Lattices of Radicals of Ω-Groups

2006 ◽  
Vol 13 (03) ◽  
pp. 381-404
Author(s):  
G. L. Booth ◽  
Q. N. Petersen ◽  
S. Veldsman
Keyword(s):  
Complete Lattice ◽  
Modular Lattice ◽  
Universal Class ◽  
Atomic Lattice ◽  
Universal Classes ◽  
Definition Of ◽  
Associative Rings

Snider initiated the study of lattices of the class of radicals, in the sense of Kurosh and Amitsur, of associative rings. Various authors continued the investigation in more general universal classes. Recently, Fernández-Alonso et al. studied the lattice of all preradicals in R-Mod. Our definition of a preradical is weaker than theirs. In this paper, we consider the lattices of ideal maps 𝕀, preradical maps ℙ, Hoehnke radical maps ℍ and Plotkin radical maps 𝔹 in any universal class of Ω-groups (of the same type). We show that 𝕀 is a complete and modular lattice which contains atoms. In general, 𝕀 is not atomic. 𝕀 contains ℙ as a complete and atomic sublattice, whereas ℍ and 𝔹 are not sublattices of 𝕀. In its own right, ℍ is a complete and atomic lattice and 𝔹 is a complete lattice. We identify subclasses of 𝕀, ℙ and ℍ that are sublattices or preserve the meet (or join) of these respective lattices.

On Radical Congruence Systems II

1998 ◽  
Vol 08 (03) ◽  
pp. 363-397
Author(s):  
T. E. Hall ◽  
Shuhua Zhang
Keyword(s):  
Universal Class ◽  
Universal Classes

This paper is a continuation of a paper of the same title by the first author and P. Weil. We first characterize the universal class of a radical congruence system. We then introduce the meet and the (limit) iteration of congruence systems. This enables us to generate new radical congruence systems from given congruence systems. Some interesting examples are presented. We finally determine the smallest radical congruence systems whose universal classes are N, LZ ◦ N, RZ ◦ N, and RB ◦ N respectively.

scholarly journals The special linear group for nonassociative rings

2020 ◽  
Vol 23 (2) ◽  
pp. 327-335
Author(s):  
Harry Petyt
Keyword(s):  
Linear Group ◽  
Special Linear Group ◽  
Large Classes ◽  
Special Linear ◽  
Lie Ring ◽  
Definition Of ◽  
Associative Rings

AbstractWe extend to arbitrary rings a definition of the octonion special linear group due to Baez. At the infinitesimal level, we get a Lie ring, which we describe over some large classes of rings, including all associative rings and all algebras over a field. As a corollary, we compute all the groups Baez defined.

Zariski-like topologies for lattices with applications to modules over associative rings

2019 ◽  
Vol 18 (07) ◽  
pp. 1950131
Author(s):  
Jawad Abuhlail ◽  
Hamza Hroub
Keyword(s):  
Complete Lattice ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Proper Class ◽  
Topological Properties ◽  
Left Module ◽  
Separation Axioms ◽  
Prime Submodules ◽  
Algebraic Properties ◽  
Associative Rings

We study Zariski-like topologies on a proper class [Formula: see text] of a complete lattice [Formula: see text]. We consider [Formula: see text] with the so-called classical Zariski topology [Formula: see text] and study its topological properties (e.g. the separation axioms, the connectedness, the compactness) and provide sufficient conditions for it to be spectral. We say that [Formula: see text] is [Formula: see text]-top if [Formula: see text] is a topology. We study the interplay between the algebraic properties of an [Formula: see text]-top complete lattice [Formula: see text] and the topological properties of [Formula: see text] Our results are applied to several spectra which are proper classes of [Formula: see text] where [Formula: see text] is a nonzero left module over an arbitrary associative ring [Formula: see text] (e.g. the spectra of prime, coprime, fully prime submodules) of [Formula: see text] as well as to several spectra of the dual complete lattice [Formula: see text] (e.g. the spectra of first, second and fully coprime submodules of [Formula: see text]).

