Lattice Of Subalgebras Research Articles

Modularity conditions in Leibniz algebras

In this paper, we continue the study of the subalgebra lattice of a Leibniz algebra. In particular, we find out that solvable Leibniz algebras with an upper semi-modular lattice are either almost-abelian or have an abelian ideal spanned by the elements with square zero. We also study Leibniz algebras in which every subalgebra is a weak quasi-ideal, as well as modular symmetric Leibniz algebras.

The Subalgebra Lattice of A Finite Diagonal–Free Two–Dimensional Cylindric Algebra

Diagonal–free two–dimensional cylindric algebras (Df2−algebras for short) are Boolean algebras enriched with two existential quantifiers which commute. Df2−algebras were introduced by A. Tarski, L. Chin and F. Thompson with the purpose of providing an algebraic device for the study of the first–order predicate calculus with two variables. This work is devoted to problems related to finite Df2−algebras. More precisely, we study and describe the family of subalgebras of a given finite Df2−algebra. Then, identifying the algebras of this family which are isomorphic, we provide a full description of the lattice of all non–isomorphic subalgebras of a given finite Df2−algebra.

On the subalgebra lattice of a restricted Lie algebra

In this paper we study the lattice of restricted subalgebras of a restricted Lie algebra. In particular, we consider those algebras in which this lattice is dually atomistic, lower or upper semimodular, or in which every restricted subalgebra is a quasi-ideal. The fact that there are one-dimensional subalgebras which are not restricted results in some of these conditions being weaker than for the corresponding conditions in the non-restricted case.

Lattice of intermediate subalgebras

Analogous to subfactor theory, employing Watatani's notions of index and C ∗ -basic construction of certain inclusions of C ∗ -algebras, (a) we develop a Fourier theory (consisting of Fourier transforms, rotation maps and shift operators) on the relative commutants of any inclusion of simple unital C ∗ -algebras with finite Watatani index, and (b) we introduce the notions of interior and exterior angles between intermediate C ∗ -subalgebras of any inclusion of unital C ∗ -algebras admitting a finite index conditional expectation. Then, on the lines of Bakshi et al. (Trans. Amer. Math. Soc. 371 (2019) 5973–5991), we apply these concepts to obtain a bound for the cardinality of the lattice of intermediate C ∗ -subalgebras of any irreducible inclusion as in (a), and improve Longo's bound for the cardinality of intermediate subfactors of an irreducible inclusion of type I I I factors with finite index. Moreover, we also show that for a fairly large class of inclusions of finite von Neumann algebras, the lattice of intermediate von Neumann subalgebras is always finite.

On the subalgebra lattice of a Leibniz algebra

In this paper, we begin to study the subalgebra lattice of a Leibniz algebra. In particular, we deal with Leibniz algebras whose subalgebra lattice is modular, upper semi-modular, lower semi-modular, distributive, or dually atomistic. The fact that a non-Lie Leibniz algebra has fewer one-dimensional subalgebras in general results in a number of lattice conditions being weaker than in the Lie case.

On Rees closure in some classes of algebras with an operator

In this paper we introduce the concept of Rees closure for subalgebras of universal algebras. We denote by △𝐴 the identity relation on 𝐴. A subalgebra 𝐵 of algebra 𝐴 is called a Rees subalgebra whenever 𝐵2 ∪ △𝐴 is a congruence on 𝐴. A congruence 𝜃 of algebra 𝐴 is called a Rees congruence if 𝜃 = 𝐵2 ∪△𝐴 for some subalgebra 𝐵 of 𝐴. We define a Rees closure operator by mapping arbitrary subalgebra 𝐵 of algebra 𝐴 into the smallest Rees subalgebra that contains 𝐵. It is shown that in the general case the Rees closure does not commute with the operation ∧ on the lattice of subalgebras of universal algebra. Consequently, in the general case, a lattice of Rees subalgebras is not a sublattice of lattice of subalgebras. A non-one-element universal algebra 𝐴 is called a Rees simple algebra if any Rees congruence on 𝐴 is trivial. We characterize Rees simple algebras in terms of Rees closure. Universal algebra is called an algebra with operators if it has an additional set of unary operations acting as endomorphisms with respect to basic operations. We described Rees simple algebras in some subclasses of the class of algebras with one operator and a ternary basic operation. For algebras from these classes, the structure of lattice of Rees subalgebras is described. Necessary and sufficient conditions for the lattice of Rees subalgebras of algebras from these classes to be a chain are obtained.

