We extend the concepts of splitting, reaping, and independent families to families of functions and permutations on ω and define associated cardinal characteristics s_f , s_p, r_f , r_p, if , and i_p. We study relationships among cov(M), non(M), and these cardinals. In this paper, we show that sf = non(M) = s_p, r_f = cov(M) ≤ r_p, and cov(M) ≤ i_f , i_p.

AMS subject classification: 03E17.

The aim of this paper is to investigate the relation between the strong and the "weak" or intuitionistic negation in Nelson algebras. To do this, we define the variety of Kleene algebras with intuitionistic negation and explore the Kalman's construction for pseudocomplemented distributive lattices. We also study the centered algebras of this variety.

AMS subject classification: 06D16, 03G25.

We define a procedure for translating a given first-order formula to an equivalent modal formula, if one exists, by using tableau-based bisimulation invariance test. A previously developed tableau procedure tests bisimulation invariance of a given first-order formula, and therefore tests whether that formula is equivalent to the standard translation of some modal formula. Using a closed tableau as the starting point, we show how an equivalent modal formula can be effectively obtained.

AMS subject classification: 03B45.

We investigate an expansion of positive intuitionistic logic obtained by adding a constant Ω  introduced by Lloyd Humberstone. Our main results include a sound and strongly complete axiomatization, some comparisons to other expansions of intuitionistic logic obtained by adding actuality and empirical negation, and an algebraic semantics. We also brie y discuss its connection to classical logic.

AMS subject classification: 03B20, 03B50.

We provide some statements equivalent in ZFC to GCH, and also to GCH above a given cardinal. These statements express the validity of the notions of replete and well-replete car- dinals, which are introduced and proved to be specially relevant to the study of cardinal exponentiation. As a byproduct, a structure theorem for linear orderings is proved to be equivalent to GCH: for every linear ordering L, at least one of L and its converse is universal for the smaller well-orderings.

AMS subject classification: 03E05.

AMS subject classification: 03C64, 12L12.