tag:blogger.com,1999:blog-86778555106746729462018-02-21T20:34:55.811-08:00Lisp AIjhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.comBlogger232125tag:blogger.com,1999:blog-8677855510674672946.post-49078326288654179122018-02-21T20:34:00.003-08:002018-02-21T20:34:55.832-08:00Scattered total ordersjhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-61575656831552790762018-02-14T21:10:00.002-08:002018-02-17T07:12:05.090-08:00Order topologyGiven a totally ordered set, we can form an open topology on that set from the set of open rays consisting of all the points that are either strictly greater or strictly less then a given point. The order topology also contains the open intervals of the set. The first concept that can be derived from the order topology is that of an isolated point. An isolated point is a singleton set of the order topology. A discrete total order consists entirely of isolated points. A point is considered to be near isolated if it always contains an isolated point in any open set containing it. <br /><br />A scattered topological space is strictly near isolated. These scattered topological points consist of both isolated points, and limits of isolated points. Scattered topologies include discrete topologies as a special case. In this sense, they are somewhat of a generalization of discrete topologies. Points are often defined by the existence of a topological subspace that contains them. A scattered point is a point that is contained within some scattered topology. Scattered total orders are defined as total orders with a scattered topology. <br /><br />A topological space that contains no isolated points is dense in itself, which makes it relatively less restricted then these other types of topological spaces. The other spaces are defined based upon forbidding dense subspaces. A space can include dense subspaces as well as isolated points and be neither type of topology. A point can be characterized based upon rather it is contained in a dense in itself. Dense total orders are defined as total orders with a dense in itself topology. The real numbers themselves have a dense in itself topology. <br /><br />A metric space is defined based upon a totally ordered set of distances. The character that a metric space can take is determined by the order topology of its set of distances. If a metric space has a discrete set of distances then it is necessarily going to be a discrete metric. For example, the path metric on a graph uses only integer distances so it will necessarily only form a discrete metric. In the same sense, if a scattered set of distances is used, then the metric space will necessarily be scattered as well. As a result, being either discrete or scattered is transferred from the order topology of the distances to the metric space. In this sense, the order topology is perhaps the most fundamental concept in the theory of metric spaces. <br /><br />It is useful to define a metric space associated with a given partial order. In a locally finite order this can be defined based upon the path metric of the covering graph. If the order has a different topology, however, then it is necessary to define a different type of metric space on it. In particular, if an order has a dense topology then the path metric may no longer suffice and it will be necessary to define some other concept of distance between points. So a dense metric can be created so that it can be associated with the dense order.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-29384227624928436512018-01-30T04:41:00.001-08:002018-02-17T06:31:07.504-08:00Subgraphs of connected graphsGiven a connected graph as well a subset of the vertices of the connected graph, we can form a subgraph of the connected graph containing only the vertices in that subset. This subgraph need not be connected itself, or even non empty as it could consist of disconnected points. The interesting property of the subgraph then is actually its metric rather then its adjacency relation. We can form a metric on the subgraph by determining the distance between each point in the parent graph based upon the shortest path metric.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-10717032134703984422018-01-02T01:25:00.001-08:002018-01-02T01:25:21.588-08:002017 year in reviewjhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-30794955562345466142017-12-31T20:09:00.002-08:002018-01-17T21:07:28.444-08:00Metric properties of graphsGiven a connected graph, we can form a metric space from the graph in which the distance between any two points is determined by the shortest path between them. The shortest path between two points need not be unique, unless the graph is a geodetic graph. This metric space is always going to be a discrete space, and a very particular type of discrete space which is generated by the unit distances between its points. The most interesting metric property of any vertex in a graph is its eccentricity which is the greatest distance between the vertex and any other point. The radius and the diameter of a graph are distance related properties determined by the minimum and the maximum eccentricity respectively.