Interpreting arithmetics in the ideal lattice of a free vector lattice ℱ n

2008 ◽  
Vol 47 (1) ◽  
pp. 42-48
Author(s):  
O. A. Kuryleva
Keyword(s):  
Vector Lattice ◽  
Ideal Lattice ◽  
The Ideal

Commutative rings whose ideal lattices are complemented

2019 ◽  
Vol 12 (03) ◽  
pp. 1950039
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Orthomodular Lattice ◽  
Commutative Rings ◽  
Sufficient Condition ◽  
Ideal Lattice ◽  
Ideal Lattices ◽  
The Ideal

We characterize those commutative rings [Formula: see text] whose ideal lattice [Formula: see text] endowed with the annihilation operation is an ortholattice. Moreover, we provide an analogous characterization for the annihilator lattice [Formula: see text] endowed with the annihilation operation. Since the ideal lattice of [Formula: see text] is modular, [Formula: see text] is already an orthomodular lattice provided it is an ortholattice. However, there exist also commutative rings whose ideal lattices are complemented but the complementation differs from annihilation. We present an example of such a ring and develop a procedure producing infinitely many rings with this property. Finally, we provide a sufficient condition for double annihilation to be a homomorphism from [Formula: see text] onto [Formula: see text].

scholarly journals The ideal lattice of an MS-algebra

1988 ◽  
Vol 30 (2) ◽  
pp. 137-143 ◽  
Author(s):  
T. S. Blyth ◽  
J. C. Varlet
Keyword(s):  
Prime Ideal ◽  
Distributive Lattice ◽  
Unary Operation ◽  
Stone Algebra ◽  
Ideal Lattice ◽  
Prime Filter ◽  
De Morgan Algebra ◽  
De Morgan Algebras ◽  
The Ideal

Recently we introduced the notion of an MS-algebra as a common abstraction of a de Morgan algebra and a Stone algebra [2]. Precisely, an MS-algebra is an algebra 〈L; ∧, ∨ ∘, 0, 1〉 of type 〈2, 2, 1, 0, 0〉 such that 〈L; ∧, ∨, 0, 1〉 is a distributive lattice with least element 0 and greatest element 1, and x → x∘ is a unary operation such that x ≤ x∘∘, (x ∧ y)∘ = x∘ ∨ y∘ and 1∘ = 0. It follows that ∘ is a dual endomorphism of L and that L∘∘ = {x∘∘ x ∊ L} is a subalgebra of L that is called the skeleton of L and that belongs to M, the class of de Morgan algebras. Clearly, theclass MS of MS-algebras is equational. All the subvarieties of MS were described in [3]. The lattice Λ (MS) of subvarieties of MS has 20 elements (see Fig. 1) and its non-trivial part (we exclude T, the class of one-element algebras) splits into the prime filter generated by M, that is [M, M1], the prime ideal generated by S, that is [B, S], and the interval [K, K2 ∨ K3].

scholarly journals Atomic Matching Across Internal Interfaces

1989 ◽  
Vol 153 ◽  
Author(s):  
Karl L. Merkle
Keyword(s):  
Electron Microscopy ◽  
High Resolution ◽  
Large Fraction ◽  
High Resolution Electron Microscopy ◽  
Low Energy ◽  
Ideal Lattice ◽  
Interatomic Distances ◽  
Resolution Electron ◽  
Atomic Coordination ◽  
The Ideal

AbstractThe atomic structure of internal interfaces in dense-packed systems has been investigated by high-resolution electron microscopy (HREM). Similarities between the atomic relaxations in heterophase Interfaces and certain largeangle grain boundaries have been observed. In both types of interfaces localization of misfit leads to regions of good atomic matching within the interface separated by misfit dislocation-like defects. It appears that, whenever possible, the GB structures assume configurations in which the atomic coordination is not too much different from the ideal lattice. It is suggested that these kinds of relaxations primarily occur whenever the translational periods along the GB are large or when the interatomic distances are incommensurate. Incorporation of low index planes into the GB appears to lead to preferred, i.e. low energy structures, that can be quite dense with good atomic matching across a large fraction of the interface.

scholarly journals Lattices whose ideal lattice is Stone

1983 ◽  
Vol 26 (1) ◽  
pp. 107-112 ◽  
Author(s):  
R. Beazer
Keyword(s):  
Distributive Lattice ◽  
Dual Space ◽  
Sufficient Conditions ◽  
Distributive Lattices ◽  
Stone Algebra ◽  
Ideal Lattice ◽  
Topological Duality ◽  
Bounded Distributive Lattice ◽  
Image Position ◽  
The Ideal

An elementary fact about ideal lattices of bounded distributive lattices is that they belong to the equational class ℬω of all distributive p-algebras (distributive lattices with pseudocomplementation). The lattice of equational subclasses of ℬω is known to be a chainof type (ω+l, where ℬ0 is the class of Boolean algebras and ℬ1 is the class of Stone algebras. G. Grätzer in his book [7] asks after a characterisation of those bounded distributive lattices whose ideal lattice belongs to ℬ (n≧1). The answer to the problem for the case n = 0 is well known: the ideal lattice of a bounded lattice L is Boolean if and only if L is a finite Boolean algebra. D. Thomas [10] recently solved the problem for the case n = 1 utilising the order-topological duality theory for bounded distributive lattices and in [5] W. Bowen obtained another proof of Thomas's result via a construction of the dual space of the ideal lattice of a bounded distributive lattice from its dual space. In this paper we give a short, purely algebraic proof of Thomas's result and deduce from it necessary and sufficient conditions for the ideal lattice of a bounded distributive lattice to be a relative Stone algebra.

