Klaus Denecke

Algebra and Discrete Mathematics are essential tools also for economists. Globalization makes that goods and services are exchanged worldwide. Input-output analysis uses the advantage of a relatively stable flow of commodities and services. The essential principles can be described by input-output tables. Such tables correspond to systems of linear equations, written as single equations consisting of matrices whose entries are non-negative rational numbers. The flow itself can be described by a directed edge-valued graph where the sets of values consist of positive rational numbers. Therefore numbers, especially non-negative rational numbers, multiple-partite graphs and matrices with non-negative rational entries play an important role in input-output analysis. Thus knowledge from Algebra and Discrete Mathematics becomes more and more useful in this area. We will consider a particular set of graphs, the so-called input-output graphs and a particular set of matrices, the input-output matrices, define a partial binary operation on each of these sets and prove that we obtain weakly isomorphic weakly partial semigroups. The result of considering this situation can be applied to multiple-step production processes, to material reqirements planning and to internal cost allocation of a business. The results show that Algebra and Discrete Mathematics are quite useful for mathematical modeling in Economy.

Das Buch enth lt Beitr ge, die zwischen 2017 und 2022 in der Monatszeitschrift Der Uhu ver ffentlicht wurden. Die Artikel berdecken ein breites Spektrum von unter-schiedlichen Themen, die aber dennoch miteinander verbunden sind, weil sie versuchen, die gesellschaft-liche Realit t dieser f nf Jahre ad quat zu beschreiben. Es werden politische Fragen behandelt, die unsere Ge-sellschaft spalten und existentiell bedrohen: prek re Besch ftigung, die zunehmenden Unterschiede zwi-schen Armen und Reichen und die daraus erwachsende Bildungsungerechtigkeit, die Corona-Pandemie und den Vormarsch der Rechtspopulisten. Sie versuchen, die Menschen f r sich zu gewinnen, indem sie einfache L -sungen f r komplexe Probleme versprechen und nicht auf Wissen, Fakten und Vernunft setzen. Durch ihre N -he zu faschistischen Gruppierungen sch ren sie Rassen-hass und Fremdenfeindlichkeit. Rechtspopulisten geben keine Antworten auf die dringlichsten Fragen unserer Zeit. Der von Menschen verursachte Klimawandel wird in Abrede gestellt. All dies tr gt dazu bei, von den wirklichen Ursachen abzulenken. Von der milit rischen Zur ckhaltung haben sich die Gro m chte nicht erst seit Beginn des Krieges zwischen der Nato und Russland in der Ukraine verabschiedet. Die Ursachen sozialer Ungerechtigkeit, von Krisen und Kriegen sind ungebremstes Wachstum und Streben nach h chsten Profitraten, also im kapitalistischen System begr ndet. Wie kann man unsere Welt ver ndern, und sie f r alle Erdenbewohner besser machen? Es ist h chste Zeit, sich auf alternative Gesellschaftsmodelle zu verst ndigen. Nachdenken allein gen gt nicht. Man muss handeln, um die Welt friedlicher und lebenswerter zu gestalten. Wir berichten ber Organisationen und einzelne Personen, die durch ihr pers nliches Engagement Probleme berwinden und damit Mut machen. Wissenschaft und Technik haben das Leben der Menschen st rker ver ndert als selbst die k hnsten Utopien es vorausahnen lie en. Das Leben der rmsten haben sie kaum verbessert. Kunst, Musik und Literatur versch nern und bereichern das Alltagsleben. Leider sind ihre Werke vielen Menschen unserer Erde nicht zug nglich.

Words are strings of letters from a fixed alphabet. Sets of words are said to be formal languages. Natural languages, but also programming languages, are examples of such formal languages. On the set of all words the concatenation is a binary associative operation which produces a new word from any two given words. Therefore, there is a semigroup defined on the set of all words on an alphabet. This semigroup is an algebra of type (2), i.e., it has one binary operation satisfying the associative identity. Many properties of words and formal languages can be described by the algebraic properties of the word semigroup. To get languages of more expressive power, words can be generalized to terms using one more alphabet consisting of operation symbols. The combination of n+1 terms to a new term can be described by an (n+1)-ary superposition operation. This superposition operation satisfies the superassociative identity, a generalization of the associative identity. A clone is a multi-based algebraic structure with (n+1)-ary superposition operations as fundamental operations and satisfying the superassociative itentity. Clones of terms take over the role of word semigroups and describe the properties of terms and sets of terms which are also called tree languages. In this book we generalize the superposition operations to partial many-sorted operations. If the superassociative law is satisfied as a weak identity we obtain partial clones. The properties of several important kinds of terms such as linear terms and linear tree languages can be described by partial clones.

