotes the restriction of $R$ to the set $X$ and $\mathit{RS}_X$ is the corresponding set of rough sets. We introduce the disjoint representation and the formal concept representation of rough sets, and show that they are Dedekind--MacNeille completions of $\mathit{RS}$.Comment: Revised version (28 pages, 1 figure

Abstract. In this paper, the ordered set of rough sets determined by a quasiorder relation R is investigated. We prove that this ordered set is a complete, completely distributive lattice. We show that on this lattice can be defined three different kinds of complementation operations, and we describe its completely join-irreducible elements. We also characterize the case in which this lattice is a Stone lattice. Our results generalize some results of J. Pomyka la and J. A. Pomyka la (1988) and M. Gehrke and E. Walker (1992) in case R is an equivalence.

ABSTRACT. We prove that the extent partitions of a formal context K := (G, M, I) can be constructed from the box extents of it, which form a complete atomistic lattice. K is called a one-object extension of the subcontext (H, M, J) if it is obtained by adding a new element with attributes in M to the set H. We investigate the interplay between the box extents of (H, M, J) and those of its one-object extension K, and describe those extent partitions of (H, M, J) which can be extended to K. Extent partitions and classification systemsA partition of a set G is a family of nonempty, mutually disjoint subsets of G whose union is G, i.e. a familywhere T is some index set) withThe sets G t are called the classes of the partition π.2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 06B23; Secondary 0B05, 06B15. K e y w o r d s: concept lattice, extent partition, box extent, one-object extension. The research of the second and third author was carried out as part of the TAMOP-4.2.1. B-10/2/KONV., and respectively TAMOP 4.2

Intervals in binary or n-ary relations or other discrete structures generalize the concept of interval in an ordered set. They are defined abstractly as closed sets of a closure system on a set V, satisfying certain axioms. Decompositions are partitions of V whose blocks are intervals, and they form an algebraic semimodular lattice. Latticetheoretical properties of decompositions are explored, and connections with particular types of intervals are established.