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1	Homology from posets Jones, Philip Robert January 1999 (has links) No description available. 510
2	On Independence, Matching, and Homomorphism Complexes Hough, Wesley K. 01 January 2017 (has links) First introduced by Forman in 1998, discrete Morse theory has become a standard tool in topological combinatorics. The main idea of discrete Morse theory is to pair cells in a cellular complex in a manner that permits cancellation via elementary collapses, reducing the complex under consideration to a homotopy equivalent complex with fewer cells. In chapter 1, we introduce the relevant background for discrete Morse theory. In chapter 2, we define a discrete Morse matching for a family of independence complexes that generalize the matching complexes of suitable "small" grid graphs. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups. Furthermore, we determine the Euler characteristic for these complexes and prove that several of their homology groups are non-zero. In chapter 3, we introduce the notion of a homomorphism complex for partially ordered sets, placing particular emphasis on maps between chain posets and the Boolean algebras. We extend the notion of folding from general graph homomorphism complexes to the poset case, and we define an iterative discrete Morse matching for these Boolean complexes. We provide formulas for enumerating the number of critical cells arising from this matching as well as for the Euler characteristic. We end with a conjecture on the optimality of our matching derived from connections to 3-equal manifolds discrete Morse theory independence complexes matching complexes homomorphism complexes Boolean algebras Discrete Mathematics and Combinatorics Geometry and Topology
3	Games on Boolean algebras / Igre na Bulovim algebrama Šobot Boris 07 September 2009 (has links) <p>The method of forcing is widely used in set theory to obtain various consistency proofs. Complete Boolean algebras play the main role in applications of forcing. Therefore it is useful to define games on Boolean algebras that characterize their properties important for the method. The most investigated game is Jech’s distributivity game, such that the first player has the winning strategy iff the algebra is not (ω, 2)-distributive. We define another game characterizing the collapsing of the continuum to ω, prove several sufficient conditions for the second player to have a winning strategy, and obtain a Boolean algebra on which the game is undetermined. </p> / <p>Forsing je metod &scaron;iroko kori&scaron;ćen u teoriji skupova za dokaze konsistentnosti. Kompletne  Bulove algebre igraju glavnu ulogu u primenama forsinga. Stoga je korisno definisati igre na Bulovim algebrama koje karakteri&scaron;u njihove osobine od značaja za taj metod. Najbolje proučena je Jehova igra, koja ima osobinu da prvi igrač ima pobedničku strategiju akko algebra nije (ω, 2)-distributivna. U tezi defini&scaron;emo jo&scaron; jednu igru, koja karakteri&scaron;e kolaps kontinuuma na ω, dokazujemo nekoliko dovoljnih uslova da bi drugi igra&scaron; imao pobedničku strategiju, i konstrui&scaron;emo Bulovu algebru na kojoj je igra neodređena.</p>
4	Sequential Topologies on Boolean Algebras / Sekvencijalne topologije na Bulovim algebrama Pavlović Aleksandar 13 January 2009 (has links) <p>A priori limit operator>. maps sequence of a set X into a subset of X.<br />There exists maximal topology on X such that for each sequence x there holds<br />>.(x) C limx. The space obtained in such way is always sequential.<br />If a priori limit operator each sequence x which satisfy lim sup x = lim inf x<br />maps into {lim sup x}, then we obtain the sequential topology Ts.  If a priori 'limit<br />operator maps each sequence x into {lim sup x}, we obtain topology denoted by<br />aT. Properties of these topologies, in general, on class of Boolean algebras with<br />condition (Ii) and on class of weakly-distributive b-cc algebras are investigated.<br />Also, the relations between these classes and other classes of Boolean algebras are<br />considered.</p> / <p>A priori limit operator A svakom nizu elemenata skupa X dodeljuje neki<br />podskup skupa X. Tada na skupu X postoji maksimalna topologija takva da za<br />svaki niz x vazi A(X) c lim x. Tako dobijen prostor je uvek sekvencijalan.<br />Ako a priori limit operator svakom nizu x koji zadovoljava uslov lim sup x =<br />liminfx dodeljuje skup {limsupx} onda se, na gore opisan nacin, dobija tzv.<br />sekvencijalna topologija Ts. Ako a priori limit operator svakom nizu x dodeljuje<br />{lim sup x}, dobija se topologija oznacena sa OT.  Ispitivane su osobine ovih<br />topologija, generalno, na klasi Bulovih algebri koje zadovoljavaju uslov (Ii) ina<br />klasi slabo-distributivnih i b-cc algebri, kao i odnosi ovih klasa prema drugim<br />klasama Bulovih algebri.</p>
