Free Lattices Research Articles

The Primary Electroviscous Effect in Surfactant Free Polystyrene Lattices. II. The Effect of increasing Concentrations of Quaternary Ammonium Counterions of different Ionic Sizes

The Primary Electroviscous Effect in Surfactant Free Polystyrene Lattices. II. The Effect of increasing Concentrations of Quaternary Ammonium Counterions of different Ionic Sizes

The skeletons of free distributive lattices

The skeletons of free distributive lattices are studied by methods of formal concept analysis; in particular, a specific closure system of sublattices is elaborated to clarify the structure of the skeletons. Up to five generators, the skeletons are completely described.

Free lattices over join-trivial partial lattices

Free lattices over join-trivial partial lattices

Infinite intervals in free lattices

if u/v is any infinite interval in FL(X), the free lattice with generating set X, and \(|{\text{Y}}| \leqslant |{\text{X}}| + \aleph _0 \) then u/v contains a sublattice isomorphic to FL(Y). This result answers a question posed by Ralph Freese and J. B. Nation in their paper “Covers in Free Lattices” [2] and the principal techniques used in proving this result all come from that paper. The most difficult step in the argument is to show that there are incomparable elements in any infinite interval.

The order dimension of relatively free lattices

If V is a nontrivial lattice variety and λ is an infinite cardinal, then the order dimension of FV(λ) is the smallest cardinal κ such that λ≤2κ.

Finitely Presented Lattices: Canonical Forms and the Covering Relation

A canonical form for elements of a lattice freely generated by a partial lattice is given. This form agrees with Whitman’s canonical form for free lattices when the partial lattice is an antichain. The connection between this canonical form and the arithmetic of the lattice is given. For example, it is shown that every element of a finitely presented lattice has only finitely many minimal join representations and that every join representation can be refined to one of these. An algorithm is given which decides if a given element of a finitely presented lattice has a cover and finds them if it does. An example is given of a nontrivial, finitely presented lattice with no cover at all.

Finitely presented lattices: canonical forms and the covering relation

A canonical form for elements of a lattice freely generated by a partial lattice is given. This form agrees with Whitman’s canonical form for free lattices when the partial lattice is an antichain. The connection between this canonical form and the arithmetic of the lattice is given. For example, it is shown that every element of a finitely presented lattice has only finitely many minimal join representations and that every join representation can be refined to one of these. An algorithm is given which decides if a given element of a finitely presented lattice has a cover and finds them if it does. An example is given of a nontrivial, finitely presented lattice with no cover at all.

The word and generator problems for lattices

We present polynomial-time algorithms for the uniform word problem and for the generator problem for lattices. The algorithms are derived from novel, prooftheoretic approaches. We prove that both problems are log-space complete for P , but can be solved in deterministic logarithmic space in the case of free lattices. We also show that the more general problem of testing whether a given open sentence is true in all lattices is co-NP complete.

Free lattice algorithms

In the late 1930s Phillip Whitman gave an algorithm for deciding for lattice terms v and u if v≤u in the free lattice on the variables in v and u. He also showed that each element of the free lattice has a shortest term representing it and this term is unique up to commutivity and associativity. He gave an algorithm for finding this term. Almost all the work on free lattices uses these algorithms. Building on the work of Ralph McKenzie, J. B. Nation and the author have developed very efficient algorithms for deciding if a lattice term v has a lower cover (i.e., if there is a w with w covered by v, which is denoted by w<v) and for finding them if it does. This paper studies the efficiency of both Whitman's algorithm and the algorithms of Freese and Nation. It is shown that although it is often quite fast, the straightforward implementation of Whitman's algorithm for testing v≤u is exponential in time in the worst case. A modification of Whitman's algorithm is given which is polynomial and has constant minimum time. The algorithms of Freese and Nation are then shown to be polynomial.

The structure of free abelian l-groups and free vector lattices

This paper gives the prime and regular subgroup structures of free abelianl-groups and free vector lattices of finite rank. For the free abelianl-group\(A_n\) of rankn, the prime structure of\(A_n\) is 2 ω copies of the prime structure of\(A_{n - 1}\) and ω many copies of the prime structure of\(A_{n - 1} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{x} \mathbb{Z}\). For the free vector lattice\(V_n\) or rankn, the prime structure is 2 ω many copies of the prime structure of\(V_{n - 1} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{x} \mathbb{R}\).