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-85770343631962228952017-08-24T22:09:00.000-07:002017-08-24T22:09:02.058-07:00Geodetic graphsGeodetic graphs have the particular property that each pair of points has a unique shortest path between them, or geodesic. Trees are a special case of geodetic graphs in which each pair of points has a unique path in general between them. Block graphs are sometimes called clique trees, they include trees and also happen to be the chordal geodetic graphs.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-56052052114160438422017-04-03T01:44:00.001-07:002017-04-03T03:01:02.912-07:00Partition logicThe logic of set partitions is a fundamental part of the analysis of ordering relations. The set partitions necessarily form a lattice which has certain properties corresponding to its join and meet operations. At the same time, the elements themselves form a Moore family depending upon which direction you intend to take the ordering. The Moore family then necessarily comes with a corresponding closure operation.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-86973928549166975592017-01-01T18:53:00.001-08:002017-03-29T06:05:05.027-07:002016 year in reviewThe methods of propositional logic can be applied to situations that arise from boolean algebra. Boolean formulas are applicable to any boolean algebra structure rather it is booleans themselves or sets, classes, and other predicate like structures. Set theoretic operations like union, intersection, difference, and symmetric difference can be expressed in terms of boolean formulas. These set theoretic operations are not unlike the logical connectives defined by these same boolean formulas. <br /><br />At the same time, a wide variety of constraint satisfaction problems like sudoku can be expressed in terms of boolean formulas. Boolean formulas defined by clauses that have no more then two elements are equivalent to implication graphs. Backtracking search methods can be used to solve constraint satisfaction problems like sudoku and implications can be used to narrow down the search space for any partial candidate solution in the backtracking search. <br /><br />The exact solutions to constraint satisfaction problems can therefore be deduced. In the vast majority of cases exact constraint satisfaction methods are not applicable and we need to use heuristics and statistics instead to get approximate solutions to optimization problems.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-91821513344474518002016-12-16T21:12:00.002-08:002016-12-16T21:12:52.285-08:00Boolean algebra equationsBoolean algebra equations can be built from meet, join, and complementation based upon the distributive law and the fact that complementation is an involution. These boolean algebra formulas can be put into conjuctive normal form or disjunctive normal form based upon which lattice operation is going to be at the highest level of nesting. These boolean algebra equations are applicable to any boolean algebra rather it is truth values or entire sets. Propositional logic formulas then can be defined based upon the principles of boolean algebra.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-19076398071697041902016-09-24T23:11:00.001-07:002016-09-24T23:11:08.687-07:00Propositional logic formulasPropositional logic deals with the truth values of propositions and how they are related through logical connectives. In general propositional logic formulas can be represented using conjunctive normal form or disjunctive normal form. These propositional logic formulas have a wide variety of general applications. In particular, numerous constraint satisfaction problems can be described as boolean satisfiability problems in propositional logic. Constraint satisfaction is a vital area of interest in propositional logic.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-15487855266858347482016-06-30T23:22:00.001-07:002016-06-30T23:22:20.077-07:00Families of kuratowski pairsFamilies of kuratowski pairs which are set systems that define a pair between two elements fully characterize correspondences between sets. It is by using families of kuratowski pairs that we can deal with questions of relation algebra. Each family of kuratowski pairs necessarily has rank at most two because each kuratowski pair has a maximum cardinality of two. This means that these are rank limited set systems.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-54434887406112223172016-06-15T22:19:00.001-07:002016-06-15T22:20:19.533-07:00The rank of set systemsGiven a set of sets then the rank of that set system is the maximum cardinality of the sets it contains as members. We can determine certain characteristics of a set system based upon their rank. Set systems of rank at most one are rank complete which means they are like a set which may or may not contain the empty set. Set systems of rank at most two include as a particular case set systems that define comparability. This process can be extended further to determine properties of the rank of higher set systems. <br /><br />The rank property can also be used to understand sets of sets of sets. Given a sets of sets of sets then the rank is the maximum cardinality of a set of sets contained in the set of sets of sets. We can deduce that families of kuratowski pairs necessarily have a rank of at most two because the pairs never have more then two members. The families of kuratowski pairs effectively generalize correspondences between sets. Kuratowski pairs also have the property that their union has at most two elements so the rank of the set of unions is at most two which makes the ranking property even stronger for them.