Primary elements in Prüfer lattices

In this paper we study primary elements in Prüfer lattices and characterize -lattices in terms of Prüfer lattices. Next we study weak ZPI-lattices and characterize almost principal element lattices and principal element lattices in terms of ZPI-lattices.     

Weak complemented and weak invertible elements in <Emphasis Type="Italic">C</Emphasis>-lattices

In this paper, we prove that an indecomposable M-lattice is either a principal element domain or a special principal element lattice. Next, we introduce weak complemented elements and characterize reduced M-lattices in terms of weak complemented elements. We also study weak invertible elements and locally weak invertible elements in C-lattices and characterize reduced Prüfer lattices, WI-lattices, reduced almost principal element lattices, and reduced principal element lattices in terms of locally weak invertible elements.     

Laskerian Lattices

In this paper we investigate prime divisors, B _w-primes and zs-primes in C-lattices. Using them some new characterizations are given for compactly packed lattices. Next, we study Noetherian lattices and Laskerian lattices and characterize Laskerian lattices in terms of compactly packed lattices.     

2-Absorbing and Weakly 2-Absorbing Elements in Multiplicative Lattices

In this article we study weakly 2-absorbing elements and 2-absorbing elements in C-lattices. Next we characterize C-lattices in which every nonzero proper element is a (weakly) 2-absorbing element. We also establish a new characterization for principal element domains in terms of 2-absorbing elements.     

An Analogue of Distributivity for Ungraded Lattices

In this paper, we study lattices that posess both the properties of being extremal (in the sense of Markowsky) and of being left modular (in the sense of Blass and Sagan). We call such lattices trim and show that they posess some additional appealing properties, analogous to those of a distributive lattice. For example, trimness is preserved under taking intervals and suitable sublattices. Trim lattices satisfy a weakened form of modularity. The order complex of a trim lattice is contractible or homotopic to a sphere; the latter holds exactly if the maximum element of the lattice is a join of atoms. Any distributive lattice is trim, but trim lattices need not be graded. The main example of ungraded trim lattices are the Tamari lattices and generalizations of them. We show that the Cambrian lattices in types A and B defined by Reading are trim; we conjecture that all Cambrian lattices are trim.     

∗-Prime Group Algebras

An associative algebra R over a field K is said to be right ?-prime if for every nonzero r ? R, there exists a finitely generated subalgebra S of R such that rSt = 0 implies t = 0. Clearly, strongly prime implies ?-prime and ?-prime implies prime. A large number of examples of group algebras are given which show that the concept of ?-prime lies strictly between prime and strongly prime. A complete characterization of ?-prime group algebras is given. It is proved that a group algebra KG of the group G over the field K is ?-prime if and only if Λ⁺(G) = (1). Intersection theorems play an important role in the study. In the process, a new intersection theorem for ?-prime group algebras is obtained. Elementwise characterization of the ?-prime radical is given and its relation with some well-known radicals is discussed.     

The lattice of strict completions of a finite poset

For a given finite poset , we construct strict completions of P which are models of all finite lattices L such that the set of join-irreducible elements of L is isomorphic to P. This family of lattices, , turns out to be itself a lattice, which is lower bounded and lower semimodular. We determine the join-irreducible elements of this lattice. We relate properties of the lattice to properties of our given poset P, and in particular we characterize the posets P for which . Finally we study the case where is distributive. Received October 13, 2000; accepted in final form June 13, 2001.     

Counting Finite Lattices

The correct values for the number of all unlabeled lattices on n elements are known for . We present a fast orderly algorithm generating all unlabeled lattices up to a given size n. Using this algorithm, we have computed the number of all unlabeled lattices as well as that of all labeled lattices on an n-element set for each . Received April 4, 2000; accepted in final form November 2, 2001. RID="h1" ID="h1" Presented by R. Freese.     

A structure theorem for dismantlable lattices and enumeration

The concept of `adjunct' operation of two lattices with respect to a pair of elements is introduced. A structure theorem namely, `A finite lattice is dismantlable if and only if it is an adjunct of chains' is obtained. Further it is established that for any adjunct representation of a dismantlable lattice the number of chains as well as the number of times a pair of elements occurs remains the same. If a dismantlable lattice L has n elements and n+k edges then it is proved that the number of irreducible elements of L lies between n-2k-2 and n-2. These results are used to enumerate the class of lattices with exactly two reducible elements, the class of lattices with n elements and upto n+1 edges, and their subclasses of distributive lattices and modular lattices. This revised version was published online in June 2006 with corrections to the Cover Date.     

Reducible classes of finite lattices

In this paper we study a notion of reducibility in finite lattices. An element x of a (finite) lattice L satisfying certain properties is deletable if L-x is a lattice satisfying the same properties. A class of lattices is reducible if each lattice of this class admits (at least) one deletable element (equivalently if one can go from any lattice in this class to the trivial lattice by a sequence of lattices of the class obtained by deleting one element in each step). First we characterize the deletable elements in a pseudocomplemented lattice what allows to prove that the class of pseudocomplemented lattices is reducible. Then we characterize the deletable elements in semimodular, modular and distributive lattices what allows to prove that the classes of semimodular and locally distributive lattices are reducible. In conclusion the notion of reducibility for a class of lattices is compared with some other notions like the notion of order variety.     