Associative algebras with a distributive lattice of subalgebras

Associative algebras with a distributive lattice of subalgebras

Associative Algebras with a Distributive Lattice of Subalgebras

We give a full description of associative algebras over an arbitrary field, whose subalgebra lattice is distributive. All such algebras are commutative, their nil-radical is at most two-dimensional, and the factor algebra with respect to the nil-radical is an algebraic extension of the base field.

Turing Degrees and Automorphism Groups of Substructure Lattices

The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the noncomputable countable structures, computably enumerable structures are one of the most important objects of investigation in computable model theory. Here we focus on the lattice structure of computably enumerable substructures of a given canonical computable structure. In particular, for a Turing degree d, we investigate the groups of d-computable automorphisms of the lattice of d-computably enumerable vector spaces, of the interval Boolean algebra Bη of the ordered set of rationals, and of the lattice of d-computably enumerable subalgebras of Bη. For these groups, we show that Turing reducibility can be used to substitute the group-theoretic embedding. We also prove that the Turing degree of the isomorphism types for these groups is the second Turing jump d′′ of d.

Restricted Lie algebras having a distributive lattice of restricted subalgebras

Let L be a restricted Lie algebra over a field of characteristic p>0. We investigate the structure of L when its lattice of restricted subalgebras satisfies some prescribed properties. In particular, we establish when is distributive. The special form that this result takes when is either the field with p elements or an algebraically closed field is also discussed. Furthermore, we establish when is Boolean or the two-elements lattice.

A Residuated Lattice of L-Fuzzy Subalgebras of a Mono-Unary Algebra

Given a complete residuated lattice [Formula: see text] and a mono-unary algebra [Formula: see text], it is well known that [Formula: see text] and the residuated lattice [Formula: see text] of [Formula: see text]-fuzzy subsets of [Formula: see text] satisfy the same residuated lattice identities. In this paper, we show that [Formula: see text] and the residuated lattice [Formula: see text] of [Formula: see text]-fuzzy subalgebras of [Formula: see text] satisfy the same residuated lattice identities if and only if the Heyting algebra [Formula: see text] of subuniverses of [Formula: see text] is a Boolean algebra. We also show that [Formula: see text] is a Boolean algebra (respectively, an [Formula: see text]-algebra) if and only if [Formula: see text] is a Boolean algebra (respectively, an [Formula: see text]-algebra) and [Formula: see text] is a Boolean algebra.

Isomorphisms of Lattices of Subalgebras of Semifields of Positive Continuous Functions

We consider the lattice of subalgebras of a semifield U(X) of positive continuous functions on an arbitrary topological space X and its sublattice of subalgebras with unity. We prove that each isomorphism of the lattices of subalgebras with unity of semifields U(X) and U(Y) is induced by a unique isomorphism of the semifields. The same result holds for lattices of all subalgebras excluding the case of the double-point Tychonoff extension of spaces.

Subalgebra lattices of totally reflexive sub-preprimal algebras

For a set Q of relations on a finite set A, the set $$\mathrm{Pol}_{A}Q$$ , of all operations on A preserving all relations in Q, is a clone. The set of all clones on a given set forms a lattice under inclusion. Each of its maximal elements can be represented as $$\mathrm{Pol}_{A}\{\rho \}$$ where $$\rho $$ is one of the six types of relations determined by Ivo G. Rosenberg. Four types of them are totally reflexive. These relations are bounded orders, non-trivial equivalence relations, central relations, and universal relations. An algebra $$\underline{A}$$ is said to be totally reflexive sub-preprimal if its clone of term operations is where $$\rho _{1}$$ and $$\rho _{2}$$ are totally reflexive relations. We describe the subalgebra lattices of such algebras.