A Characterization of Sharply Transferable Lattices

1980 ◽  
Vol 32 (1) ◽  
pp. 145-154 ◽  
Author(s):  
G. Grätzer ◽  
C. R. Platt
Keyword(s):  
Ideal Lattice ◽  
Infinite Case ◽  
The Ideal

A lattice L is called transferable if and only if, whenever L can be embedded in the ideal lattice I(K) of a lattice K, L can be embedded in K. L is called sharply transferable if and only if, for every lattice embedding ψ(x) , there exists an embedding such that for x, y ϵ L, if and only if x ≤ y. Finite sharply transferable lattices were characterized in [3]. In this paper we extend the characterization to the infinite case. We begin by revising some of the terminology of [3].1.1. Definition, (a) Let 〈P; ≧〉 be a poset and X, Y ⊆ P. Then X dominates Y (written X Dom Y) if and only if, for every y ∈ F, there exists x ∈ X such that y ∈ x. Dually, X supports Y (written X Spp Y) if and only if, for every y ∈ Y, there exists x ∈ X such that x ≦ y.

scholarly journals On-line computations of the ideal lattice of posets

1995 ◽  
Vol 29 (3) ◽  
pp. 227-244 ◽  
Author(s):  
Claude Jard ◽  
Guy-Vincent Jourdan ◽  
Jean-Xavier Rampon
Keyword(s):  
Ideal Lattice ◽  
On Line ◽  
The Ideal

High Resolution Tem Applied to Nanoscale Structure Studies

1994 ◽  
Vol 332 ◽  
Author(s):  
L. Beltran Del Rio ◽  
M.Jose Yacaman ◽  
S. Tehuacanero ◽  
A. Gomez
Keyword(s):  
Crystal Structure ◽  
High Resolution ◽  
General Trend ◽  
Ideal Lattice ◽  
Metallic Particles ◽  
Image Processing Techniques ◽  
High Resolution Tem ◽  
Nanoscale Structure ◽  
Processing Techniques ◽  
The Ideal

ABSTRACTIn this work, digital image processing techniques are used to study the structure of small metallic particles imaged under high resolution conditions. An algorithm is devised to extract directly from the micrographs the coordinates of columns of atoms in such a way that the crystal structure of the particles and their boundaries can be determined. For distorted regions (such as grain boundaries) the actual positions can be compared to the ones in the ideal lattice so that a general trend of the distortion field can be elucidated.

Lattice Statics of a Hydrogen Centre in Magnesium

1985 ◽  
Vol 40 (5) ◽  
pp. 439-449
Author(s):  
B. Bratschek ◽  
B. Rager ◽  
H.-J. Volkert
Keyword(s):  
Hydrogen Atom ◽  
Hexagonal Lattice ◽  
Harmonic Approximation ◽  
Nearest Neighbors ◽  
Interstitial Site ◽  
Force Constants ◽  
Ideal Lattice ◽  
Lattice Statics ◽  
Tetrahedral Interstitial Site ◽  
The Ideal

In this work we consider the static perturbation of the hexagonal lattice of magnesium by one stored hydrogen atom. Equations are given to evaluate the equilibrium positions of the ions in the disturbed crystal. The effect of the hydrogen atom is described by a so-called Kanzaki force, an additional force in the ideal lattice. The force between two ions of the ideal lattice is assumed to obey the harmonic approximation. The force constants are calculated form an effective ion-ion potential. The static Green tensor, which is the inverse of the tensor of force constants, is calculated and with its aid the equations of lattice statics are transformed. The transformed equations are solved under the assumption of a Kanzaki force acting only upon the nearest and next-nearest neighbors of the hydrogen atom at a tetrahedral interstitial site. Finally we calculate the volume change Δ V and compare the result with the experimental value.

scholarly journals On annihilators in BL-algebras

2016 ◽  
Vol 14 (1) ◽  
pp. 324-337 ◽  
Author(s):  
Yu Xi Zou ◽  
Xiao Long Xin ◽  
Peng Fei He
Keyword(s):  
Boolean Algebra ◽  
Nonempty Subset ◽  
Sufficient Condition ◽  
Ideal Lattice ◽  
Necessary And Sufficient Condition ◽  
Finite Product ◽  
Brouwerian Lattice ◽  
Necessary And Sufficient ◽  
Complemented Lattice ◽  
The Ideal

AbstractIn the paper, we introduce the notion of annihilators in BL-algebras and investigate some related properties of them. We get that the ideal lattice (I(L), ⊆) is pseudo-complemented, and for any ideal I, its pseudo-complement is the annihilator I⊥ of I. Also, we define the An (L) to be the set of all annihilators of L, then we have that (An(L); ⋂,∧An(L),⊥,{0}, L) is a Boolean algebra. In addition, we introduce the annihilators of a nonempty subset X of L with respect to an ideal I and study some properties of them. As an application, we show that if I and J are ideals in a BL-algebra L, then $J_I^ \bot $ is the relative pseudo-complement of J with respect to I in the ideal lattice (I(L), ⊆). Moreover, we get some properties of the homomorphism image of annihilators, and also give the necessary and sufficient condition of the homomorphism image and the homomorphism pre-image of an annihilator to be an annihilator. Finally, we introduce the notion of α-ideal and give a notation E(I ). We show that (E(I(L)), ∧E, ∨E, E(0), E(L) is a pseudo-complemented lattice, a complete Brouwerian lattice and an algebraic lattice, when L is a BL-chain or a finite product of BL-chains.

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