M-Solid Varieties of Algebras provides a complete and systematic introduction to the fundamentals of the hyperequational theory of universal algebra, offering the newest results on M-solid varieties of semirings and semigroups. The book aims to develop the theory of M-solid varieties as a system of mathematical discourse that is applicable in several concrete situations. It applies the general theory to two classes of algebraic structures, semigroups and semirings. Both these varieties and their subvarieties play an important role in computer science. A unique feature of this book is the use of Galois connections to integrate different topics. Galois connections form the abstract framework not only for classical and modern Galois theory, involving groups, fields and rings, but also for many other algebraic, topological, ordertheoretical, categorical and logical theories. This concept is used throughout the whole book, along with the related topics of closure operators, complete lattices, Galois closed subrelations and conjugate pairs of completely additive closure operators.

M-Solid Varieties of Algebras provides a complete and systematic introduction to the fundamentals of the hyperequational theory of universal algebra, offering the newest results on M-solid varieties of semirings and semigroups. The book aims to develop the theory of M-solid varieties as a system of mathematical discourse that is applicable in several concrete situations. It applies the general theory to two classes of algebraic structures, semigroups and semirings. Both these varieties and their subvarieties play an important role in computer science. A unique feature of this book is the use of Galois connections to integrate different topics. Galois connections form the abstract framework not only for classical and modern Galois theory, involving groups, fields and rings, but also for many other algebraic, topological, ordertheoretical, categorical and logical theories. This concept is used throughout the whole book, along with the related topics of closure operators, complete lattices, Galois closed subrelations and conjugate pairs of completely additive closure operators.

Die Themen aus Algebra und Diskreter Mathematik, die Informatiker in erster Linie benotigen, finden sich in dieser leicht verstandlichen Einfuhrung. Mathematik wird Studienanfangern als elementares Werkzeug zur Darstellung, Beschreibung, Abstraktion und Symbolisierung vermittelt. Die Bedeutung algebraischer Strukturen in der Kodierungstheorie, in der Automatentheorie und in der Theorie Formaler Sprachen wird in besonderem Masse verdeutlicht. Das Buch enthalt zahlreiche Beispiele und Ubungsaufgaben mit kompletten Losungen.

Over the past 20 years, the emergence of clone theory, hyperequational theory, commutator theory and tame congruence theory has led to a growth of universal algebra both in richness and in applications, especially in computer science. Yet most of the classic books on the subject are long out of print and, to date, no other book has integrated these theories with the long-established work that supports them. Universal Algebra and Applications in Theoretical Computer Science introduces the basic concepts of universal algebra and surveys some of the newer developments in the field. The first half of the book provides a solid grounding in the core material. A leisurely pace, careful exposition, numerous examples, and exercises combine to form an introduction to the subject ideal for beginning graduate students or researchers from other areas. The second half of the book focuses on applications in theoretical computer science and advanced topics, including Mal'cev conditions, tame congruence theory, clones, and commutators. The impact of the advances in universal algebra on computer science is just beginning to be realized, and the field will undoubtedly continue to grow and mature. Universal Algebra and Applications in Theoretical Computer Science forms an outstanding text and offers a unique opportunity to build the foundation needed for further developments in its theory and in its computer science applications.

Theories and results on hyperidentities have been published in various areas of the literature over the last 18 years. Hyperidentities and Clones integrates these into a coherent framework for the first time. The author also includes some applications of hyperidentities to the functional completeness problem in multiple-valued logic and extends the general theory to partial algebras. The last chapter contains exercises and open problems with suggestions for future work in this area of research. Graduate students and mathematical researchers will find Hyperidentities and Clones a thought-provoking and illuminating text that offers a unique opportunity to study the topic in one source.