ON BASE RADICAL OPERATORS FOR CLASSES OF FINITE ASSOCIATIVE RINGS

2018 ◽  
Vol 98 (2) ◽  
pp. 239-250 ◽  
Author(s):  
R. G. MCDOUGALL ◽  
L. K. THORNTON
Keyword(s):  
Maximum Order ◽  
Semisimple Class ◽  
Hereditary Radical ◽  
Complete Listing ◽  
Universal Classes ◽  
General Relations ◽  
Associative Rings

In this paper, class operators are used to give a complete listing of distinct base radical and semisimple classes for universal classes of finite associative rings. General relations between operators reveal that the maximum order of the semigroup formed is 46. In this setting, the homomorphically closed semisimple classes are precisely the hereditary radical classes and hence radical–semisimple classes, and the largest homomorphically closed subclass of a semisimple class is a radical–semisimple class.

On the interpretation of Aristotelian syllogistic

1956 ◽  
Vol 21 (2) ◽  
pp. 137-147 ◽  
Author(s):  
J. C. Shepherdson
Keyword(s):  
Embedding Theorem ◽  
Equivalence Classes ◽  
Universal Class ◽  
Preliminary Step ◽  
Prove Theorem ◽  
Decidable Theory ◽  
Universal Classes

The main purpose of this note is to prove (theorem 11, § 5) that, in any interpretation of the formalisation of Aristotelian syllogistic given by Łukasiewicz [4], it is always possible to associate with each element a a non-null sub-class φ(a) of some ‘universal’ class V in such a way that ‘Aab’ (all a are b), ‘Iab’ (some a are b) are equivalent respectively to ‘φ(a) is contained in φ(b)’, ‘φ(a) has a non-null intersection with φ(b)’. Similarly (theorem 6, §4) we show that in Wedberg's system [14] with primitives ‘Aab’, ‘a’ (not a), it is possible to find a mapping a → φ(a) as above such that ‘Aab’ is equivalent to ‘φ(a) is contained in φ(b)’ and φ(a‘ is equal to φ(a)’, the complement of φ(a) with respect to V. Thus, if we make the preliminary step of identifying elements a, b such that Aab and Aba both hold (i.e. taking equivalence classes with respect to the relation Aab & Aba), we are left with essentially only one kind of interpretation for these systems, namely the ‘normal’ interpretation by classes. Slupecki [11], [12] has proved that Łukasiewicz's system is a complete and decidable theory of the relations of inclusion and intersection of non-null classes, and Wedberg [14] has proved that his system is a complete and decidable theory of the relation of inclusion and the operation of complementation for nonnull, non-universal classes. Using the above-mentioned embedding theorem, we are able to obtain (theorems 9, 6, §§ 5, 4) very simple proofs of these results.

scholarly journals Characterization of Birkhoff’s Conditions by Means of Cover-Preserving and Partially Cover-Preserving Sublattices

2016 ◽  
Vol 45 (3/4) ◽  
Author(s):  
Marcin Łazarz
Keyword(s):  
Modular Lattice ◽  
Continuous Lattice ◽  
Atomic Lattice ◽  
Discrete Lattices ◽  
Upper Continuous Lattice ◽  
Upper Continuous ◽  
Strongly Atomic Lattice ◽  
Atomic Lattices

In the paper we investigate Birkhoff’s conditions (Bi) and (Bi*). We prove that a discrete lattice L satisfies the condition (Bi) (the condition (Bi*)) if and only if L is a 4-cell lattice not containing a cover-preserving sublattice isomorphic to the lattice S*7 (the lattice S7). As a corollary we obtain a well known result of J. Jakub´ık from [6]. Furthermore, lattices S7 and S*7 are considered as so-called partially cover-preserving sublattices of a given lattice L, S7 ≪ L and S7 ≪ L, in symbols. It is shown that an upper continuous lattice L satisfies (Bi*) if and only if L is a 4-cell lattice such that S7 ≪/ L. The final corollary is a generalization of Jakubík’s theorem for upper continuous and strongly atomic lattices. Keywords: Birkhoff’s conditions, semimodularity conditions, modular lattice, discrete lattices, upper continuous lattice, strongly atomic lattice, cover-preserving sublattice, cell, 4-cell lattice.  

