The lattice and semigroup structure of multipermutations

We study the algebraic properties of binary relations whose underlying digraph is smooth, that is, has no source or sink. Such objects have been studied as surjective hyper-operations (shops) on the corresponding vertex set, and as binary relations that are defined everywhere and whose inverse is also defined everywhere. In the latter formulation, they have been called multipermutations. We study the lattice structure of sets (monoids) of multipermutations over an [Formula: see text]-element domain. Through a Galois connection, these monoids form the algebraic counterparts to sets of relations closed under definability in positive first-order logic without equality. We show one side of this Galois connection, and give a simple dichotomy theorem for the evaluation problem of positive first-order logic without equality on the class of structures whose preserving multipermutations form a monoid closed under inverse. These problems turn out either to be in [Formula: see text]or to be [Formula: see text]-complete. We go on to study the monoid of all multipermutations on an [Formula: see text]-element domain, under usual composition of relations. We characterize its Green relations, regular elements and show that it does not admit a generating set that is polynomial on [Formula: see text].

Consumerism Qualification: Doctrine and Case Law

А number of factors influence the qualification of relations as consumer relations, however, neither a scientific literature nor law enforcement practice has formed a common list of signs defining the relationship to the subject of regulation of consumer protection legislation. The main purpose of the article is to identify signs that qualify the relationship as consumer. Achieving the goal of the article is carried out by solving the following tasks: determining the influence of the subject composition of relations, the nature of their activities, as well as other signs on the qualification of relations in order to apply legislation on consumer protection. The article was prepared using general scientific methods (systemic, logical) and special legal methods (comparative legal, formal legal). The author comes to the conclusion that for the proper qualification of public relations in order to apply legislation on the protection of consumer rights, it is necessary to take into account both the subject composition and the purpose of purchasing goods. The formal affiliation of a person to a particular category of entities cannot definitely indicate the goals of a person entering into a relationship. In particular, an individual entrepreneur has a dual status, being, on the one hand, a citizen, and on the other, a subject of entrepreneurial activity. The article also concluded that when determining the purpose of acquiring a product (work, service), it is necessary to take into account both direct signs expressed in the nature of the person's activity and indirect ones determined by the quantity of the purchased goods and its technical and functional characteristics. To assess indirect features, the author suggests referring to the category of «average person» formed in the general theory of law.

Using the whole image and restrictions in the research of properties of some signature operations in Table Algebra

The features of the whole image relative to many binary relation, and restrictions on a binary relation on the set for some of the signature operations of Table Algebra are used in the work. Constructions of the whole image and restrictions are of general interest for Mathematics, and Table Algebra is a modern analogue of Codd's well-known Relational Algebra. It forms the theoretical foundation of modern query language databases. Elements of the carrier of Table Algebra specify relational table data structures, and signature operations are based on the basic table manipulations in Relational Algebra and SQL-like languages. The following results in the research of the features of the whole image were obtained: interconnections between the whole image and restrictions were found; the monotony and distribution of the whole image and restrictions on unions, a criterion of their emptiness and interconnections with first and second projection relations were proved; the whole image of the composition of relations and composition restrictions were found; the distribution of restriction on intersection of sets was set; the estimates of the distribution of the whole image of intersection and difference of sets were given; criteria for distribution of the whole image relative to the intersection and differences of sets were found. In addition, the clues were provided with the help of the whole image and restrictions on some of the signature operations of Table Algebra: intersection, union, difference, projection and joining. These representations allowed us to obtain some features of these operations, which derive directly from the features of the whole image and restrictions. It is supposed to get similar views on other signature operations of Table Algebras and to allocate their features arising from such representation. The obtained results can be used in the theory of Table Algebra as an approach to the research of the features of their signature operations, this can be used in query optimization in relational databases.

From segregation to protagonism. The great lesson of young blind people in Italy.