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-29882358536803304642016-05-25T01:11:00.002-07:002016-05-25T01:11:31.844-07:00Relation algebraWe can establish a theory of binary relations in terms of their ordering as well as composition and transposition. The composition of binary relations generalizes the notion of composition of functions. Functional binary relations merely have a functional dependency from their input argument to the output argument. The inverse functional binary relations are their transpose. One to one binary relations are both functional and inverse functional. The composition of one to one binary relations is a one to one binary relation. Symmetric binary relations are equal to their transposition. The involutions are symmetric as well as functional. These properties allow us to better understand binary relations in terms of composition and transposition.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-77523611971250437692016-03-12T22:14:00.002-08:002016-03-12T22:14:37.531-08:00Incidence order of graphsThe incidence order of a graph is a type of partial order that forbids three different partial orders [1 1 1], [2 2], and [3 1]. If these three partial orders are forbidden as suborders then the partial order may be embedded in the incidence order of a graph. This defines the suborder closure of the class of incidence orders of graphs. In order for a partial order to be an incidence order of a graph all of its non-minimal elements must also have exactly two parts. <br /><br />If this final condition is met then the join representations of the partial order correspond to a dependency family. There is a complementary condition produced by the transposition of the incidence order of a graph. Only those graphs which have locally at most two elements meet this suborder condition in their transposition. All graphs can be produced from the join representations of a partial order that meets these conditions.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-32139903957675873962016-01-02T10:15:00.001-08:002016-03-03T02:38:16.452-08:002015 year in reviewWithin the year 2015 it was realized that it can be useful to study pure set theory. By studying pure set theory we are also dealing with pure ontology. We are classifying sets entirely in terms of their own properties as sets. We can use set systems in order to define preorder containment families and alexandrov families which correspond to one another. We can also define dependency families in accordance with this process. Dependency families can be defined as the join representations of certain preorder containment families. Though we can deal with sets of sets as a particular case we can also deal with sets of sets of sets. These allows us to deal with structures which we could not effectively define simply with sets of sets. In this way we are still dealing with pure ontology as we are defining sets entirely in terms of sets. As we create a pure ontology dealing with sets and classes defined entirely in terms of themselves we can later extend this ontology to deal with particulars but in the end the pure ontology is the foundation of this.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-17036399517767553212015-08-31T04:12:00.001-07:002015-09-25T13:02:54.284-07:00Threshold graphs and interval ordersThreshold graphs like interval orders have the property that each element is determined by some vertex invariant in all substructures. In threshold graphs all elements are determined by degree. In interval orders all elements are determined entirely by their interval representations. Both of these properties are maintained in forbidden substructures of each of these structures. Weak orders are the special class of interval orders whose elements are determined in terms of degrees. Threshold orders are interval orders with threshold comparability.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-34732961139952196522015-05-30T20:59:00.001-07:002015-09-25T12:47:45.655-07:00Weakly chordal graphsThe weakly chordal graphs are a subclass of the perfect graphs that exclude all holes and antiholes. The weakly chordal graphs are an important class of graphs in order theory. Well the class of weakly chordal graphs does not include all comparability graphs or all cocomparability graphs it does include all permutation graphs which are the graphs that are both comparability and cocomparability. This means it also includes the cographs which also happen to be permutation graphs. <br /><br />The chordal graphs as well as their complements the cochordal graphs are weakly chordal. The trivially perfect graphs are precisely those graphs that are both chordal and cographs. The cotrivially perfect graphs are precisely those graphs that are both cochordal and cographs. The trivially perfect graphs and the cotrivially perfect graphs can both be expressed entirely in terms of preorders. Only weakly chordal graphs can be described entirely in terms of preorders in this sense.