In this thesis after recalling some basic definitions and theorems in category theory, lattice theory and topology we first introduce the so called Stone duality of the category of boolean lattices and the category of boolean topological spaces. Then we introduce its generalization, the so called Priestley duality of the category of bounded distributive lattices and the category of total order disconnected topological spaces. Then we introduce the M3[.] lattice construction and prove that for every bounded distributive lattice L there is an isomorphism from the lattice M3[L] to the lattice of all continuous monotone maps from the Priestley space of L to the lattice M3 with discrete topology. Finally we introduce the so called boolean power, which we generalize to the so called priestley power and we prove that for every natural number n ≥ 3 and every bounded distributive lattice L there is an isomorphism from the lattice Mn to the priestley power of the lattice Mn by the lattice L. 1

To any Frobenius superalgebra A we associate an oriented Frobenius Brauer category and an affine oriented Frobenius Brauer categeory. We define natural actions of these categories on categories of supermodules for general linear Lie superalgebras gl_m|n(A) with entries in A. These actions generalize those on module categories for general linear Lie superalgebras and queer Lie superalgebras, which correspond to the cases where A is the ground field and the two-dimensional Clifford superalgebra, respectively. We include background on monoidal supercategories and Frobenius superalgebras and discuss some possible further directions.

Past research has shown that accessing a memory allows faster subsequent access to the memory activated as well as to related information (priming). There has been much research devoted to implicit category priming (unintentional priming of a category of information), but this research has not determined the number of categories that can be implicitly primed simultaneously. The goal of the present quantitative study was to address that gap. Twenty participants (ages 27-54 years, M=44 years), who volunteered through an online participant pool, were presented with 2 tasks over the Internet. A scrambled phrase task implicitly primed 5 unrelated categories and a lexical decision (LD) task measured the priming (mean time between tasks = 42 seconds). Resulting primed and unprimed LD response latency distributions were strongly, positively skewed, which obscured individual priming effects. Gaussian parameters were extracted to overcome this skew, and the distributions were created for analysis. Dunnett's multiple comparisons post-hoc test following a 1-way ANOVA showed that 2 of the 5 categories remained significantly primed. Follow-up research should determine the reliability of this value. This value, and its range (to be identified in follow-up studies) would provide a means for comparing lesson efficacy and teacher performance. The results of this research also replicate previous research demonstrating long-term implicit category priming.

The present study was designed to examine the relationship between Mednick's Remote Associates Test and Pettigrew's Category-Width Test. The sample was made up of 36 educational psychology students enrolled in the Utah State Summer School. The correlation for males and females was made independently. Although the males showed a higher correlation between the two tests than did the females (males, .296 and females, .058), the correlations were not statistically significant. No full explanation was given for the lack of correlation between the category-width scores and the RAT scores except for the qualification that the RAT may be measuring a different type of creativity than what the category-width test is measuring.

Indiana University-Purdue University Indianapolis (IUPUI) / In this dissertation, a potential outcomes (PO) based framework is developed for causally interpretable treatment effect parameters in the multinomial dependent variable regression framework. The specification of the relevant data generating process (DGP) is also derived. This new framework simultaneously accounts for the potential endogeneity of the treatment and loosens inter-category covariance restrictions on the multinomial outcome model (e.g., the independence from irrelevant alternatives restriction). Corresponding consistent estimators for the “deep parameters” of the DGP and the treatment effect parameters are developed and implemented (in Stata). A novel approach is proposed for assessing the inter-category covariance flexibility afforded by a particular multinomial modeling specification [e.g. multinomial logit (MNL), multinomial probit (MNP), and nested multinomial logit (NMNL)] in the context of our general framework. This assessment technique can serve as a useful tool for model selection. The new modeling/estimation approach developed in this dissertation is quite general. I focus here, however, on the NMNL model because, among the three modeling specifications under consideration (MNL, MNP and NMNL), it is the only one that is both computationally feasible and is relatively unrestrictive with regard to inter-category covariance. Moreover, as a logical starting point, I restrict my analyses to the simplest version of the model – the trinomial (three-category) NMNL with an endogenous treatment (ET) variable conditioned on individual-specific covariates only. To identify potential computational issues and to assess the statistical accuracy of my proposed NMNL-ET estimator and its implementation (in Stata), I conducted a thorough simulation analysis. I found that conventional optimization techniques are, in this context, generally fraught with convergence problems. To overcome this, I implement a systematic line search algorithm that successfully resolves this issue. The simulation results suggest that it is important to accommodate both endogeneity and inter-category covariance simultaneously in model design and estimation. As an illustration and as a basis for comparing alternative parametric specifications with respect to ease of implementation, computational efficiency and statistical performance, the proposed model and estimation method are used to analyze the impact of substance abuse/dependence on the employment status using the National Epidemiologic Survey on Alcohol and Related Conditions (NESARC) data.

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Given a symmetric Frobenius superalgebra A equipped with a compatible involution, we define the associated Frobenius Brauer category B(A) and affine Frobenius Brauer category AB(A), generalizing the plain Brauer category B and affine Brauer category AB. We define the orthosymplectic Lie superalgebra osp m|2n(A) and a functor from B(A) to osp m|2n(A)-mod, the category of supermodules over osp m|2n(A). We also define a functor from AB(A) to the endofunctor supercategory of osp m|2n(A)-mod.We prove that these two functors are well-defined and use the former functor to prove a basis result for B(A, δ), a specialized version of B(A). Prior to defining these categories and functors, we provide the background information on super-mathematics and Frobenius superalgebras needed to understand the new results.

Using the the dependently-typed programming language Agda, we formalise orders, with attention to the theory of Galois Connections, and showcase it by formalising a few results of the category of algebraic contexts with relational homomorphisms presented by Jipsen (2012) and Moshier (2013). We aim to exhibit an internal theory of Galois Connections and Closure operators where the ambient space need not have a notion of meets (intersections), which are the usual medium in presenting antisymmetry of partial orders. Instead we consider `symmetric quotients' as being the relational counterpart of propositional calculus' primitive connective: equivalence. We argue that it as a more natural primitive than meet --- especially its connection with certain proof heuristics regarding posets. Moreover, not only do we constrain ourselves to an unconventional set of primitive operators, but in fact we discard the familiar setting of relations and sets in favour of the more general setting of ordered categories with converse (OCCs) --- in fact, a large portion does not require identities and so semigroupoids may be used instead. / Thesis / Master of Science (MSc)

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