On Baer Invariants of Triples of Groups

In this paper, we develop the theory of Baer invariants for triples of groups. First, we focus on the general properties of the Baer invariant of triples. Second, we prove that the Baer invariant of a triple preserves direct limits of directed systems of triples of groups. Moreover, we present a structure for the nilpotent multiplier of a triple of the free product in some cases. Finally, we give some conditions in which the Baer invariant of a triple is a torsion group.Comment: 9 page

Latin directed triple systems

Abstract It is well known that given a Steiner triple system then a quasigroup can be formed by defining an operation · by the identities x · x = x and x · y = z where z is the third point in the block containing the pair {x, y}. The same is true for a Mendelsohn triple system where the pair (x, y) is considered to be ordered. But it is not true in general for directed triple systems. However directed triple systems which form quasigroups under this operation do exist. We call these Latin directed triple systems and in this paper begin the study of their existence and properties

Types of directed triple systems

We introduce three types of directed triple systems. Two of these, Mendelsohn directed triple systems and Latin directed triple systems, have previously appeared in the literature but we prove further results about them. The third type, which we call skewed directed triple systems, is new and we determine the existence spectrum to be v ≡ 1 (mod 3), v ≠ 7, except possibly for v = 22, as well as giving enumeration results for small orders

The spectrum of bicyclic antiautomorphisms of directed triple systems

AbstractA transitive triple, (a,b,c), is defined to be the set {(a,b),(b,c),(a,c)} of ordered pairs. A directed triple system of order v, DTS(v), is a pair (D,β), where D is a set of v points and β is a collection of transitive triples of pairwise distinct points of D such that any ordered pair of distinct points of D is contained in precisely one transitive triple of β. An antiautomorphism of a directed triple system, (D,β), is a permutation of D which maps β to β−1, where β−1={(c,b,a)|(a,b,c)∈β}. In this paper we complete the necessary and sufficient conditions for the existence of a directed triple system of order v admitting an antiautomorphism consisting of two cycles

Large sets of extended directed triple systems with even orders

AbstractFor three types of triples: unordered, cyclic and transitive, the corresponding extended triple, extended triple system and their large sets are introduced. The existence of LESTS(υ) and LEMTS(υ) were completely solved. In this paper, we shall discuss the existence problem of LEDTS(υ) and give the following conclusion: there exists an LEDTS(υ) for any even υ except υ=4. The existence of LEDTS(υ) with odd order υ will be discussed in another paper, we are working at it

The intersection spectrum of Skolem sequences and its applications to lambda fold cyclic triple systems, together with the Supplement

A Skolem sequence of order n is a sequence S_n=(s_{1},s_{2},...,s_{2n}) of 2n integers containing each of the integers 1,2,...,n exactly twice, such that two occurrences of the integer j in {1,2,...,n} are separated by exactly j-1 integers. We prove that the necessary conditions are sufficient for existence of two Skolem sequences of order n with 0,1,2,...,n-3 and n pairs in same positions. Further, we apply this result to the fine structure of cyclic two, three and four-fold triple systems, and also to the fine structure of lambda-fold directed triple systems and lambda-fold Mendelsohn triple systems. For a better understanding of the paper we added more details into a "Supplement".Comment: The Supplement for the paper "The intersection spectrum of Skolem sequences and its applications to lambda fold cyclic triple systems" is available here. It comes right after the paper itsel

Using RDF to Model the Structure and Process of Systems

Many systems can be described in terms of networks of discrete elements and their various relationships to one another. A semantic network, or multi-relational network, is a directed labeled graph consisting of a heterogeneous set of entities connected by a heterogeneous set of relationships. Semantic networks serve as a promising general-purpose modeling substrate for complex systems. Various standardized formats and tools are now available to support practical, large-scale semantic network models. First, the Resource Description Framework (RDF) offers a standardized semantic network data model that can be further formalized by ontology modeling languages such as RDF Schema (RDFS) and the Web Ontology Language (OWL). Second, the recent introduction of highly performant triple-stores (i.e. semantic network databases) allows semantic network models on the order of $10^9$ edges to be efficiently stored and manipulated. RDF and its related technologies are currently used extensively in the domains of computer science, digital library science, and the biological sciences. This article will provide an introduction to RDF/RDFS/OWL and an examination of its suitability to model discrete element complex systems.Comment: International Conference on Complex Systems, Boston MA, October 200