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-55019398098493894342015-04-30T22:32:00.002-07:002015-09-25T12:32:20.100-07:00Comparisons between setsWe can produce certain comparisons between sets such as based upon the intersection size. This is what leads to the classes of independent, linear, and antidisjoint families of sets. The independent families have no intersection between elements. Linear families have no intersection larger then one. And the antidisjoint families are complementary to the independent families in the sense that no elements lack intersection. Pairs of sets can only either be comparable or incomparable which is what leads to chain families and sperner families. In chain families each pair of sets is comparable. In sperner families each pair of sets is incomparable. Laminar families are produced by the union of independence and comparability as each element is either comparable or it has empty intersection. In uniform families each element has the same size. Each of these comparisons between sets corresponds to a clique family. The dependence condition of antidisjoint families produces line graphs and the comparability condition produces comparability graphs and cocomparability graphs of subclass containment orders.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-47263768948461256072015-03-30T19:49:00.001-07:002015-09-25T12:24:03.100-07:00Forbidden subfamiliesGiven a set theory we can classify families of sets based upon forbidden subfamilies. As with most classes determined by forbidden substructures the classification of families based upon forbidden subfamilies is characterized based upon the size of the largest forbidden subfamily. If the largest forbidden subfamily is of size one then the class is going to be a power set of some family often determined by cardinality. If the largest forbidden subfamily is of size two then the class is going to be a clique family because it is generated by the dependencies between pairs of elements. Another issue is the smallest forbidden subfamily. All families with size greater then the forbidden size will be accepted automatically. The two issues of the largest and smallest size of a forbidden substructure help to determine nature of subclass closed families.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-56105428049056987342015-03-28T19:41:00.001-07:002015-04-25T03:32:21.926-07:00Algebraic set theoryThe theory of sets can be described by a boolean algebra equipped with a singleton function which converts between any atom in the boolean algebra into its corresponding member value. Sets can then be described within this algebraic structure entirely in terms of the join operation of the boolean algebra and the singleton function. This allows us to describe sets entirely in terms of algebraic set theory. <br /><br />The only element in a boolean algebra which does not contain any atoms is the lower bound element of the boolean algebra. By using this lower bound element, the join operation of the boolean algebra, and the singleton function we can produce elements of the pure elements of the algebra. These pure elements correspond to the pure sets which we generally encounter in set theory. Some algebras of sets are limited to only pure elements and others are not.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-38013682035168974782015-02-28T22:01:00.001-08:002015-04-25T02:44:34.847-07:00Atoms of a latticeGiven a distributive lattice then we can determine that the atoms of that lattice are those elements that cover the lower bound of that lattice. In other words they are the singleton elements of that lattice. For example in the case of the lattice of sets under inclusion the atoms of that lattice are the singleton sets. The atoms of a lattice play an interesting role in the description of that lattice. <br /><br />In the case of the lattice of sets we find that any given entity can be converted to a singleton set containing only that entity and likewise any singleton set containing only a single entity can be converted into back into that entity. This means that the function to convert entities into singleton sets is a one to one correspondence. This singleton function is not described by the order on the sets so it is outside of the underlying order theory. A lattice ordered structure can be extended with such a singleton function to be produced a more advanced structure of this sort.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-8207479051975794552015-01-01T19:28:00.001-08:002015-04-25T02:18:39.211-07:002014 year in reviewWithin the year 2014 I maintained my focus on order theory and then I began to examine set systems which are equivalent to suborders of a boolean algebra. Being a suborder each set system is partially ordered and each partial order can be embedded into a boolean algebra often in a variety of ways. In particular, given a partial order we can convert that partial order into the set system by taking the set of all sets of elements of principal ideals of that partial order. We can also convert that partial order into a distributive lattice by taking the set of lower sets though that does not preserve order. In this way the theory of set systems subsumes the theory of partial orders. Everything we can say about partial orders can also be expressed in terms of set