Strong sequences and partition relations

The first result in partition relations topic belongs to Ramsey (1930). Since that this topic has been still explored. Probably the most famous partition theorem is Erdös-Rado theorem (1956). On the other hand in 60’s of the last century Efimov introduced strong sequences method, which was used for proving some famous theorems in dyadic spaces. The aim of this paper is to generalize theorem on strong sequences and to show that it is equivalent to generalized version of well-known Erdös-Rado theorem. It will be also shown that this equivalence holds for singulars. Some applications and conclusions will be presented too.

JV-Algèbres et JH*-Algèbres

RésuméIn the present article we generalize Theorem 2.3 of [6] in the case of JV algebras without a unit element and we obtain as a consequence that the multiplicativity of the involution ((xy)* = y*x*) in the definition of a JH*-algebra is redundant (see [3]). We end this paper with a theorem on unital JH*-algebra which is a nonassociative extension of the main result in [4].

Pseudo-idempotents in semigroups of functions

The aim of this paper is to generalize Theorem 2.10 (i) of [2]. As stated in [2] this theorem deals with the semigroup of all selfmaps on a discrete space and provides a characterization of H-classes which contain an idempotent. We will generalize this theorem to the case of other semigroups of functions on a discrete space, some semigroups of continuous functions on non-discrete topological spaces, and one semigroup of binary relations. The results in this paper form the main part of chapter 3 of [1]. Some results will be quoted from [1] without proof; the required proofs can easily be supplied by the reader.