Definability of semifields of continuous positive functions by the lattices of their subalgebras

We consider the lattice of subalgebras of a semifield of continuous positive functions on an arbitrary topological space and its sublattice of subalgebras with unity. The main result of the paper is the proof of the definability of any semifield both by the lattice and by its sublattice . Bibliography: 12 titles.

The Conrad Program: From ℓ-groups to algebras of logic

A number of research articles have established the significant role of lattice-ordered groups (ℓ-groups) in logic. The fact that underpins these studies is the realization that important algebras of logic may be viewed as ℓ-groups with a modal operator. These connections are just the tip of the iceberg. The purpose of the present article is to lay the groundwork for, and provide significant initial contributions to, the development of a Conrad type approach to the study of algebras of logic. The term Conrad Program refers to Paul Conrad's approach to the study of ℓ-groups, which analyzes the structure of individual or classes of ℓ-groups by primarily using strictly lattice theoretic properties of their lattices of convex ℓ-subgroups. The present article demonstrates that large parts of the Conrad Program can be profitably extended in the setting of e-cyclic residuated lattices – that is residuated lattices that satisfy the identity x\\e≈e/x. An indirect benefit of this work is the introduction of new tools and techniques in the study of algebras of logic, and the enhanced role of the lattice of convex subalgebras of a residuated lattice.

Hereditary [formula omitted]-subalgebra lattices

We investigate the connections between order and algebra in the hereditary C*-subalgebra lattice H(A) and *-annihilator ortholattice P(A)⊥. In particular, we characterize ∨-distributive elements of H(A) as ideals, answering a 25 year old question, allowing the quantale structure of H(A) to be completely determined from its lattice structure. We also show that P(A)⊥ is separative, allowing for C*-algebra type decompositions which are completely consistent with the original von Neumann algebra type decompositions.

Joins of subalgebras and normals in 0-regular varieties

In any 0-normal variety (0-regular variety in which {0} is a subalgebra), every congruence class containing 0 is a subalgebra. These “normal subalgebras” of a fixed algebra constitute a lattice, isomorphic to its congruence lattice. We are interested in those 0-normal varieties for which the join of two normal subalgebras in the lattice of normal subalgebras of an algebra equals their join in the lattice of subalgebras, as happens with groups and rings. We characterise this property in terms of a Mal’cev condition, and use examples to show it is strictly stronger than being ideal determined but strictly weaker than being 0-coherent (classically ideal determined) and does not imply congruence permutability.

The subalgebra lattice of a finite algebra

Abstract The aim of this paper is to characterize pairs (L, A), where L is a finite lattice and A a finite algebra, such that the subalgebra lattice of A is isomorphic to L. Next, necessary and sufficient conditions are found for pairs of finite algebras (of possibly distinct types) to have isomorphic subalgebra lattices. Both of these characterizations are particularly simple in the case of distributive subalgebra lattices. We do not restrict our attention to total algebras only, but we consider the more general case of partial algebras. Moreover, we use connections between algebras and hypergraphs to solve these problems.

A STRONG PROPERTY OF THE WEAK SUBALGEBRA LATTICE FOR LOCALLY FINITE ALGEBRAS OF FINITE TYPE

The aim of this paper is to show that the weak subalgebra lattice uniquely determines the subalgebra lattice for locally finite algebras of a fixed finite type. However, this algebraic result turns out to be a very particular case of the following hypergraph result (which is interesting itself): A total directed hypergraph D of finite type is uniquely determined, in the class of all the directed hypergraphs of this type, by its skeleton up to the orientation of some pairwise edge-disjoint directed hypercycles and hyperpaths. The skeleton of D is a hypergraph obtained from D by omitting the orientation of all edges.

Finite-dimensional simple lie algebras with subalgebra lattice of length 3

Lie algebras with subalgebra lattice of length 2 are well known. For studying the subalgebra lattice of a greater length one needs some information on Lie algebras with subalgebra lattice of length 3. We show that there exist four types of finite-dimensional simple Lie algebras over a field of characteristic 0 or a perfect field of prime characteristic greater than 5 such that the length of their subalgebra lattices equals 3.