5	Weakly Dense Subsets of Homogeneous Complete Boolean Algebras Bozeman, Alan Kyle 08 1900 (has links) The primary result from this dissertation is following inequality: d(B) ≤ min(2^< wd(B),sup{λ^c(B): λ < wd(B)}) in ZFC, where B is a homogeneous complete Boolean algebra, d(B) is the density, wd(B) is the weak density, and c(B) is the cellularity of B. Chapter II of this dissertation is a general overview of homogeneous complete Boolean algebras. Assuming the existence of a weakly inaccessible cardinal, we give an example of a homogeneous complete Boolean algebra which does not attain its cellularity. In chapter III, we prove that for any integer n > 1, wd_2(B) = wd_n(B). Also in this chapter, we show that if X⊂B is κ—weakly dense for 1 < κ < sat(B), then sup{wd_κ(B):κ < sat(B)} = d(B). In chapter IV, we address the following question: If X is weakly dense in a homogeneous complete Boolean algebra B, does there necessarily exist b € B\{0} such that {x∗b: x ∈ X} is dense in B\|b = {c € B: c ≤ b}? We show that the answer is no for collapsing algebras. In chapter V, we give new proofs to some well known results concerning supporting antichains. A direct consequence of these results is the relation c(B) < wd(B), i.e., the weak density of a homogeneous complete Boolean algebra B is at least as big as the cellularity. Also in this chapter, we introduce discernible sets. We prove that a discernible set of cardinality no greater than c(B) cannot be weakly dense. In chapter VI, we prove the main result of this dissertation, i.e., d(B) ≤ min(2^< wd(B),sup{λ^c(B): λ < wd(B)}). In chapter VII, we list some unsolved problems concerning this dissertation. homogeneous complete Boolean algebras weak density cellularity weakly dense sets cardinal functions Algebra, Boolean.
6	Matroids on Complete Boolean Algebras Higgs, Denis Arthur 10 1900 (has links) The approach to a theory of non-finitary matroids, as outlined by the author in [20], is here extended to the case in which the relevant closure operators are defined on arbitrary complete Boolean algebras, rather than on the power sets of sets. As a preliminary to this study, the theory of derivatives of operators on complete Boolean algebras is developed and the notion, having interest in its own right, of an analytic closure operator is introduced . The class of B-matroidal closure operators is singled out for especial attention and it is proved that this class is closed under Whitney duality. Also investigated is the class of those closure operators which are both matroidal and topological. / Thesis / Doctor of Philosophy (PhD)
7	On the Complexity of Boolean Unification Baader, Franz 19 May 2022 (has links) Unification modulo the theory of Boolean algebras has been investigated by several autors. Nevertheless, the exact complexity of the decision problem for unification with constants and general unification was not known. In this research note, we show that the decision problem is complete for unification with constants and PSPACE-complete for general unification. In contrast, the decision problem for elementary unification (where the terms to be unified contain only symbols of the signature of Boolean algebras) is 'only' NP-complete. info:eu-repo/classification/ddc/004 ddc:004
