scholarly journals SUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, II.

2003 ◽  
Vol 13 (05) ◽  
pp. 543-564 ◽  
Author(s):  
MARINA SEMENOVA ◽  
FRIEDRICH WEHRUNG
Keyword(s):  
Positive Integer ◽  
Partially Ordered Set ◽  
Convex Sets ◽  
Ordered Set ◽  
Finitely Based ◽  
Locally Finite ◽  
Partially Ordered ◽  
Finitely Based Variety ◽  
Atomistic Lattice ◽  
Convex Subsets

For a positive integer n, we denote by SUB (respectively, SUBn) the class of all lattices that can be embedded into the lattice Co(P) of all order-convex subsets of a partially ordered set P (respectively, P of length at most n). We prove the following results: (1) SUBn is a finitely based variety, for any n≥1. (2) SUB2 is locally finite. (3) A finite atomistic lattice L without D-cycles belongs to SUB if and only if it belongs to SUB2; this result does not extend to the nonatomistic case. (4) SUBn is not locally finite for n≥3.

scholarly journals SUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, III: THE CASE OF TOTALLY ORDERED SETS

2004 ◽  
Vol 14 (03) ◽  
pp. 357-387 ◽  
Author(s):  
MARINA SEMENOVA ◽  
FRIEDRICH WEHRUNG
Keyword(s):  
Positive Integer ◽  
Convex Sets ◽  
Ordered Set ◽  
Subdirectly Irreducible ◽  
Finitely Based ◽  
Ordered Sets ◽  
Locally Finite ◽  
Varieties Of Lattices ◽  
Irreducible Member ◽  
Subdirectly Irreducible Member

For a partially ordered set P, let Co(P) denote the lattice of all order-convex subsets of P. For a positive integer n, we denote by [Formula: see text] (resp., SUB(n)) the class of all lattices that can be embedded into a lattice of the form [Formula: see text] where <Ti|i∈I> is a family of chains (resp., chains with at most n elements). We prove the following results: (1) Both classes [Formula: see text] and SUB(n), for any positive integer n, are locally finite, finitely based varieties of lattices, and we find finite equational bases of these varieties. (2) The variety [Formula: see text] is the quasivariety join of all the varieties SUB(n), for 1≤n<ω, and it has only countably many subvarieties. We classify these varieties, together with all the finite subdirectly irreducible members of [Formula: see text]. (3) Every finite subdirectly irreducible member of [Formula: see text] is projective within [Formula: see text], and every subquasivariety of [Formula: see text] is a variety.

scholarly journals Essential ideals of incidence algebras

2000 ◽  
Vol 68 (2) ◽  
pp. 252-260 ◽  
Author(s):  
Eugene Spiegel
Keyword(s):  
Commutative Ring ◽  
Left Ideal ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Locally Finite ◽  
Partially Ordered ◽  
Incidence Algebra ◽  
Incidence Algebras ◽  
Essential Ideal ◽  
Finite Partially Ordered Set

AbstractIt is determined when there exists a minimal essential ideal, or minimal essential left ideal, in the incidence algebra of a locally finite partially ordered set defined over a commutative ring. When such an ideal exists, it is described.

scholarly journals AUTOMORPHISMS OF POWERS OF LINEARLY ORDERED SETS

2006 ◽  
Vol 14 (01) ◽  
pp. 77-85 ◽  
Author(s):  
CAROL L. WALKER ◽  
ELBERT A. WALKER
Keyword(s):  
Positive Integer ◽  
Partially Ordered Set ◽  
Automorphism Groups ◽  
Ordered Set ◽  
Unit Interval ◽  
Ordered Sets ◽  
Real Numbers ◽  
Main Interest ◽  
Partially Ordered ◽  
Usual Order

Let S be a bounded, partially ordered set, and n a positive integer. We investigate automorphism groups of Sn and of S[n], the non-decreasing n-tuples of Sn. Our main interest is in the case where S is the unit interval of real numbers with the usual order.

