scholarly journals Endpoints inT0-Quasimetric Spaces: Part II

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Collins Amburo Agyingi ◽  
Paulus Haihambo ◽  
Hans-Peter A. Künzi
Keyword(s):  
Complete Lattice ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partial Orders ◽  
Macneille Completion ◽  
Partially Ordered ◽  
Irreducible Elements ◽  
Quasimetric Space

We continue our work on endpoints and startpoints inT0-quasimetric spaces. In particular we specialize some of our earlier results to the case of two-valuedT0-quasimetrics, that is, essentially, to partial orders. For instance, we observe that in a complete lattice the startpoints (resp., endpoints) in our sense are exactly the completely join-irreducible (resp., completely meet-irreducible) elements. We also discuss for a partially ordered set the connection between its Dedekind-MacNeille completion and theq-hyperconvex hull of its naturalT0-quasimetric space.

On the Partially Ordered Set of Prime Ideals of a Distributive Lattice

1971 ◽  
Vol 23 (5) ◽  
pp. 866-874 ◽  
Author(s):  
Raymond Balbes
Keyword(s):  
Distributive Lattice ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Complete Characterization ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
Partially Ordered ◽  
Irreducible Elements

For a distributive lattice L, let denote the poset of all prime ideals of L together with ∅ and L. This paper is concerned with the following type of problem. Given a class of distributive lattices, characterize all posets P for which for some . Such a poset P will be called representable over. For example, if is the class of all relatively complemented distributive lattices, then P is representable over if and only if P is a totally unordered poset with 0, 1 adjoined. One of our main results is a complete characterization of those posets P which are representable over the class of distributive lattices which are generated by their meet irreducible elements. The problem of determining which posets P are representable over the class of all distributive lattices appears to be very difficult.

Arithmetic Properties Of Freely α-Generated Lattices

1962 ◽  
Vol 14 ◽  
pp. 476-481 ◽  
Author(s):  
Bjarni Jónsson
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Canonical Representation ◽  
Free Lattice ◽  
Partially Ordered ◽  
Binary Operations ◽  
Arithmetic Properties ◽  
Irreducible Elements

In § 1 we give a characterization of a lattice L that is freely α-generated by a given partially ordered set P. In § 2 we obtain a representation of an element of such a lattice as a sum (product) of additively (multiplicatively) irreducible elements which, although not unique, has some of the desirable features of the canonical representation, in Whitman (2), of an element of a free lattice. The usefulness of this representation is illustrated in § 3, where some further arithmetic properties of these lattices are derived.We use + and . for the binary operations of lattice addition and multiplication, and Σ and II for the corresponding operations on arbitrary sets and sequences of lattice elements. In other respects the terminology will be the same as in Crawley and Dean (1).

A RING-THEORETIC BASIS FOR LOGIC PROGRAMMING

1990 ◽  
Vol 01 (01) ◽  
pp. 23-48 ◽  
Author(s):  
V.S. SUBRAHMANIAN
Keyword(s):  
Logic Programming ◽  
Algebraic Structure ◽  
Complete Lattice ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Truth Values ◽  
Partially Ordered

Investigations into the semantics of logic programming have largely restricted themselves to the case when the set of truth values being considered is a complete lattice. While a few theorems have been obtained which make do with weaker structures, to our knowledge there is no semantical characterization of logic programming which does not require that the set of truth values be partially ordered. We derive here semantical results on logic programming over a space of truth values that forms a commutative pseudo-ring (an algebraic structure a bit weaker than a ring) with identity. This permits us to study the semantics of multi-valued logic programming having a (possibly) non-partially ordered set of truth values.

The k-Normal Completion of Function Lattices

1965 ◽  
Vol 17 ◽  
pp. 669-675 ◽  
Author(s):  
Henry B. Cohen
Keyword(s):  
Lower Bounds ◽  
Complete Lattice ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Upper Bounds ◽  
Partially Ordered ◽  
Normal Completion ◽  
Set Intersection ◽  
The Family ◽  
Point Set

A subset G of a non-empty partially ordered set C is called normal if it coincides with the set of all upper bounds of the set of lower bounds of G. This is equivalent to stipulating that G be the set of all upper bounds of some subset of C called a set of generators for G. When ordered by inclusion, the family of all normal subsets of C forms a complete lattice with maximum C and minimum empty or singleton. The meet operation is simply point set intersection; whence, the meet of a family Gi of normal subsets is the set of upper bounds of ∪ Fi where Fi generates Gi for each i. A normal subset is called proper if it is neither void nor C, and the proper normal subsets. of C form a boundedly complete lattice.

