The cardinal power of a set A is denoted by IA . Two sets A,, A2 are said to e almost disjoint if IA1 U A2I < lA i l (i = 1, 2) . e call B a transversal of the disjoint non-empty sets A,, (v E M) if Bc U A,, and B intersects each A,. (v E M) in a singleton . D1 An old and well known theorem of W . SIERPINSKI is that an infinite set of po-ver m contains more than m subsets of power m which are p...

Systems of equations with sets of integers as unknowns are considered, with the operations of union, intersection and addition of sets, S + T = {m + n | m ∈ S, n ∈ T}. These equations were recently studied by the authors (“On equations over sets of integers”, STACS 2010 ), and it was shown that their unique solutions represent exactly the hyperarithmetical sets. In this paper it is demonstrated...

Let M = (L , ∗) be a GL-monoid. An M-valued preordered set is an L-subset endowed with a reflexive and M-transitive L-relation, it is essentially a category enriched in a quantaloid generated by M. This paper presents a study of M-valued preordered sets with emphasis on symmetrization and the Cauchy completion. The main result states that symmetrization and the Cauchy completion of M-valued pre...

The matching property of nested sets, which has been studied in [6], is extended to generalized nested sets, defined in [9]. Pairs of matching generalized nested sets on [m] are constructed and used for the creation of all generalized planar permutations on [m].

Using techniques from TRO equivalence of masa bimodules we prove various transference results: We show that when $$\alpha $$ is a group homomorphism which pushes forward the Haar measure G to absolutely continuous with respect on H, then $$(\alpha \times \alpha )^{-1}$$ preserves sets compact operator synthesis, and conversely onto. also similar preservation results for Ditkin M-sets, obtaining...

This paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. It incorporates the most important settings of lattice-valued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$-fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$-fuzzy topology on $L$-fuzzy sets of C.~Guido. Moreover, its respe...

A well known consequence of the Borsuk-Ulam theorem is that if the d-dimensional sphere S is covered with less than d + 2 open sets, then there is a set containing a pair of antipodal points. In this paper we provide lower and upper bounds on the minimum number of open sets, not containing a pair of antipodal points, needed to cover the d-dimensional sphere n times, with the additional property...

Feed-forward networks are generally trained to represent functions or many-toone (m-o) mappings. In this paper however a feed-forward network with modified training algorithm is considered to represent multi-valued or one-tomany (o-m) mappings. The o-m mapping is viewed as an m-o mapping where the values corresponding to a value of the independent variable are sets. Thus the problem of represen...

The primary objective of this paper is to study core sets in the setting m-subharmonic functions on class (non-compact) Kähler manifolds. Core are investigated different aspects by considering various classes plurisubharmonic functions. One crucial concepts studying structure kind pseudoconcavity. In a more general way, we will have defined with respect functions, which call m-core our setting,...