scholarly journals TWO APPROACHES TO MÖBIUS INVERSION

2011 ◽  
Vol 85 (1) ◽  
pp. 68-78
Author(s):  
I-CHIAU HUANG
Keyword(s):  
Partially Ordered Set ◽  
Inversion Formula ◽  
Ordered Set ◽  
Lagrange Inversion ◽  
Locally Finite ◽  
Partially Ordered ◽  
Möbius Inversion ◽  
Möbius Inversion Formula ◽  
Schauder Bases ◽  
Finite Partially Ordered Set

AbstractThe Möbius inversion formula for a locally finite partially ordered set is realized as a Lagrange inversion formula. Schauder bases are introduced to interpret Möbius inversion.

scholarly journals Some remarks concerning semi-T1/2 spaces

Filomat ◽  
2014 ◽  
Vol 28 (1) ◽  
pp. 21-25 ◽  
Author(s):  
Vitalij Chatyrko ◽  
Sang-Eon Han ◽  
Yasunao Hattori
Keyword(s):  
Positive Integer ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered ◽  
Basic Properties

In this paper we prove that each subspace of an Alexandroff T0-space is semi-T1/2. In particular, any subspace of the folder Xn, where n is a positive integer and X is either the Khalimsky line (Z, ?K), the Marcus-Wyse plane (Z2, ?MW) or any partially ordered set with the upper topology is semi-T1/2. Then we study the basic properties of spaces possessing the axiom semi-T1/2 such as finite productiveness and monotonicity.

Arithmetical Functions and Distributivity

1970 ◽  
Vol 13 (4) ◽  
pp. 491-496 ◽  
Author(s):  
P. J. McCarthy
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Locally Finite ◽  
Arithmetical Functions ◽  
Partially Ordered ◽  
Incidence Functions ◽  
Finite Partially Ordered Set

In this note we shall present a result about incidence functions on a locally finite partially ordered set, a result which is related to theorems of Lambek [2] and Subbarao [6]. Our terminology and notation will be that of Smith [4, 5] and Rota [7].Let (L, ≤) be a partially ordered set which is locally finite in the sense that for all x, y ∊ L the interval [x, y] = {z | x ≤ z ≤ y} is finite. Denote by A(L, ≤) the set of functions f from L × L into some field, which is fixed once and for all, such that f(x, y) = 0 whenever x ≰ y.

The Periodic Radical of Group Rings and Incidence Algebras

1998 ◽  
Vol 41 (4) ◽  
pp. 481-487 ◽  
Author(s):  
M. M. Parmenter ◽  
E. Spiegel ◽  
P. N. Stewart
Keyword(s):  
Group Ring ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Sufficient Conditions ◽  
Group Rings ◽  
Locally Finite ◽  
Partially Ordered ◽  
Incidence Algebras ◽  
Finite Partially Ordered Set ◽  
Necessary And Sufficient

AbstractLet R be a ring with 1 and P(R) the periodic radical of R. We obtain necessary and sufficient conditions for P(RG) = 0 when RG is the group ring of an FC group G and R is commutative. We also obtain a complete description of when I(X, R) is the incidence algebra of a locally finite partially ordered set X and R is commutative.

scholarly journals Almost Tiling of the Boolean Lattice with Copies of a Poset

10.37236/6636 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
István Tomon
Keyword(s):  
Positive Integer ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Minimal Element ◽  
Boolean Lattice ◽  
Partially Ordered

Let $P$ be a partially ordered set. If the Boolean lattice $(2^{[n]},\subset)$ can be partitioned into copies of $P$ for some positive integer $n$, then $P$ must satisfy the following two trivial conditions:(1) the size of $P$ is a power of $2$,(2) $P$ has a unique maximal and minimal element.Resolving a conjecture of Lonc, it was shown by Gruslys, Leader and Tomon that these conditions are sufficient as well.In this paper, we show that if $P$ only satisfies condition (2), we can still almost partition $2^{[n]}$ into copies of $P$. We prove that if $P$ has a unique maximal and minimal element, then there exists a constant $c=c(P)$ such that all but at most $c$ elements of $2^{[n]}$ can be covered by disjoint copies of $P$.

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.

Sign in / Sign up

Export Citation Format

Share Document