Prime, irreducible elements and coatoms in posets

2013 ◽  
Vol 63 (6) ◽  
Author(s):  
Weifeng Zhou ◽  
Qingguo Li ◽  
Lankun Guo
Keyword(s):  
Generating Set ◽  
Finite Poset ◽  
Irreducible Elements ◽  
Boolean Poset ◽  
Prime Elements

AbstractIn this paper, some properties of prime elements, pseudoprime elements, irreducible elements and coatoms in posets are investigated. We show that the four kinds of elements are equivalent to each other in finite Boolean posets. Furthermore, we demonstrate that every element of a finite Boolean poset can be represented by one kind of them. The example presented in this paper indicates that this result may not hold in every finite poset, but all the irreducible elements are proved to be contained in each order generating set. Finally, the multiplicative auxiliary relation on posets and the notion of arithmetic poset are introduced, and some properties about them are generalized to posets.

scholarly journals The “k = 1” Case of a Problem of Greene and Kleitman from 1976: Join-Irreducible Elements in the Lattice of Sperner 1-Families

2021 ◽  
Author(s):  
Jonathan David Farley
Keyword(s):  
Finite Poset ◽  
Irreducible Elements

Let k ≥ 1. A Sperner k-family is a maximum-sized subset of a finite poset that contains no chain with k + 1 elements. In 1976 Greene and Kleitman defined a lattice-ordering on the set Sk(P) of Sperner k-families of a fifinite poset P and posed the problem: “Characterize and interpret the join- and meet-irreducible elements of Sk(P),” adding, “This has apparently not been done even for the case k = 1.”In this article, the case k = 1 is done.

scholarly journals Existence of prime elements in rings of generalized power series

2001 ◽  
Vol 66 (3) ◽  
pp. 1206-1216 ◽  
Author(s):  
Daniel Pitteloud
Keyword(s):  
Abelian Group ◽  
Power Series ◽  
Prime Ideal ◽  
Additive Group ◽  
Convex Subgroup ◽  
Generalized Power Series ◽  
Irreducible Elements ◽  
Closed Fields ◽  
Proper Convex ◽  
Prime Elements

AbstractThe field K((G)) of generalized power series with coefficients in the field K of characteristic 0 and exponents in the ordered additive abelian group G plays an important role in the study of real closed fields. Conway and Gonshor (see [2, 4]) considered the problem of existence of non-standard irreducible (respectively prime) elements in the huge “ring” of omnific integers, which is indeed equivalent to the existence of irreducible (respectively prime) elements in the ring K((G≤0)) of series with non-positive exponents. Berarducci (see [1]) proved that K((G≤0)) does have irreducible elements, but it remained open whether the irreducibles are prime i.e., generate a prime ideal. In this paper we prove that K((G≤0)) does have prime elements if G = (ℝ, +) is the additive group of the reals, or more generally if G contains a maximal proper convex subgroup.

Studying the Krull dimension of finite lattices under the prism of matrices

Filomat ◽  
2017 ◽  
Vol 31 (10) ◽  
pp. 2901-2915 ◽  
Author(s):  
T. Dube ◽  
D.N. Georgiou ◽  
A.C. Megaritis ◽  
F. Sereti
Keyword(s):  
Partially Ordered Set ◽  
Incidence Matrix ◽  
Ordered Set ◽  
Finite Lattice ◽  
Krull Dimension ◽  
Reduction Algorithm ◽  
Partially Ordered ◽  
Finite Poset ◽  
Finite Lattices ◽  
Prime Elements

The Krull dimension of a finite lattice (X;6) is equal to the height of the poset of join prime elements of X minus 1. To every partially ordered set we assign an order-matrix, and we use these ordermatrices to characterize the join prime elements of finite lattices. In addition, we present a reduction algorithm for the computation of the height of a finite poset. The algorithm is based on the concept of the incidence matrix. Our main objective, ultimately, is to use these processes to calculate the Krull dimension of any given finite lattice.

