Generalization of Scott's formula for retractions from generalized Alexandroff's cube

In the paper [2] the following theorem is shown: Theorem (Th. 3,5, [2]), If α=0 or δ=∞ or α?δ, then a closure space X is an absolute extensor for the category of 〈α, δ〉 -closure spaces iff a contraction of X is the closure space of all 〈α, δ〉-filters in an 〈α, δ〉-semidistributive lattice. In the case when α=ω and δ=∞, this theorem becomes Scott's theorem: Theorem ([7]). A topological space X is an absolute extensor for the category of all topological spaces iff a contraction of X is a topological space of “Scott's open sets” in a continuous lattice. On the other hand, when α=0 and δ=ω, this theorem becomes Jankowski's theorem: Theorem ([4]). A closure space X is an absolute extensor for the category of all closure spaces satisfying the compactness theorem iff a contraction of X is a closure space of all filters in a complete Heyting lattice. But for separate cases of α and δ, the Theorem 3.5 from [2] is proved using essentialy different methods. In this paper it is shown that this theorem can be proved using, for retraction, one uniform formula. Namely it is proved that if α= 0 or δ= ∞ or α ? δ and \(F_{\alpha ,\delta } \left( L \right) \subseteq B_{\alpha ,\delta }^\mathfrak{n} \) and if L is an 〈α, δ〉-semidistributive lattice, then the function $$r:{\text{ }}B_{\alpha ,\delta }^\mathfrak{n} \to F_{\alpha ,\delta } \left( L \right)$$ such that for x ε ? ( \(\mathfrak{n}\) ): (*) $$r\left( x \right) = inf_L \left\{ {l \in L|\left( {\forall A \subseteq L} \right)x \in C\left( A \right) \Rightarrow l \in C\left( A \right)} \right\}$$ defines retraction, where C is a proper closure operator for \(B_{\alpha ,\delta }^\mathfrak{n} \) . It is also proved that the formula (*) defines retraction for all 〈α, δ〉, whenever L is an 〈α, δ〉 -pseudodistributive lattice. Moreover it is proved that when α=ω and δ=∞, the formula (*) defines identical retraction to the formula given in [7], and when α = 0 and δ=ω, the formula (*) defines identical retraction to the formula given in [4].     

Interpolation and amalgamation properties in varieties of equivalential algebras

Important positive as well as negative results on interpolation property in fragments of the intuitionistic propositional logic (INT) were obtained by J. I. Zucker in [6]. He proved that the interpolation theorem holds in purely implicational fragment of INT. He also gave an example of a fragment of INT for which interpolation fails. This fragment is determined by the constant falsum (), well known connectives: implication () and conjunction (), and by a ternary connective defined as follows: (p, q, r)= ^df (pq)(pr).Extending this result of J. I. Zucker, G. R. Renardel de Lavalette proved in [5] that there are continuously many fragments of INT without the interpolation property.This paper is meant to continue the research mentioned above. To be more precise, its aim is to answer questions concerning interpolation and amalgamation properties in varieties of equivalential algebras, particularly in the variety determined by the purely equivalential fragment of INT.     

Q-ultrafilters and normal ultrafilters in B-algebras

The first part of the paper deals with some subclasses of B-algebras and their applications to the semantics of SCI _B , the Boolean strengthening of the sentential calculus with identity (SCI). In the second part a generalization of the McKinsey-Tarski construction of well-connected topological Boolean, algebras to the class of B-algebras is given.     

Development and validation of adolescent-perceived microsystem scales: Social support,daily hassles,and involvement

Developed and validated instruments for urban and culturally diverse adolescents to assess their self-reported transactions with family, peer, school, and neighborhood microsystems for the constructs of social support, daily hassles, and involvement. The sample of 998 youth were from schools in three Eastern cities with high percentages of economically disadvantaged youth. Data were collected before and after the transition to junior high school or to senior high school. Blacks constituted 26%, whites 26%, and Latinos 37% of the sample. Factor analyses confirmed and enhanced the hypothesized four-factor microsystem factor structure for support, hassles, and involvement; internal consistency and stability coefficients were consistent with these structures. In general, the microsystem factors were common across gender, ethnicity, and age. However, when group differences did occur on these demographic variables, they tended to validate the salience of microsystem specificity. In contrast to the total scores, the microsystem-specific factors yielded more meaningful and differential information with regard to demographic differences and the mediating processes across a school transition.     

Ideal objects as models in science

Three main concepts of model in science are distinguished: (1) semantical model of a theory; (2) real model of another real thing; (3) mathematical model of a real thing. The last concept is the most important for the empirical sciences. The mathematical model is not identical with a theory: it is an ideal object which is directly described by the theory. We have here an intermediate level between reality and theory.     

From the Profession

Studies in East European Thought -     

Treatment outcome in personality disorder: The effect of social,psychological and behavioural variables

The study followed-up 245 patients referred to the Henderson Hospital (194 admitted for treatment and 51 not admitted) and for whom information had been collected on their social background and certain psychological variables allowing a classification and typology of personality disorder to be worked out. The study indicates that the therapeutic community treatment is effective with selected individuals showing the antisocial behaviour associated with such disorder. In particular this treatment is of benefit to the offenders with only one conviction and who are able to persevere with treatment for a period of 6 months for treatment to be maximally effective. The variation of psychological types within the broad category of personality disorder was demonstrated and this has a bearing on the outcome of treatment, the extrapunitive neurotic being of poorest prognosis with or without treatment. Yet even in this group, therapeutic community treatment can be effective.