LINEAR ORDERS REALIZED BY CE EQUIVALENCE RELATIONS

AbstractLetEbe a computably enumerable (c.e.) equivalence relation on the setωof natural numbers. We say that the quotient set$\omega /E$(or equivalently, the relationE)realizesa linearly ordered set${\cal L}$if there exists a c.e. relation ⊴ respectingEsuch that the induced structure ($\omega /E$; ⊴) is isomorphic to${\cal L}$. Thus, one can consider the class of all linearly ordered sets that are realized by$\omega /E$; formally,${\cal K}\left( E \right) = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly ordered sets realized byE. One can also define the following pre-order$\le _{lo}$on the class of all c.e. equivalence relations:$E_1 \le _{lo} E_2$if every linear order realized byE1is also realized byE2. Following the tradition of computability theory, thelo-degrees are the classes of equivalence relations induced by the pre-order$\le _{lo}$. We study the partially ordered set oflo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among thelo-degrees.</jats:p

A Lopez-Escobar Theorem for Continuous Domains

We prove an effective version of the Lopez-Escobar theorem for continuous domains. Let $Mod(\tau)$ be the set of countable structures with universe $\omega$ in vocabulary $\tau$ topologized by the Scott topology. We show that an invariant set $X \subseteq Mod(\tau)$ is $\Pi^0_\alpha$ in the effective Borel hierarchy of this topology if and only if it is definable by a $\Pi^p_\alpha$ - formula, a positive $\Pi^0_\alpha$ formula in the infinitary logic $L_{\omega_1,\omega}$. As a corollary of this result we obtain a new pullback theorem for positive computable embeddings: Let $K$ be positively computably embeddable in $K'$ by $\Phi$, then for every $\Pi^p_\alpha$ formula $\xi$ in the vocabulary of $K'$ there is a $\Pi^p_\alpha$ formula $\xi^\star$ in the vocabulary of $K$ such that for all $A \in K$, $A \models \xi^\star$ if and only if $\Phi(A) \models \xi$. We use this to obtain new results on the possibility of positive computable embeddings into the class of linear orderings.Comment: 17 page

Biological effectiveness of Pseudomonas fluorescens strains against bacterial and fungal diseases of spring rape in field conditions

The influence of four phytopathogens Fusarium oxysporum, Botrytis cinerea, Alternaria brassicicola , and Xanthomonas campestris on the biometric parameters of spring rape plants in field conditions was evaluated against the background of the use of the biological preparation Binoram, including live antagonistic bacteria Pseudomonas fluorescens. A significant decrease in economically important parameters of plants as a result of infection with tested phytopathogens and a positive effect on plant indicators of treatment with the biological preparation Binoram in comparison with the infected control is shown

Business administration as a basis for development of global entrepreneurship

The purpose of the article is to determine the role and meaning of business administration in development of global entrepreneurship. The work uses the proprietary methodology of evaluation of global competitiveness of enterprise and the proprietary methodology of evaluation of efficiency of business administration of enterprise, as well as method of correlation analysis. The authors determine the role of business administration in management of factors of development of global enterprise and offer the proprietary structural and logical model of organization of the process of business administration of global enterprise as a perspective tool for provision of global competitiveness of business and development of global entrepreneurship on the whole. The authors come to the conclusion that business administration is a basis for development of global entrepreneurship and plays an important role in this process, which consists in managing other internal and external factors of functioning of global business.peer-reviewe