Disjoint union is a partial binary operation returning the union of two sets if they are disjoint and undefined otherwise. A disjoint-union partial algebra of sets is a collection of sets closed under disjoint unions, whenever they are defined. We provide a recursive first-order axiomatisation of the class of partial algebras isomorphic to a disjointunion partial algebra of sets but prove that ...

The Bermond-Thomassen conjecture states that, for any positive integer r, a digraph of minimum out-degree at least 2r−1 contains at least r vertex-disjoint directed cycles. Thomassen proved that it is true when r = 2, and very recently the conjecture was proved for the case where r = 3. It is still open for larger values of r, even when restricted to (regular) tournaments. In this paper, we pre...

The Bermond-Thomassen conjecture states that, for any positive integer r, a digraph of minimum out-degree at least 2r − 1 contains at least r vertex-disjoint directed cycles. Thomassen proved that it is true when r = 2, and very recently the conjecture was proved for the case where r = 3. It is still open for larger values of r, even when restricted to (regular) tournaments. In this paper, we p...

The purpose of this article is two-fold. In the first place, we prove that a set is the image of a non empty closed convex subset of a real Banach space under an onto Fredholm operator of positive index if and only if it can be written as the union of {Dn : n ∈ N}, a non-decreasing family of non empty, closed, convex and bounded sets such that Dn + Dn+2 ⊆ 2Dn+1 for every n ∈ N. The second part ...

The Bermond-Thomassen conjecture states that, for any positive integer r, a digraph of minimum out-degree at least 2r − 1 contains at least r vertex-disjoint directed cycles. Thomassen proved that it is true when r = 2, and very recently the conjecture was proved for the case where r = 3. It is still open for larger values of r, even when restricted to (regular) tournaments. In this paper, we p...

In the paper a short proof is given for Kneser's conjecture. The proof is based on Borsuk's theorem and on a theorem of Gale. Recently, LOV~SZ has given a proof for Kneser's conjecture [4]. He used Borsuk's theorem. This fact gave the author the idea of the proof we present here. THEOREM (Kneser's conjecture [3]). If the n-tuples of a set of 2n + k elements are partitioned into k + 1 classes, t...

Arenas et al. [1] introduced the notion of lambda-closed sets as a generalization of locally closed sets. In this paper, we introduce the notions of lambda-locally closed sets, Lambda_lambda-closed sets and lambda_g-closed sets and obtain some decompositions of closed sets and continuity in topological spaces.

A family H of sets is said to be hereditary if all subsets of any set in H are in H; in other words, H is hereditary if it is a union of power sets. A family A is said to be intersecting if no two sets in A are disjoint. A star is a family whose sets contain at least one common element. An outstanding open conjecture due to Chvátal claims that among the largest intersecting sub-families of any ...

Algebraic structures and lattice structures of rough sets are basic and important topics in rough sets theory. In this paper we pointed out that a basic problem had been overlooked, that is the closeness of union and intersection operations of rough approximation pairs, i.e. (lower approximation, upper approximation). We present that the union and intersection operations of rough approximation ...

We show that there is some absolute constant c > 0, such that for any union-closed family F ⊆ 2, if |F| ≥ ( 1 2 − c)2n, then there is some element i ∈ [n] that appears in at least half of the sets of F . We also show that for any union-closed family F ⊆ 2, the number of sets which are not in F that cover a set in F is at most 2, and provide examples where the inequality is tight.