Special Pedagogy is a continuous composition of relations, actions and projects. Special Pedagogy should not have the presumption to consider worthy only those questions to which we already know the answer. It should learn to live with questions that do not have ready prepared answers. Its task is to seek answers without being sure of finding them. Special Pedagogy owes much of its knowledge and wealth of experience to the cultural presence of persons with disabilities: in this paper, thanks to the efforts of people with disabilities and their fight for civil rights in Italy, we present the historical and socio-cultural path from segregation (institutionalization) to the inclusion in the society of blind people, with a particular attention paid to the adolescenthood. 

Relations and Functions on Union-Soft Sets

In this article, the theoretical aspects of union-soft sets by extending the notions of equivalence relations, partition, composition of relations, and function to the framework of union-soft sets are introduced. Further the Cartesian product, the relation between union-soft sets, induced relations from the universal set and the attribute set with examples are discussed. Moreover, the composition of union-soft set relationships with examples and some related theorems are demonstrated. Finally, the concepts of a union-soft set function and their respective composition function are examined.

O trabalho comum através do aprendizado dos afetos na Rede de Atenção Psicossocial / Learning about affects through common-work in Psychosocial Attention Networks

ResumoO presente artigo propõe pensar o trabalho das Redes de Atenção Psicossocial a partir da perspectiva de Baruch de Espinosa e de Fernand Deligny. A proposta é mostrar o quanto a questão do aprendizado dos afetos, apresentado por Spinoza é determinante em um trabalho comum entre profissionais de saúde e usuários da Rede. Os profissionais de saúde, ao aprenderem a respeito da capacidade de pensar e agir dos usuários, descobrem do que eles próprios são capazes. A rede se constitui através da conexão entre diferentes serviços de saúde e entre profissionais de diferentes áreas. Trata-se da construção de uma maneira de agir e viver junto, o que diz respeito a uma ideia de rede que vamos encontrar na obra de Fernand Deligny.Palavras-chave: RAPS; Trabalho Comum; Afetos; Espinosa; Deligny. AbstractThis article proposes to understand the Psychosocial Attention Network from the points of view of Baruch Spinoza and Fernand Deligny where the network is understood as a composition of relations between various health services and professionals from different fields. We seek to show how the learning of affects, as outlined by Spinoza, is key to common-work between health professionals and network users. Once health professionals recognise network users’ ability to think and act for themselves, professionals can come to appreciate what users can actually do on their own. What emerges as most relevant is the constitution of modes of acting and living together and how these relate to the idea of network as elaborated in the works of Deligny.Keywords: RAPS; Common Work; Affects; Spinoza; Deligny.

On factorization systems for surjective quandle homomorphisms

We study and compare two factorization systems for surjective homomorphisms in the category of quandles. The first one is induced by the adjunction between quandles and trivial quandles, and a precise description of the two classes of morphisms of this factorization system is given. In doing this we observe that a special class of congruences in the category of quandles always permute in the sense of the composition of relations, a fact that opens the way to some new universal algebraic investigations in the category of quandles. The second factorization system is the one discovered by E. Bunch, P. Lofgren, A. Rapp and D. N. Yetter. We conclude with an example showing a difference between these factorization systems.

Generality of proofs and its Brauerian representation

The generality of a derivation is an equivalence relation on the set of occurrences of variables in its premises and conclusion such that two occurrences of the same variable are in this relation if and only if they must remain occurrences of the same variable in every generalization of the derivation. The variables in question are propositional or of another type. A generalization of the derivation consists in diversifying variables without changing the rules of inference.This paper examines in the setting of categorial proof theory the conjecture that two derivations with the same premises and conclusions stand for the same proof if and only if they have the same generality. For that purpose generality is defined within a category whose arrows are equivalence relations on finite ordinals, where composition is rather complicated. Several examples are given of deductive systems of derivations covering fragments of logic, with the associated map into the category of equivalence relations of generality.This category is isomorphically represented in the category whose arrows are binary relations between finite ordinals, where composition is the usual simple composition of relations. This representation is related to a classical representation result of Richard Brauer.