8	Dicomplemented Lattices / A Contextual Generalization of Boolean Algebras / Treillis Dicomplementes. Une Generalisation Contextuelle des Algebres de Boole. / Dikomplementaere Verbaende. Eine Kontextuelle Verallgemeinerung Boolescher Algebren Kwuida, Leonard 23 October 2004 (has links) (PDF) Das Ziel dieser Arbeit ist es die mathematische Theorie der Begriffsalgebren zu entwickeln. Wir betrachten dabei hauptsaechlich das Repraesentationsproblem dieser vor Kurzem eingefuehrten Strukturen. Motiviert durch die Suche nach einer geeigneten Negation sind die Begriffsalgebren entstanden. Sie sind nicht nur fuer die Philosophie oder die Wissensrepraesentation von Interesse, sondern auch fuer andere Felder, wie zum Beispiel Logik oder Linguistik. Das Problem Negationen geeignet einzufuehren, ist sicher eines der aeltesten der wissenschaftlichen oder philosophischen Gemeinschaft und erregt auch zur Zeit die Aufmerksamkeit vieler Wissenschaftler. Verschiedene Typen von Logik (die sich sehr stark durch die eigefuehrte Negation unterscheiden) unterstreichen die Wichtigkeit dieser Untersuchungen. In dieser Arbeit beschaeftigen wir uns hauptsaechlich mit der kontextuellen Logik, eine Herangehensweise der Formalen Begriffsanalyse, basierend auf der Idee, den Begriff als Einheit des Denkens aufzufassen. / The aim of this investigation is to develop a mathematical theory of concept algebras. We mainly consider the representation problem for this recently introduced class of structures. Motivated by the search of a &quot;negation&quot; on formal concepts, &quot;concept algebras&quot; are of considerable interest not only in Philosophy or Knowledge Representation, but also in other fields as Logic or Linguistics. The problem of negation is surely one of the oldest problems of the scientific and philosophic community, and still attracts the attention of many researchers. Various types of Logic (defined according to the behaviour of the corresponding negation) can attest this affirmation. In this thesis we focus on &quot;Contextual Logic&quot;, a Formal Concept Analysis approach, based on concepts as units of thought. Begriffsalgebren Boolesche Algebren Boolesche Begriffsalgebren Darstellungssatz Formale Begriffsanalyse Kongruenzen Kontextuelle Logik Merkmalexploration Negation Skeleton Strukturtheory Tripel Konstruktion Varietaeten dikomplementaere Verbae Boolean Concept Logic Boolean algebras Concept algebras Contextual logic Formal Concept Analysis Negation Representation attribute exploration congruence dicomplemented lattices skeletons structure theory triple construction varieties ddc:510 rvk:SK 130 Boolesche Algebra Formale Begriffsanalyse Strukturtheorie Verbandstheorie
9	Dicomplemented Lattices: A Contextual Generalization of Boolean Algebras Kwuida, Leonard 29 June 2004 (has links) Das Ziel dieser Arbeit ist es die mathematische Theorie der Begriffsalgebren zu entwickeln. Wir betrachten dabei hauptsaechlich das Repraesentationsproblem dieser vor Kurzem eingefuehrten Strukturen. Motiviert durch die Suche nach einer geeigneten Negation sind die Begriffsalgebren entstanden. Sie sind nicht nur fuer die Philosophie oder die Wissensrepraesentation von Interesse, sondern auch fuer andere Felder, wie zum Beispiel Logik oder Linguistik. Das Problem Negationen geeignet einzufuehren, ist sicher eines der aeltesten der wissenschaftlichen oder philosophischen Gemeinschaft und erregt auch zur Zeit die Aufmerksamkeit vieler Wissenschaftler. Verschiedene Typen von Logik (die sich sehr stark durch die eigefuehrte Negation unterscheiden) unterstreichen die Wichtigkeit dieser Untersuchungen. In dieser Arbeit beschaeftigen wir uns hauptsaechlich mit der kontextuellen Logik, eine Herangehensweise der Formalen Begriffsanalyse, basierend auf der Idee, den Begriff als Einheit des Denkens aufzufassen. / The aim of this investigation is to develop a mathematical theory of concept algebras. We mainly consider the representation problem for this recently introduced class of structures. Motivated by the search of a &quot;negation&quot; on formal concepts, &quot;concept algebras&quot; are of considerable interest not only in Philosophy or Knowledge Representation, but also in other fields as Logic or Linguistics. The problem of negation is surely one of the oldest problems of the scientific and philosophic community, and still attracts the attention of many researchers. Various types of Logic (defined according to the behaviour of the corresponding negation) can attest this affirmation. In this thesis we focus on &quot;Contextual Logic&quot;, a Formal Concept Analysis approach, based on concepts as units of thought. info:eu-repo/classification/ddc/510 ddc:510

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