scholarly journals ON COMPLETE CONGRUENCE LATTICES OF COMPLETE MODULAR LATTICES

1991 ◽  
Vol 01 (02) ◽  
pp. 147-160 ◽  
Author(s):  
R. FREESE ◽  
G. GRÄTZE ◽  
E. T. SCHMIDT
Keyword(s):  
Complete Lattice ◽  
Modular Lattice ◽  
Modular Lattices ◽  
Congruence Lattices ◽  
Congruence Relations

The lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In 1988, the second author announced the converse: every complete lattice L can be represented as the lattice of complete congruence relations of some complete lattice K. In this paper we improve this result by showing that K can be chosen to be a complete modular lattice.

scholarly journals On coatoms of the lattice of matric-extensible radicals

2005 ◽  
Vol 72 (3) ◽  
pp. 403-406
Author(s):  
Halina France-Jackson
Keyword(s):  
Matrix Ring ◽  
Natural Numbers ◽  
Universal Class ◽  
Maximal Elements ◽  
Associative Rings

A radical α in the universal class of all associative rings is called matric-extensible if for all natural numbers n and all rings A, A ∈ α if and only if Mn(A) ∈ α, where Mn(A) denotes the n × n matrix ring with entries from A. We show that there are no coatoms, that is, maximal elements in the lattice of all matric-extensible radicals of associative rings.

Some Examples of Complemented Modular Lattices

1962 ◽  
Vol 5 (2) ◽  
pp. 111-121 ◽  
Author(s):  
G. Grätzer ◽  
Maria J. Wonenburger
Keyword(s):  
Boolean Algebra ◽  
Modular Lattice ◽  
Complete Boolean Algebra ◽  
Modular Lattices ◽  
Additive Property ◽  
Von Neumann ◽  
Continuous Geometry ◽  
Definition Of ◽  
Homogeneous Basis

Let L be a complemented, χ-complete modular lattice. A theorem of Amemiya and Halperin (see [l], Theorem 4.3) asserts that if the intervals [O, a] and [O, b], a, bεL, are upper χ-continuous then [O, a∪b] is also upper χ-continuous. Roughly speaking, in L upper χ-continuity is additive. The following question arises naturally: is χ-completeness an additive property of complemented modular lattices? It follows from Corollary 1 to Theorem 1 below that the answer to this question is in the negative.A complemented modular lattice is called a Von Neumann geometry or continuous geometry if it is complete and continuous. In particular a complete Boolean algebra is a Von Neumann geometry. In any case in a Von Neumann geometry the set of elements which possess a unique complement form a complete Boolean algebra. This Boolean algebra is called the centre of the Von Neumann geometry. Theorem 2 shows that any complete Boolean algebra can be the centre of a Von Neumann geometry with a homogeneous basis of order n (see [3] Part II, definition 3.2 for the definition of a homogeneous basis), n being any fixed natural integer.

On the lattice of biorder ideals of regular rings

2019 ◽  
Vol 13 (06) ◽  
pp. 2050103
Author(s):  
R. Akhila ◽  
P. G. Romeo
Keyword(s):  
Regular Semigroup ◽  
Significant Role ◽  
Modular Lattice ◽  
Regular Ring ◽  
Biordered Set ◽  
Definition Of ◽  
Regular Rings

The study of biordered set plays a significant role in describing the structure of a regular semigroup and since the definition of regularity involves only the multiplication in the ring, it is natural that the study of semigroups plays a significant role in the study of regular rings. Here, we extend the biordered set approach to study the structure of the regular semigroup [Formula: see text] of a regular ring [Formula: see text] by studying the idempotents [Formula: see text] of the regular ring and show that the principal biorder ideals of the regular ring [Formula: see text] form a complemented modular lattice and certain properties of this lattice are studied.

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