Birkhoff-like sheaf representation for varieties of lattice expansions

Given a variety we study the existence of a class such that S1 every A can be represented as a global subdirect product with factors in and S2 every non-trivial A is globally indecomposable. We show that the following varieties (and its subvarieties) have a class satisfying properties S1 and S2: p-algebras, distributive double p-algebras of a finite range, semisimple varieties of lattice expansions such that the simple members form a universal class (bounded distributive lattices, De Morgan algebras, etc) and arithmetical varieties in which the finitely subdirectly irreducible algebras form a universal class (f-rings, vector groups, Wajsberg algebras, discriminator varieties, Heyting algebras, etc). As an application we obtain results analogous to that of Nachbin saying that if every chain of prime filters of a bounded distributive lattice has at most length 1, then the lattice is Boolean.We wish to thank Lic. Alfredo Guerin and Dr. Daniel Penazzi for helping us with linguistics aspects. We are indebted to the referee for several helpful suggestions. We also wish to thank Professor Mick Adams for providing us with several reprints and useful e-mail information on the subject.Suported by CONICOR and SECyT (UNC).     

A rating formulation for ordered response categories

A rating response mechanism for ordered categories, which is related to the traditional threshold formulation but distinctively different from it, is formulated. In addition to the subject and item parameters two other sets of parameters, which can be interpreted in terms of thresholds on a latent continuum and discriminations at the thresholds, are obtained. These parameters are identified with the category coefficients and the scoring function of the Rasch model for polychotomous responses in which the latent trait is assumed uni-dimensional. In the case where the threshold discriminations are equal, the scoring of successive categories by the familiar assignment of successive integers is justified. In the case where distances between thresholds are also equal, a simple pattern of category coefficients is shown to follow.This work was conducted in part in the first half of 1977 while the author was on study leave at the Danish Institute for Educational Research. The Institute provided required research facilities while The University of Western Australia provided financial support.     

A note on analyzing ordered alternatives

A general method of devising stepwise multiple testing procedures with fixed experimentwise error is applied to the problem of non-parametric randomized block design with ordered alternatives. In addition, the method is applied to other models with ordered alternatives.The author wishes to thank the referees for helpful suggestions.     

The Variety of Lattice-Ordered Monoids Generated by the Natural Numbers

We study the variety Var() of lattice-ordered monoids generated by the natural numbers. In particular, we show that it contains all 2-generated positively ordered lattice-ordered monoids satisfying appropriate distributive laws. Moreover, we establish that the cancellative totally ordered members of Var() are submonoids of ultrapowers of and can be embedded into ordered fields. In addition, the structure of ultrapowers relevant to the finitely generated case is analyzed. Finally, we provide a complete isomorphy invariant in the two-generated case.     

Soins pénalement ordonnés et positionnements professionnels : des pratiques au carrefour de la Santé et de la Justice

At the crossroad of Health and Justice, penally ordered care underlies professional practices, the articulation of which can turn out to be delicate. Here we tackle the relevance and practical application of these measures. For that purpose, semi-directive interviews and field observations have been conducted with probation agents, judges (who follow up the way sentences are executed) and psychologists. The professionals’ comments and experiences show the complexity of these measures, which intend to make “care” into a recidivism prevention tool. Misconceptions appear about theory, practices and interprofessional relationships. The idea of “care” as a sole medical term appears to be ineffective and calls for the creation of new practices.     

Effect of cooling rate on size-dependent atomic ordering of CoPt nanoparticles

We report on the effect of cooling rate on the size-dependent atomic ordering of CoPt nanoparticles using aberration corrected high-resolution transmission electron microscopy. It was found that cooling rate plays a crucial role in promoting atomic ordering during the cooling process after annealing. Nanoparticles of ≈3?nm in diameter show the A1-disordered phase after annealing at 873?K for 1?h followed by rapid cooling (110?K/min), while the L1₀-ordered phase is obtained when the cooling rate is slow (1.5?K/min). The disordered phase is also obtained by rapid cooling after annealing at 973?K for 1?h. These results unambiguously indicate that the order–disorder transformation temperature is reduced to a temperature at least lower than 873?K for CoPt nanoparticles smaller than 3?nm in diameter. The slow cooling process promotes the atomic ordering, which resulted in an enhancement of magnetic coercivity as high as 2200?Oe. This study demonstrates that hard magnetic properties of the CoPt nanoparticles can be improved by controlling the cooling rate after heat treatments.     

New polytypes of long-period stacking ordered structures in a near-equilibrium Mg97Zn1Y2 alloy

By atomic-scale high-angle annular dark-field scanning transmission electron microscopy, the long-period stacking ordered (LPSO) structures in a near-equilibrium Mg₉₇Zn₁Y₂ (at.%) alloy have been characterised. In addition to 18R and 14H, new polytypes of LPSO structures are analysed and determined as 60R, 78R, 26H, 96R, 38H, 40H, 108H and 246R. All of these LPSO structures feature AB′C′A building blocks with two Mg layers and three Mg layers sandwiched between them. The Bravais lattices and space groups of new polytypes of LPSO structures were easily determined via the newly introduced method. A structural relationship between the LPSOs is proposed.     

The Universal Group of a Heyting Effect Algebra

A Heyting effect algebra (HEA) is a lattice-ordered effect algebra that is at the same time a Heyting algebra and for which the Heyting center coincides with the effect-algebra center. Every HEA is both an MV-algebra and a Stone-Heyting algebra and is realized as the unit interval in its own universal group. We show that a necessary and sufficient condition that an effect algebra is an HEA is that its universal group has the central comparability and central Rickart properties. Presented by Daniele Mundici     

The countable homogeneous universal model of B 2

We give a detailed account of the Algebraically Closed and Existentially Closed members of the second Lee class B ₂ of distributive p-algebras, culminating in an explicit construction of the countable homogeneous universal model of B ₂. The axioms of Schmid [7], [8] for the AC and EC members of B ₂ are reduced to what we prove to be an irredundant set of axioms. The central tools used in this study are the strong duality of Clark and Davey [3] for B ₂ and the method of Clark [2] for constructing AC and EC algebras using a strong duality. Applied to B ₂, this method transfers the entire discussion into an equivalent dual category X ₂ of Boolean spaces which carry a pair of tightly interacting orderings. The doubly ordered spaces of X ₂ prove to be much more readily constructed and analyzed than the corresponding algebras in B ₂.     

On Categorical Equivalences of Commutative BCK-algebras

A commutative BCK-algebra with the relative cancellation property is a commutative BCK-algebra (X;*,0) which satisfies the condition: if a ≤ x, a ≤ y and x * a = y * a, then x = y. Such BCK-algebras form a variety, and the category of these BCK-algebras is categorically equivalent to the category of Abelian ℓ-groups whose objects are pairs (G, G ₀), where G is an Abelian ℓ-group, G ₀ is a subset of the positive cone generating G ⁺ such that if u, v ∈ G ₀, then 0 ∨ (u - v) ∈ G ₀, and morphisms are ℓ-group homomorphisms h: (G, G ₀) → (G′,G′₀) with f(G ₀) ⫅ G′₀. Our methods in particular cases give known categorical equivalences of Cornish for conical BCK-algebras and of Mundici for bounded commutative BCK-algebras (= MV-algebras). This revised version was published online in June 2006 with corrections to the Cover Date.