Materialism and Legal Historiography, from Bachelard to Benjamin

2019 ◽  
pp. 1-19
Author(s):  
Christopher Tomlins
Keyword(s):  
Walter Benjamin ◽  
Historical Materialism ◽  
First Century ◽  
Gaston Bachelard ◽  
Spatial Justice ◽  
Legal Studies ◽  
New Materialisms ◽  
The World ◽  
Prime Elements ◽  
Cultural Turn

As the linguistic/cultural turn of the last fifty years has begun to ebb, sociolegal and legal-humanist scholarship has seen an accelerating return to materiality. This chapter asks what relationship may be forthcoming between the “new materialisms” and “vibrant matter” of recent years, and the older materialisms—both historical and literary, both Marxist and non-Marxist—that held sway prior to post-structuralism. What impact might such a relationship have on the forms, notably “spatial justice,” that materiality is assuming in contemporary legal studies? To attempt answers, the chapter turns to two figures from more than half a century ago: Gaston Bachelard—once famous, now mostly forgotten; and Walter Benjamin—once largely forgotten, now famous. A prolific and much-admired writer between 1930 and 1960, Bachelard pursued two trajectories of inquiry: a dialectical and materialist and historical (but non-Marxist) philosophy of science; and a poetics of the material imagination based on inquiry into the literary reception and representation of the prime elements—earth, water, fire, and air. Between the late 1920s and 1940, meanwhile, Benjamin developed an idiosyncratic but potent form of historical materialism dedicated to “arousing [the world] from its dream of itself.” The chapter argues that by mobilizing Bachelard and Benjamin for scholarship at the intersection of law and the humanities, old and new materialisms can be brought into a satisfying conjunction that simultaneously offers a poetics for spatial justice and lays a foundation for a materialist legal historiography for the twenty-first century.

scholarly journals A generating set for the canonical ideal of HKG-curves

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Aristides Kontogeorgis ◽  
Ioannis Tsouknidas
Keyword(s):  
Generating Set ◽  
Canonical Ideal

A Control System for a High-Quality Generating Set Equipped With a Two-Shaft Gas Turbine

1991 ◽  
Vol 113 (2) ◽  
pp. 290-295 ◽  
Author(s):  
H. Kumakura ◽  
T. Matsumura ◽  
E. Tsuruta ◽  
A. Watanabe
Keyword(s):  
Control System ◽  
Gas Turbine ◽  
Gas Turbines ◽  
High Quality ◽  
Constant Frequency ◽  
Generating Set ◽  
Generating Sets ◽  
Power Turbine ◽  
Computer Centers ◽  
Supply Systems

A control system has been developed for a high-quality generating set (150-kW) equipped with a two-shaft gas turbine featuring a variable power turbine nozzle. Because this generating set satisfies stringent frequency stability requirements, it can be employed as the direct electric power source for computer centers without using constant-voltage, constant-frequency power supply systems. Conventional generating sets of this kind have normally been powered by single-shaft gas turbines, which have a larger output shaft inertia than the two-shaft version. Good frequency characteristics have also been realized with the two-shaft gas turbine, which provides superior quick start ability and lower fuel consumption under partial loads.

scholarly journals Intersection Conductance and Canonical Alternating Paths: Methods for General Finite Markov Chains

2014 ◽  
Vol 23 (4) ◽  
pp. 585-606
Author(s):  
RAVI MONTENEGRO
Keyword(s):  
Markov Chains ◽  
Cayley Graph ◽  
Mixing Time ◽  
Symmetric Case ◽  
Generating Set ◽  
Finite Markov Chains

We extend the conductance and canonical paths methods to the setting of general finite Markov chains, including non-reversible non-lazy walks. The new path method is used to show that a known bound for the mixing time of a lazy walk on a Cayley graph with a symmetric generating set also applies to the non-lazy non-symmetric case, often even when there is no holding probability.

A Note on Parabolic Subgroups and the Steenrod Algebra Action

2008 ◽  
Vol 15 (04) ◽  
pp. 689-698
Author(s):  
Nondas E. Kechagias
Keyword(s):  
Steenrod Algebra ◽  
Parabolic Subgroups ◽  
Generating Set ◽  
Modular Invariants ◽  
Dickson Algebra

The ring of modular invariants of parabolic subgroups has been described by Kuhn and Mitchell using Dickson algebra generators. We provide a new generating set which is closed under the Steenrod algebra action along with the relations between these elements.

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