Boolean Hierarchies of Partitions over a Reducible Base

The Boolean hierarchy of partitions was introduced and studied by Kosub and Wagner, primarily over the lattice of NP-sets. Here, this hierarchy is treated over lattices with the reduction property, showing that it has a much simpler structure in this instance. A complete characterization is given for the hierarchy over some important lattices, in particular, over the lattices of recursively enumerable sets and of open sets in the Baire space.     

Dilworth's principal elements

Principal elements were introduced in multiplicative lattices by R. P. Dilworth, following an earlier but less successful attempt in the joint work of Ward and Dilworth. As suggested by their name, principal elements are the analogue in multiplicative lattices of principal ideals in (commutative) rings. Principal elements are the cornerstone on which the theory of multiplicative lattices and abstract ideal theory now largely rests. In this paper, we obtain some new results regarding principal elements and extend some others. In addition, we try to convey what is known and what is not known about the subject. We conclude with a fairly extensive (but likely not exhaustive) bibliography on principal elements.Dedicated to R. P. DilworthPresented by E. T. Schmidt.     

Studies on the squaring construction

The code formulas for the iterated squaring construction for a finite group partition chain, derived by Forney [2], are extended to the case with a bi-infinite group partition chain, and the derivation presented here is much simpler than the one given by Forney for the finite case. It is also proven that the iterated squaring construction indeed generates the Reed-Muller codes. Moreover, the generalization of the code formulas to the bi-infinite case is used to derive code formulas for the lattices Λ(r,n) andRΛ(r,n), which correct some errors in [2]. Further, Gaussian integer lattices are discussed. A definition of their dual lattices is given, which is more general than the definition given by Forney [1]. Using this definition, some interesting properties of dual lattices and the squaring construction are obtained and then formulas of the duals of the Barnes-Wall lattices and their principal sublattices are derived, and one assumption from the derivation given by Forney [2] can be eliminated.     

2-Join decomposition lattices

In this paper we characterize principal element lattices and Dedekind domains in terms of 2-join decomposition lattices. Received March 2, 1999; accepted in final form July 10, 2000.     

A splitting theorem for the Medvedev and Muchnik lattices

This is a contribution to the study of the Muchnik and Medvedev lattices of non‐empty Π⁰₁ subsets of 2^ω. In both these lattices, any non‐minimum element can be split, i. e. it is the non‐trivial join of two other elements. In fact, in the Medvedev case, ifP > _M Q, then P can be split above Q. Both of these facts are then generalised to the embedding of arbitrary finite distributive lattices. A consequence of this is that both lattices have decidible ?‐theories.     

Generalized Bosbach and Rie?an states based on relative negations in residuated lattices

Bosbach and Rie?an states on residuated lattices both are generalizations of probability measures on Boolean algebras. Recently, two types of generalized Bosbach states on residuated lattices were introduced by Georgescu and Mure?an through replacing the standard MV-algebra in the original definition with arbitrary residuated lattices as codomains. However, several interesting problems there remain still open. The first part of the present paper gives positive answers to these open problems. It is proved that every generalized Bosbach state of type II is of type I and the similarity Cauchy completion of a residuated lattice endowed with an order-preserving generalized Bosbach state of type I is unique up to homomorphisms preserving similarities, where the codomain of the type I state is assumed to be Cauchy-complete. Consequently, many existing results about generalized Bosbach states can be further strengthened. The second part of the paper introduces the notion of relative negation (with respect to a given element, called relative element) in residuated lattices, and then many issues with the canonical negation such as Glivenko property, semi-divisibility, generalized Rie?an state of residuated lattices can be extended to the context of such relative negations. In particular, several necessary and sufficient conditions for the set of all relatively regular elements of a residuated lattice to be special residuated lattices are given, of which one extends the well-known Glivenko theorem, and it is also proved that relatively generalized Rie?an states vanishing at the relative element are uniquely determined by their restrictions on the MV-algebra consisting of all relatively regular elements when the domain of the states is relatively semi-divisible and the codomain is involutive.     

Layer-projective lattices. I

The class of layer-projective lattices is singled out. For example, it contains the lattices of subgroups of finite Abelianp-groups, finite modular lattices of centralizers that are indecomposable into a finite sum, and lattices of subspaces of a finite-dimensional linear space over a finite field that are invariant with respect to a linear operator with zero eigenvalues. In the class of layer-projective lattices, the notion of type (of a lattice) is naturally introduced and the isomorphism problem for lattices of the same type is posed. This problem is positively solved for some special types of layer-projective lattices. The main method is the layer-wise lifting of the coordinates. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 170–182, February, 1998.     

One simple result concerning imprimitivity relations of monoids of transformations and its applications

A subsetX of an algebraA is called conditional if every congruence relation onA is uniquely determined by its congruence classes containing elements ofX. In terms of transformation monoids, it is found in which case one can be sure that a given subsetX is conditional. Some applications of this results to semigroups and lattices are demonstrated. These applications lead to known results as well as to new ones. Particularly, some sufficient conditions are obtained for semigroups and lattices to be harmonic.