systems. As a result of this fact, set systems became a primary concern of mine in the past year. <br /><br />Through my exploration of set systems as suborders of boolean algebras I later came to consider suborders of other distributive lattices. In particular, we can consider suborders of distributive lattices that are defined by multiset inclusion. In this way we can examine the idea of multiset systems in addition to set systems. An interesting facet of the theory of multiset systems is that we can convert any ordered collection into a multiset system regardless of cardinality just as we can convert any ordered collection into a set system using the kuratowski pair. After considering the theory of suborders of distributive lattices like these we can generalize and then consider the theory of suborders of a partial order in general. <br /><br />As the notion of set systems as suborders of a boolean algebra has been explored my ontology has continued to expand as these notions have been explored. As a result an ontology of set systems including a wide variety of classes of set systems has been produced including classes of set systems which are defined entirely by their order characteristics. In this way the ontology has expanded to deal with sets as well as sets whose members are all sets. These are distinguished from objects which are not ontologically classified as sets and sets which contain elements that are not classified as sets. As an ontology deals with sets and classes itself it is sensible that one of the most basic things to be classified in an ontology are such sets and classes. This is the approach that I am now taking to ontology engineering.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-88710973269960492902014-12-31T19:50:00.001-08:002015-01-01T00:54:55.218-08:00Structural specializationThere are a variety of cases in which it makes sense to preorder a set based upon the elements that it contains. One example is that with the set representation of a multiset the elements of the set include special multiple elements which are dependent upon previous multiples. Also algebraic structures like graphs have a specialization preorder associated with them in which edge elements are dependent upon the vertices that they contain. Multigraphs are a combination of these two notions as they can have multiples of edges which are dependent upon previous multiples of edges which are then dependent upon vertices. <br /><br />A standard specialization relation can be provided that combines all these different notions of structures on sets into a single unified relation. This combined specialization relation will allow for all elements of distributive lattices to be treated like sets, multisets, and algebraic structures to be treated in a uniform way. This will then improve the handling of multisets and related structures like multigraphs so that they can be treated as a first class object in the algebra system.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-89321699575695526032014-12-16T20:31:00.002-08:002014-12-16T20:40:04.390-08:00Multiset systemsOne of the critical problems of set theory is how to represent sequences as sets. Well Kuratowski solved the problem for sequences of size two through the set theoretic definition of the ordered pair there is no obvious solution to the problem of representing sequences of arbitrary size as set systems because sequences may contain repeated elements. I have thought about this problem so much in terms of set systems that I did not consider the possibility of using a multiset system instead. <br /><br />In order to represent any sequence all we need to do is use a multiset system containing the multiset of elements up to a point in the sequence for each point in the sequence. This solution is so simple I am surprised I did not hear about this before. It is probably because multisets are far too often pushed aside for sets but no more. From now on I am going to make full use of multisets when I think about mathematics.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0tag:blogger.com,1999:blog-8677855510674672946.post-21056667270913424972014-11-26T21:59:00.001-08:002014-12-02T16:31:18.780-08:00Disjoint union closed familiesThe disjoint union closed families are precisely those families for which it is the case that the union of any two disjoint sets in the family is contained within the family. The disjoint union closed families generalize both the union closed families which are both disjoint union closed and nondisjoint union closed and the symmetric difference closed families. The symmetric difference of any two disjoint sets is their union so symmetric difference closed families are disjoint union closed. <br /><br />Besides the union closed families and the symmetric difference closed families the antidisjoint families are disjoint union closed. The antidisjoint families are precisely those families of sets whose every pair of sets is not disjoint. These include the nullfree chain families which also happen to have the property that they are union closed as they are chain families as well as the antidisjoint sperner families which are antidisjoint families that also contain no comparable pairs.jhunihttp://www.blogger.com/profile/05342856856194792634noreply@blogger.com0