Varieties of Lattices with Geometric Descriptions

A lattice L is spatial if every element of L is a join of completely join-irreducible elements of L (points), and strongly spatial if it is spatial and the minimal coverings of completely join-irreducible elements are well-behaved. Herrmann et al. proved in 1994 that every modular lattice can be embedded, within its variety, into an algebraic and spatial lattice. We extend this result to n-distributive lattices, for fixed n. We deduce that the variety of all n-distributive lattices is generated by its finite members, thus it has a decidable word problem for free lattices. This solves two problems stated by Huhn in 1985. We prove that every modular (resp., n-distributive) lattice embeds within its variety into some strongly spatial lattice. Every lattice which is either algebraic modular spatial or bi-algebraic is strongly spatial. We also construct a lattice that cannot be embedded, within its variety, into any algebraic and spatial lattice. This lattice has a least and a largest element, and it generates a locally finite variety of join-semidistributive lattices.     

A note on zip rings

A ring R is called right zip if the right annihilator r _R (S) of a subset S of R is zero, r _R (X)=0 for a finite subset X of S. In this note we will prove that for any u.p.-monoid M a right uniform ring R is right zip if and only if the monoid ring R[M] is right zip.     

Idempotent elements of the semigroup corresponding to an X-semilattice of unions of the class Σ16(X, 6)

It is known that if we know all XI-subsemilattices of a given X-semilattice of unions, then we can determine all idempotent elements of the semigroup, and the structure of idempotent elements is characterized. In this work, we find idempotent elements of the semigroup corresponding to X-semilattices of unions of the class ??₁₆(X, 6). Moreover, we give formulas for the number of idempotent elements, where X is finite.     

On some chains of fuzzy sets

A partial order relation σ is defined in the set $\text{F}$(X) of the fuzzy sets in X. If this ordering is induced in the subset F(X) of the measurable fuzzy sets in the set X with totally finite positive measure, then fσg implies that the entropy of the fuzyy set f is not less than the entropyof g. By means of this ordering a lattice L on $\text{F}$(X) is defined and a lattice structure is induced in the set of infinite chains in L. Furthermore the set F′(X) of the fuzzy sets of F(X) which assume value in a finite subset of the real interval [0,1] is considered and the following properties are stated: any chain of elements of F′(X) is an infinite sequence of functions convergent in the mean to an integrable function, and the entropy is a valuation of bounded variation on the sublattice of L whose elements are in F′(X). The chains on L can offer a model of a cognitive process in a fuzzy environment when their elements are determined by a sequence of decisions. The limit property traduces the determinism of a such procedure.     

Slim Semimodular Lattices. I. A Visual Approach

A finite lattice L is called slim if no three join-irreducible elements of L form an antichain. Slim lattices are planar. After exploring some elementary properties of slim lattices and slim semimodular lattices, we give two visual structure theorems for slim semimodular lattices.     

On some subsets defined with respect to weak structures

We define and study the properties of some subsets of X with respect to a weak structure on X and generalize some already established results.     

Pointwise bornological spaces

With each metric space (X,d) we can associate a bornological space (X,Bd) where Bd is the set of all subsets of X with finite diameter. Equivalently, Bd is the set of all subsets of X that are contained in a ball with finite radius. If the metric d can attain the value infinite, then the set of all subsets with finite diameter is no longer a bornology. Moreover, if d is no longer symmetric, then the set of subsets with finite diameter does not coincide with the set of subsets that are contained in a ball with finite radius. In this text we will introduce two structures that capture the concept of boundedness in both symmetric and non-symmetric extended metric spaces.     

An extremal problem for finite topologies and distributive lattices

Let (r₁, r₂, …) be a sequence of non-negative integers summing to n. We determine under what conditions there exists a finite distributive lattice L of rank n with r_i join-irreducibles of rank i, for all i = 1, 2, …. When L exists, we give explicit expressions for the greatest number of elements L can have of any given rank, and for the greatest total number of elements L can have. The problem is also formulated in terms of finite topological spaces.     

