Classical Mereology

The aim of this chapter is to provide a comprehensive introduction to classical mereology. It examines this theory by providing a clear and perspicuousaxiom system that isolates several important elements of any mereological theory. The chapter examines, algebraic, and set-theoretic models of classical mereology, sketching proofs of their equivalence. The new axiom system facilitates algebraic comparisons, showing that models of these axioms are complete Boolean algebras without a bottom element. Then set-theoretic models are presented, and are shown to satisfy the axioms. The chapter explains the important relationship between models and powersets, and the role of Stone’s Representation Theorem in this connection. Finally, a number of significant rival axiom systems using different mereological primitives are introduced.

Array of stacked leaky-wave antennas in groove gap waveguide technology

The design of an array of stacked leaky-wave antennas in groove gap waveguide technology is presented in this work. The proposed array is formed by simply stacking a number of leaky-wave antennas one on top of the other and feeding all of them with uniform amplitude and phase. The inter-element distance is studied in order to avoid grating lobes and to maximize the directivity. A feeding network based on vertical coupling is designed, where the input port feeds the bottom element, and then the energy is equally coupled to the other elements. To obtain maximum directivity the phase is corrected at each element separately. The central frequency of the proposed design is 28 GHz. With this technique of stacking the elements a pencil beam is achieved, i.e. the radiated energy is focalized in the two main planes. The designed array with four elements achieves an enhancement of + 5 dB, reaching 24.5 dBi of directivity in comparison to 19.6 dBi of directivity of the single leaky-wave antenna made in this technology. A prototype is manufactured and measured and its results are presented and compared with the simulations.

Solving Generalized Equations with Bounded Variables and Multiple Residuated Operators

This paper studies the resolution of sup-inequalities and sup-equations with bounded variables such that the sup-composition is defined by using different residuated operators of a given distributive biresiduated multi-adjoint lattice. Specifically, this study has analytically determined the whole set of solutions of such sup-inequalities and sup-equations. Since the solvability of these equations depends on the character of the independent term, the resolution problem has been split into three parts distinguishing among the bottom element, join-irreducible elements and join-decomposable elements.

LATTICE-ORDERED ABELIAN GROUPS AND PERFECT MV-ALGEBRAS: A TOPOS-THEORETIC PERSPECTIVE

We establish, generalizing Di Nola and Lettieri’s categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-interpretability holding for particular classes of formulas: irreducible formulas, geometric sentences, and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang’s MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element.

Generalized Caratheodory Extension Theorem on Fuzzy Measure Space

Lattice-valued fuzzy measures are lattice-valued set functions which assign the bottom element of the lattice to the empty set and the top element of the lattice to the entire universe, satisfying the additive properties and the property of monotonicity. In this paper, we use the lattice-valued fuzzy measures and outer measure definitions and generalize the Caratheodory extension theorem for lattice-valued fuzzy measures.

Flat algebras and the translation of universal Horn logic to equational logic

We describe which subdirectly irreducible flat algebras arise in the variety generated by an arbitrary class of flat algebras with absorbing bottom element. This is used to give an elementary translation of the universal Horn logic of algebras, partial algebras, and more generally still, partial structures into the equational logic of conventional algebras. A number of examples and corollaries follow. For example, the problem of deciding which finite algebras of some fixed type have a finite basis for their quasi-identities is shown to be equivalent to the finite identity basis problem for the finite members of a finiteiy based variety with definable principal congruences.

The Constructive Lift Monad

The lift monad is the construction which to a poset freely adjoins a bottom<br />element to it, or equivalently (from the classical viewpoint), the construction which freely adjoins suprema for subsets with at most one element. In constructive mathematics (i.e. inside a topos), these two constructions are no longer equivalent, since the equivalence is based on the boolean reasoning that a set with at most one element either is a singleton {x}, or is empty.<br />Likewise based on boolean reasoning is the proof of two important properties of the lift monad T :<br />1) If a poset C has filtered suprema, then so does TC.<br />2) Every poset with a bottom element ? is "free", i.e. comes about by<br />applying T to some poset (namely the original poset less the bottom).<br />Both these properties fail to hold constructively, if the lift monad is interpreted<br />as "adding a bottom"; see Remark below. If, on the other hand,<br />we interpret the lift monad as the one which freely provides supremum for<br />each subset with at most one element (which is what we shall do), then the first property holds; and we give a necessary and sufficient condition that the second does. Finally, we shall investigate the lift monad in the context of (constructive) locale theory. I would like to thank Bart Jacobs for guiding me to the literature on Z-systems; to Gonzalo Reyes for calling my attention to Barr's work on totally connected spaces; to Steve Vickers for some pertinent correspondence.<br />I would like to thank the Netherlands Science Organization (NWO) for supporting my visit to Utrecht, where a part of the present research was carried out, and for various travel support from BRICS.

A HOL Basis for Reasoning about Functional Programs

Domain theory is the mathematical theory underlying denotational semantics. This thesis presents a formalization of domain theory in the Higher Order Logic (HOL) theorem proving system along with a mechanization of proof functions and other tools to support reasoning about the denotations of functional programs. By providing a fixed point operator for functions on certain domains which have a special undefined (bottom) element, this extension of HOL supports the definition of recursive functions which are not also primitive recursive. Thus, it provides an approach to the long-standing and important problem of defining non-primitive recursive functions in the HOL system.<br /> <br />Our philosophy is that there must be a direct correspondence between elements of complete partial orders (domains) and elements of HOL types, in order to allow the reuse of higher order logic and proof infrastructure already available in the HOL system. Hence, we are able to mix domain theoretic reasoning with reasoning in the set theoretic HOL world to advantage, exploiting HOL types and tools directly. Moreover, by mixing domain and set theoretic reasoning, we are able to eliminate almost all reasoning about the bottom element of complete partial orders that makes the LCF theorem prover, which supports a first order logic of domain theory, difficult and tedious to use. A thorough comparison with LCF is provided.<br /> <br />The advantages of combining the best of the domain and set theoretic worlds in the same system are demonstrated in a larger example, showing the correctness of a unification algorithm. A major part of the proof is conducted in the set theoretic setting of higher order logic, and only at a late stage of the proof domain theory is introduced to give a recursive definition of the algorithm, which is not primitive recursive. Furthermore, a total well-founded recursive unification function can be defined easily in pure HOL by proving that the unification algorithm (defined in domain theory) always terminates; this proof is conducted by a non-trivial well-founded induction. In such applications, where non-primitive recursive HOL functions are defined via domain theory and a proof of termination, domain theory constructs only appear temporarily.

Determination Of Nitrogen and Other Elements in Plant Material By X-Ray Fluorescence

In the early years, the element range covered by the Wavelength Dispersive X-Ray fluorescence (WDXRF) technique was restricted by the quality of vacuum, crystals and detectors in the system. The lightest element detectable was Aluminium (Z = 13). The area of light element analysis had obvious potential for extension over the years. It created the barrier for XRP coverage of the Periodic Table because the naturally softer radiations lay out of range of the technique. In the middle of the 1980's, new technology became available for the manufacture of synthetic crystals with artificially created 2d spacings to suit the X-ray wavelength range of interest. These hyered synthetic microstructures (also called “multilayer crystals”) allowed more sensitivity to be achieved in this part of the spectrum, and the bottom element limit soon moved down towards lighter elements.