On the Ring of Quotients of a Boolean Ring

1959 ◽  
Vol 2 (1) ◽  
pp. 25-29 ◽  
Author(s):  
B. Brainerd ◽  
J. Lambek
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Macneille Completion ◽  
Boolean Ring ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Boolean Rings ◽  
Ring Of Quotients

Two important mathematical constructions are: the construction of the rational s from the integers and the construction of the reals from the rationals. The first process can be carried out for any ring, producing its maximal ring of quotients [4, 5]. The second process can be carried out for any partially ordered set producing its Dedekind-MacNeille completion [2, p. 58]. We will show that for Boolean rings, which are both rings and partially ordered sets, the two constructions coincide.

Information Retrieval Systems, an Algebraic Approach I1

1981 ◽  
Vol 4 (3) ◽  
pp. 551-603
Author(s):  
Zbigniew Raś
Keyword(s):  
Information Retrieval ◽  
Boolean Algebra ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Algebraic Approach ◽  
Partially Ordered ◽  
Retrieval Systems ◽  
Information Retrieval Systems ◽  
Present Part ◽  
L Systems

This paper is the first of the three parts of work on the information retrieval systems proposed by Salton (see [24]). The system is defined by the notions of a partially ordered set of requests (A, ⩽), the set of objects X and a monotonic retrieval function U : A → 2X. Different conditions imposed on the set A and a function U make it possible to obtain various classes of information retrieval systems. We will investigate systems in which (A, ⩽) is a partially ordered set, a lattice, a pseudo-Boolean algebra and Boolean algebra. In my paper these systems are called partially ordered information retrieval systems (po-systems) lattice information retrieval systems (l-systems); pseudo-Boolean information retrieval systems (pB-systems) and Boolean information retrieval systems (B-systems). The first part concerns po-systems and 1-systems. The second part deals with pB-systems and B-systems. In the third part, systems with a partial access are investigated. The present part discusses the method for construction of a set of attributes. Problems connected with the selectivity and minimalization of a set of attributes are investigated. The characterization and the properties of a set of attributes are given.

scholarly journals Each join-completion of a partially ordered set in the solution of a universal problem

1974 ◽  
Vol 17 (4) ◽  
pp. 406-413 ◽  
Author(s):  
Jürgen Schmidt
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Unique Extension ◽  
Partially Ordered ◽  
Special Cases

The main result of this paper is the theorem in the title. Only special cases of it seem to be known so far. As an application, we obtain a result on the unique extension of Galois connexions. As a matter of fact, it is only by the use of Galois connexions that we obtain the main result, in its present generality.

scholarly journals Primitive idempotent measures on compact semitopological semigroups

1972 ◽  
Vol 13 (4) ◽  
pp. 451-455 ◽  
Author(s):  
Stephen T. L. Choy
Keyword(s):  
Partial Order ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Zero Element ◽  
Primitive Idempotent ◽  
Natural Partial Order ◽  
Partially Ordered

For a semigroup S let I(S) be the set of idempotents in S. A natural partial order of I(S) is defined by e ≦ f if ef = fe = e. An element e in I(S) is called a primitive idempotent if e is a minimal non-zero element of the partially ordered set (I(S), ≦). It is easy to see that an idempotent e in S is primitive if and only if, for any idempotent f in S, f = ef = fe implies f = e or f is the zero element of S. One may also easily verify that an idempotent e is primitive if and only if the only idempotents in eSe are e and the zero element. We let П(S) denote the set of primitive idempotent in S.

ON RIBBON PRESENTATIONS OF RIBBON KNOTS

1994 ◽  
Vol 03 (02) ◽  
pp. 223-231
Author(s):  
TOMOYUKI YASUDA
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered ◽  
Totally Ordered Set

A ribbon n-knot Kn is constructed by attaching m bands to m + 1n-spheres in the Euclidean (n + 2)-space. There are many way of attaching them; as a result, Kn has many presentations which are called ribbon presentations. In this note, we will induce a notion to classify ribbon presentations for ribbon n-knots of m-fusions (m ≥ 1, n ≥ 2), and show that such classes form a totally ordered set in the case of m = 2 and a partially ordered set in the case of m ≥ 1.

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