Almost Primitive Elements of Free lie Algebras of Small Ranks

Let K be a field, X = {x1, . . . , xn}, and let L(X) be the free Lie algebra over K with the set X of free generators. A. G. Kurosh proved that subalgebras of free nonassociative algebras are free, A. I. Shirshov proved that subalgebras of free Lie algebras are free. A subset M of nonzero elements of the free Lie algebra L(X) is said to be primitive if there is a set Y of free generators of L(X), L(X) = L(Y ), such that M ? Y (in this case we have |Y | = |X| = n). Matrix criteria for a subset of elements of free Lie algebras to be primitive and algorithms to construct complements of primitive subsets of elements with respect to sets of free generators have been constructed. A nonzero element u of the free Lie algebra L(X) is said to be almost primitive if u is not a primitive element of the algebra L(X), but u is a primitive element of any proper subalgebra of L(X) that contains it. A series of almost primitive elements of free Lie algebras has been constructed. In this paper, for free Lie algebras of rank 2 criteria for homogeneous elements to be almost primitive are obtained and algorithms to recognize homogeneous almost primitive elements are constructed.     

Comparability graphs of lattices

A theorem of N. Terai and T. Hibi for finite distributive lattices and a theorem of Hibi for finite modular lattices (suggested by R.P. Stanley) are equivalent to the following: if a finite distributive or modular lattice of rank d contains a complemented rank 3 interval, then the lattice is (d+1)-connected.In this paper, the following generalization is proved: Let L be a (finite or infinite) semimodular lattice of rank d that is not a chain (d∈N₀). Then the comparability graph of L is (d+1)-connected if and only if L has no simplicial elements, where z∈L is simplicial if the elements comparable to z form a chain.     

Monoidal intervals of clones on infinite sets

Let X be an infinite set of cardinality κ. We show that if L is an algebraic and dually algebraic distributive lattice with at most 2^κ completely join irreducibles, then there exists a monoidal interval in the clone lattice on X which is isomorphic to the lattice 1+L obtained by adding a new smallest element to L. In particular, we find that if L is any chain which is an algebraic lattice, and if L does not have more than 2^κ completely join irreducibles, then 1+L appears as a monoidal interval; also, if λ?2^κ, then the power set of λ with an additional smallest element is a monoidal interval. Concerning cardinalities of monoidal intervals these results imply that there are monoidal intervals of all cardinalities not greater than 2^κ, as well as monoidal intervals of cardinality 2^λ, for all λ?2^κ.     

A counterexample to the generalization of Sperner's theorem

It has been conjectured that the analog of Sperner's theorem on non-comparable subsets of a set holds for arbitrary geometric lattices, namely, that the maximal number of non-comparable elements in a finite geometric lattice is max w(k), where w(k) is the number of elements of rank k. It is shown in this note that the conjecture is not true in general. A class of geometric lattices, each of which is a bond lattice of a finite graph, is constructed in which the conjecture fails to hold.     

The unit graph of a left Artinian ring

Let R be a ring with non-zero identity and U(R) be the group of units of R. The unit graph of R, denoted by G(R), is a graph defined on the elements of R, and two distinct vertices r and s are adjacent if and only if r+s∈U(R). We investigate connectivity, diameter and the girth of the unit graph of a left Artinian ring. Also, by providing an algorithm, we determine when the unit graph of a finite ring is Hamiltonian.     

Yosida frames

A Yosida frame is an algebraic frame in which every compact element is a meet of maximal elements. Yosida frames are used to abstractly characterize the frame of z-ideals of a ring of continuous functions C(X), when X is a compact Hausdorff space. An algebraic frame in which the meet of any two compact elements is compact is Yosida precisely when it is “finitely subfit”; that is, if and only if for each pair of compact elements a<b, there is a z (not necessarily compact) such that a∨z<1=b∨z. This is used to prove that if L is an algebraic frame in which the meet of any two compact elements is compact, and L has disjointification and dim(L)=1, then it is Yosida. It is shown that this result fails with almost any relaxation of the hypotheses. The paper closes with a number of examples, and a characterization of the Bézout domains in which the frame of semiprime ideals is Yosida frame.     

