Thesis (Ph. D.)--University of Wisconsin--Madison, 1984. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 82-83).

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The purpose of this thesis is to explore some of the characteristics of 0-simple semigroups and completely 0-simple semigroups.

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An examination of some properties that interrelate the computational complexities of evaluating multivariate polynomial functions is presented. The kind of relationship between polynomial functions that is studied takes the form of linear transformations of the arguments and results of a polynomial function that transform it into another such function. Such transformations are a generalisation of projection (a form of reduction in algebraic complexity first introduced by Valiant, whereby variables and constants are substituted for the arguments of a polynomial function in order to transform it into another polynomial function). In particular, two restricted forms of this generalised projection are considered: firstly, those that relate a polynomial function to itself, and secondly, those that are invertable. Call these symmetries and similarities, respectively. The structure of the set of symmetries of a polynomial function is explored, and the computationally useful members of the set identified; a technique for finding all such symmetries is presented. It is shown that polynomials related by similarity have "isomorphic" sets of symmetries, and this condition may be used as a criterion for similarity. Similarity of polynomial functions is shown to be an equivalence relation, and "similar polynomials" can be seen to possess closely comparable complexities. A fast probabilistic algorithm for finding the symmetries of a polynomial function is given. The symmetries of the determinant and of the permanent (which differs from the determinant only in that all of its monomials have coefficients of +1), and those of some other polynomials, are explicitly found using the above theory. Fast algorithms using linear algebra for evaluating the determinant are known, whereas evaluating the permanent is known to be a #p-complete problem, and is apparently intractable; the reasons for this are exposed. As an easy corollary it is shown that the permanent is not preserved by any bilinear product of matrices, in con'trast to the determinant which is preserved by matrix multiplication. The result of Marcus and Minc, that the determinant cannot be transformed into the permanent by substitution of linear combinations of variables for its arguments (i.e. the permanent and determinant are not similar), also follows as an easy corollary. The relationship between symmetries and ease of evaluation is discussed.

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Language development involves the learning of multiple sets of equivalence relations. Research has shown that if certain conditional relations are directly taught for one member of a class of stimuli, then additional conditional relations often emerge for other members of that class, without direct training. There are currently very few studies which have demonstrated this research finding in individuals with autism spectrum disorder (ASD). The research design used for the present experiment was a single-subject AB cross-over design replicated across five plus five children with ASD and five plus five typically developing children. The children with ASD and the typically developing children were matched on their level of vocabulary development. Participants were randomly assigned to either a teaching order Treatment A+B or a teaching order Treatment B+A. The first experimental treatment (Treatment A+B) involved teaching responses to S1 and S2 in the order Condition A followed by Condition B. The second experimental treatment (Treatment B+A) involved teaching responses to S1 and S2 in the order Condition B followed by Condition A. Condition A involved the teaching of AB and AC (hear-select) relations, and Condition B involved the teaching of BA and CA (see-say) relations. The participants in this study were taught stimulus-response relations that involved six names and numerical representations of quantities in the range 1 to 18. Tests for the emergence of symmetry and transitivity were then conducted. The relationships between the emergence of the untaught equivalence relations and teaching condition, the entering characteristics of the children, and trials to criterion were examined. The results of this study showed that five out of ten participants with ASD demonstrated the emergence of all of the untaught equivalence relations regardless of the treatment condition. The remaining five participants with ASD showed substantial variability. Of the children in the Typically Developing Group nine of the ten demonstrated emergence of all of the untaught equivalence relations. The variables that were most strongly correlated with the emergence of untaught equivalence relations were speed of acquisition of taught relations, functional academics scores, and the chronological age of the participants. The effect of communication ability, pre-academic numeracy skill level, and the experimental treatment (the teaching order conditions) were not strongly related to the emergence of untaught equivalence relations. These findings suggest that outcomes on tests for emergence may have been a function of children’s rate of development and prior learning history. The findings of the current study are best explained by Relational Frame Theory. The implications of these findings for teaching children with ASD and other developmental disabilities, and also teaching in general are discussed.

