Related papers: Rough Sets Determined by Quasiorders
A mixed lattice is a partially ordered set with two mixed partial orderings that are linked by asymmetric upper and lower envelopes. These notions generalize the join and meet operations of a lattice. In the present paper, we study…
By the means of lower and upper fuzzy approximations we define quasiorders. Their properties are used to prove our main results. First, we characterize those pairs of fuzzy sets which form fuzzy rough sets w.r.t. a t-similarity relation…
In this paper, we give an algebraic completeness theorem for constructive logic with strong negation in terms of finite rough set-based Nelson algebras determined by quasiorders. We show how for a quasiorder $R$, its rough set-based Nelson…
We consider partially ordered sets of combinatorial structures under consecutive orders, meaning that two structures are related when one embeds in the other such that `consecutive' elements remain consecutive in the image. Given such a…
In this paper, we show that every quasiorder $R$ induces a Nelson algebra $\mathbb{RS}$ such that the underlying rough set lattice $RS$ is algebraic. We note that $\mathbb{RS}$ is a three-valued {\L}ukasiewicz algebra if and only if $R$ is…
For several types of information relations, the induced rough sets system RS does not form a lattice but only a partially ordered set. However, by studying its Dedekind-MacNeille completion DM(RS), one may reveal new important properties of…
We consider Kleene and Stone algebras defined on the completion DM(RS) of the ordered set of rough sets induced by a reflexive relation. We focus on cases where the completion forms a spatial and completely distributive lattice. We derive…
This paper pursues an investigation on groups equipped with an $L$-ordered relation, where $L$ is a fixed complete complete Heyting algebra. First, by the concept of join and meet on an $L$-ordered set, the notion of an $L$-lattice is…
Let $R$ be a not necessarily commutative ring with $1.$ In the present paper we first introduce a notion of quasi-orderings, which axiomatically subsumes all the orderings and valuations on $R$. We proceed by uniformly defining a coarsening…
Linearly repetitive cut and project sets are mathematical models for perfectly ordered quasicrystals. In a previous paper we presented a characterization of linearly repetitive cut and project sets. In this paper we extend the classical…
We characterize the order of principal congruences of a bounded lattice (also of a complete lattice and of a lattice of length 5) as a bounded ordered set. We also state a number of open problems in this new field.
A mixed lattice is a lattice-type structure consisting of a set with two partial orderings, and generalizing the notion of a lattice. Mixed lattice theory has previously been studied in various algebraic structures, such as groups and…
We answer the question what properties a collection $\mathcal{F}$ of three-valued functions on a set $U$ must fulfill so that there exists a quasiorder $\leq$ on $U$ such that the rough sets determined by $\leq$ coincide with the…
Rough sets are efficient for data pre-processing in data mining. Matroids are based on linear algebra and graph theory, and have a variety of applications in many fields. Both rough sets and matroids are closely related to lattices. For a…
Let $(W,S)$ be an arbitrary Coxeter system. For each word $\omega$ in the generators we define a partial order--called the {\sf $\omega$-sorting order}--on the set of group elements $W_\omega\subseteq W$ that occur as subwords of $\omega$.…
We characterize the order of principal congruences of a bounded lattice as a bounded ordered set. We also state a number of open problems in this new field.
We show that a lattice-ordered field (not necessarily commutative) is totally ordered if and only if each square is positive, answering a generalized question of Conrad and Dauns (Pacific J. Math. 30 (1969), 385--398) in the affirmative. As…
Brauer and Thrall conjectured that a finite-dimensional algebra over a field of bounded representation type is actually of finite representation type and a finite-dimensional algebra (over an infinite field) of infinite representation type…
The set of idempotents of a regular semigroup is given an abstract characterization as a regular biordered set in [2], and in [4] it is shown how a biordered set can be associated with a complemented modular lattice. Von Neumann has shown…
This paper has been withdrawn by the authors due to a crucial computational error. In this paper we deal with the finite case. We prove that a finite bounded ordered set can be represented as the order of principal congruences of a finite…