Relative ranks of Lipschitz mappings on countable discrete metric spaces

Let X be a countable discrete metric space and let XX denote the family of all functions on X. In this article, we consider the problem of finding the least cardinality of a subset A of XX such that every element of XX is a finite composition of elements of A and Lipschitz functions on X. It follows from a classical theorem of Sierpiński that such an A either has size at most 2 or is uncountable.We show that if X contains a Cauchy sequence or a sufficiently separated, in some sense, subspace, then |A|≤1. On the other hand, we give several results relating |A| to the cardinal d; defined as the minimum cardinality of a dominating family for NN. In particular, we give a condition on the metric of X under which |A|≥d holds and a further condition that implies |A|≤d. Examples satisfying both of these conditions include all subsets of Nk and the sequence of partial sums of the harmonic series with the usual euclidean metric.To conclude, we show that if X is any countable discrete subset of the real numbers R with the usual euclidean metric, then |A|=1 or almost always, in the sense of Baire category, |A|=d.     

A new look at the Jordan-Hölder theorem for semimodular lattices

We show that in a semimodular lattice L of finite length, from any prime interval we can reach any maximal chain C by an up- and a down-perspectivity. Therefore, C is a congruence-determining sublattice of L.     

The lattice of integer partitions

In this paper we study the lattice L_n of partitions of an integer n ordered by dominance. We show L_n to be isomorphic to an infimum subsemilattice under the component ordering of certain concave nondecreasing (n+1)-tuples. For L_n, we give the covering relation, maximal covering number, minimal chains, infimum and supremum irreducibles, a chain condition, distinguished intervals; and show that partition conjugation is a lattice antiautomorphism. L_n is shown to have no sublattice having five elements and rank two, and we characterize intervals generated by two cocovers. The Möbius function of L_n is computed and shown to be 0,1 or -1. We then give methods for studying classes of (0,1)-matrices with prescribed row and column sums and compute a lower bound for their cardinalities.     

Quasi-antichain Chermak–Delgado lattices of finite groups

The Chermak–Delgado lattice of a finite group is a dual, modular sublattice of the subgroup lattice of the group. This paper considers groups with a quasi-antichain interval in the Chermak–Delgado lattice, ultimately proving that if there is a quasi-antichain interval between subgroups L and H with L ≤ H then there exists a prime p such that H/L is an elementary abelian p-group and the number of atoms in the quasi-antichain is one more than a power of p. In the case where the Chermak–Delgado lattice of the entire group is a quasi-antichain, the relationship between the number of abelian atoms and the prime p is examined; additionally, several examples of groups with a quasi-antichain Chermak–Delgado lattice are constructed.     

Discrete reflexivity and complements of the diagonal

We prove that, for a countably compact space X of weight at most ω ₁, if ${\mathcal {P}}$ ∈{countable tightness, Fréchet–Urysohn property, sequentiality, first countability} and the closure of every discrete subspace of X has ${\mathcal {P}}$ then X has  ${\mathcal {P}}$ . Given a countably compact space X, if X ²?Δ is discretely Lindelöf then X is metrizable. We establish that a Lindelöf Σ-space X is cosmic whenever X ²?Δ is paracompact; this generalizes the respective result of Gruenhage for compact X. Furthermore, a countably compact space X is metrizable if the closure of every discrete subspace of X ² is metrizable. It turns out that many non-discretely reflexive properties behave much better in finite powers of spaces. In particular, if X is compact and $\overline{D}$ is Corson (Eberlein) compact for any discrete D?X ³ then X is a Corson (Eberlein) compact.     

An algorithm for determining Φ(L) in finite distributive lattices

Following G. Birkhoff, the Frattini sublattice Φ(L) of a latticeL is defined as the intersection of all its maximal proper sublattices. Let ?(FD) be the class of all finite distributive lattices. The main aim in this note is to provide a new but elementary characterization of elements in Φ(L),L∈?(FD), and also an extremely simple algorithm for determining the Frattini sublattice of any finite distributive lattice. By applying this algorithm, it is shown that there is a new way to determine the Frattini subalgebra of a finite Stone algebra.