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To the pair (E,c), where E is a countable Borel equivalence relation on a standard Borel space (X,A) and c a normalized Borel T-valued 2-cocycle on E, we associate a sequentially weakly closed Borel *-algebra Br*(E,c), contained in the bounded linear operators on L^2(E). Associated to Br*(E,c) is a natural (Borel) Cartan subalgebra (Definition 6.4.10) L(Bo(X)) isomorphic to the bounded Borel functions on X. Then L(Bo(X)) and its normalizer (the set of the unitaries u in Br*(E,c) such that u*fu in L(Bo(X)), f in L(Bo(X))) countably generates the Borel *-algebra Br*(E,c). In this thesis, we study Br*(E,c) and in particular prove that: i) If E is smooth, then Br*(E,c) is a type I Borel *-algebra (Definition 6.3.10). ii) If E is a hyperfinite, then Br*(E,c) is a Borel AF-algebra (Definition 7.5.1). iii) Generalizing Kumjian's definition, we define a Borel twist G over E and its associated sequentially closed Borel *-algebra Br*(G). iv) Let a Borel Cartan pair (B, Bo) denote a sequentially closed Borel *-algebra B with a Borel Cartan subalgebra Bo, where B is countably Bo-generated. Generalizing Feldman-Moore's result, we prove that any pair (B, Bo) can be realized uniquely as a pair (Br*(E,c), L(Bo(X))). Moreover, we show that the pair (Br*(E,c), L(Bo(X))) is a complete invariant of the countable Borel equivalence relation E. v) We prove a Krieger type theorem, by showing that two aperiodic hyperfinite countable equivalence relations are isomorphic if and only if their associated Borel *-algebras Br*(E1) and Br*(E2) are isomorphic.

In recent years, there has been an increasing interest in the study of human operant behavior. One area of study reflecting this interest is the study of the formation of equivalent classes of stimuli by human subjects. The focus of the present research was the study of the conditions under which classes of equivalent stimuli can be inferred to be under conditional control. In Experiment 1-A, three college students were trained to respond to a balanced five-term contingency via a visual-visual simultaneous matching-to-sample task with two choices of comparison stimuli. Probe tests showed that subjects' behavior could be described as being controlled by positive and negative stimulus relations. When the second-order stimulus was removed during subsequent probes, none of the three subjects demonstrated strong correct responses to the four-term unit relations. Also, none of the three subjects demonstrated the expected transitive relations when the second-order (five-term) stimulus was removed. In Experiment 1-B--with the same three subjects--explicit training of the four-term unit relations showed the expected transitive relations in the absence of the second-order stimulus. In Experiments 2 through 5--using a matching-to-sample task similar to that used in Experiments 1-A and 1-B--five subjects were trained to respond to comparison stimuli C and E in the presence of sample A and second-order stimulus X and to comparison stimuli Band Fin the presence of sample D and second-order stimulus X. Likewise, the subjects were trained to respond to comparison stimuli Band Fin the presence of sample A and second-order stimulus Y and to comparison stimuli C and E in the presence of sample D and second-order stimulus Y. Probe tests for transitive relations showed that four of the five subjects eventually demonstrated four three-member classes of equivalent stimuli that functioned separately under the control of the second-order stimuli. The four subjects demonstrating the classes of equivalent stimuli either a) demonstrated the transitive relations immediately orb) demonstrated the transitive relations after explict retraining of the underlying four-term unit relations. The results of all experiments together indicated that the composition of classes of equivalent stimuli can be conditionally controlled by either a) removing the second-order stimulus orb) training subjects to respond to classes of equivalent stimuli under the control of other explicit visual second-order stimuli. The results are discussed in terms of verbal behavior, emergent behavior, and conceptual development.