Finite axiomatization for some intermediate logics

LetN. be the set of all natural numbers (except zero), and letD _n ^* = {k ∈N ∶k|n} ∪ {0} wherek¦n if and only ifn=k.x f or somex∈N. Then, an ordered setD _n ^* = 〈D _n ^* , ? _n , wherex? _ny iffx¦y for anyx, y∈D _n ^* , can easily be seen to be a pseudo-boolean algebra. In [5], V.A. Jankov has proved that the class of algebras {D _n ^* ∶n∈B}, whereB =,{k ∈N∶ ? \(\mathop \exists \limits_{n \in N} \) (n > 1 ≧n ² k)is finitely axiomatizable. The present paper aims at showing that the class of all algebras {D _n ^* ∶n∈B} is also finitely axiomatizable. First, we prove that an intermediate logic defined as follows: $$LD = Cn(INT \cup \{ p_3 \vee [p_3 \to (p_1 \to p_2 ) \vee (p_2 \to p_1 )]\} )$$ finitely approximatizable. Then, defining, after Kripke, a model as a non-empty ordered setH = 〈K, ?〉, and making use of the set of formulas true in this model, we show that any finite strongly compact pseudo-boolean algebra ? is identical with. the set of formulas true in the Kripke modelH _B = 〈P(?), ?〉 (whereP(?) stands for the family of all prime filters in the algebra ?). Furthermore, the concept of a structure of divisors is defined, and the structure is shown to beH D _n ^* = 〈P (D _n ^* ), ?〉for anyn∈N. Finally, it is proved that for any strongly compact pseudo-boolean algebraU satisfying the axiomp ₃∨ [p ₃→(p₁→p₂)∨(p₂→p₁)] there is a structure of divisorsD _* ⁿ such that it is possible to define a strong homomorphism froomiH D _n ^* ontoH D _U . Exploiting, among others, this property, it turns out to be relatively easy to show that \(LD = \mathop \cap \limits_{n \in N} E(\mathfrak{D}_n^* )\) .     

Agency,communion and entitlement

In this study, the relationship between agency, communion, and the active, passive, and revenge forms of entitlement is examined. Results indicate that active entitlement was positively related to agency, negatively to communion (Study 1), and unrelated to unmitigated agency and communion (Study 2). Passive entitlement was positively related to communion (in regular and unmitigated forms) and negatively related to agency (in both forms). Revenge entitlement was positively related to agency (unmitigated and regular), and negatively related to both regular and unmitigated communal orientations. Detected relationships were independent from self‐esteem (Study 1). The findings are discussed in relation to distinctions between narcissistic and healthy entitlement, and within the context of the three‐dimensional model of entitlement.     

Distribution of Cr atoms in a strained and strain-relaxed Fe89.15Cr10.75 alloy: a Mössbauer effect study

An Fe_89.15Cr_10.75 alloy in a heavily strained (by cold rolling) state and in strain-relaxed states was studied by means of conversion electrons Mössbauer spectroscopy. Analysis of the spectra in terms of a two-shell model revealed significant differences between the studied samples, particularly in values of the hyperfine field and the distribution of Cr atoms within the first two neighbour shells. The latter is quantified in terms of short-range order parameters.