Singular covers in free lattices

A covering a>b in a lattice is called a singular cover if a is join irreducible and b is meet irreducible. A classification of the singular covers which occur in free lattices is given.

A Technique to Generate -Ary Free Lattices from Finitary Ones

Let be an infinite regular cardinal. A poset L is called an -lattice if and only if for all XL satisfying 0 &lt; |X| &lt; m, ∧ X and ∨ X exist.This paper is a part of a sequence of papers, [5], [6], [7], [8], developing the theory of -lattices. For a survey of some of these results, see [9].The -lattice is described in [6]; γ denotes the zero and γ′ the unit of . In particular, formulas for -joins and meets are given. (We repeat the essentials of this description in Section 4.)In [6] we proved the theorem stated below. Our proof was based on characterization of (the free -lattice on P) due to [1]; as a result, our proof was very computational.

Covers in Free Lattices

In this paper we study the covering relation $(u \succ v)$ in finitely generated free lattices. The basic result is an algorithm which, given an element $w \in {\text {FL}}(X)$, finds all the elements which cover or are covered by $w$ (if any such elements exist). Using this, it is shown that covering chains in free lattices have at most five elements; in fact, all but finitely many covering chains in each free lattice contain at most three elements. Similarly, all finite intervals in ${\text {FL}}(X)$ are classified; again, with finitely many exceptions, they are all one-, two- or three-element chains.

Covers in free lattices

In this paper we study the covering relation ( u ≻ v ) (u \succ v) in finitely generated free lattices. The basic result is an algorithm which, given an element w ∈ FL ( X ) w \in {\text {FL}}(X) , finds all the elements which cover or are covered by w w (if any such elements exist). Using this, it is shown that covering chains in free lattices have at most five elements; in fact, all but finitely many covering chains in each free lattice contain at most three elements. Similarly, all finite intervals in FL ( X ) {\text {FL}}(X) are classified; again, with finitely many exceptions, they are all one-, two- or three-element chains.

Selectors: A theory of formal languages, semimodular lattices, and branching and shelling processes

Combinatorial theory has made great strides in the present era thanks in large part to its symbiotic relationship with computers and computer programming. The combinatorial study of formal languages has done much to clarify what computer programming is all about, and the need to write clear and workable algorithms has often forced mathematicians to reconsider the value of their existential (nonconstructive) theories. In this paper, we see how the study of certain hereditary formal languages can lead to an understanding of all sorts of stepwise processes, those falling under the broad headings of branching and shelling processes, and also to a representation of a large class of lattices, which carry the natural logic of these processes. Our basic notion is that of a selector, a hereditary language consisting of words formed from a certain alphabet, and possessed of a unit increase property which guarantees the appearance of a dimension function and forces certain associated lattices of (closed) partial alphabets to be semimodular. A remarkable subclass of selectors consists of those which are locally free. They arise from arbitrary hereditary languages, and form the substrate for the general theory of selectors, since every selector is seen to be a quotient of a locally free selector. Locally free selectors are representable as shelling orders of abstract discs, in the same way that finite distributive lattices are represented by shelling orders (or by linear extensions) of partially ordered sets. Furthermore, every set r of linear orders on (or permutations of) a finite set can be completed to the set of bases of a locally free selector E The analogue of the theory of (Dushnik-Miller) dimension is thus available for locally free selectors, and thus for locally free semimodular lattices.

A reduced free product of distributive lattices. I

This paper was inspired by the problem whether every distributive algebraic lattice can be represented as the congruence lattice of a lattice. However, here we do not attack this problem directly. Our "conjecture" is that the problem is closely connected with the theory of free distributive lattices and free distributive products. In the present paper we start a systematic study of those questions of the theory of free distributive products that we think relevant in this context. Applications to the representation problem are given in [2]. Let D1 and D~ be finite distributive lattices such that D1 is isomorphic with a 0--V-subsemilattice of D2. This isomorphism will be denoted by 6: a( 6 D1) ~--,-a +.

Finite Transferable Lattices are Sharply Transferable

A lattice e5 is called transferable if and only if, whenever e5 can be embedded into the lattice I(3C) of all ideals of a lattice SC, e can be embedded into X itself. If for every lattice embeddingf of e into I(CC) there exists an embedding g of e into C satisfying the further condition that for x and y in L, f(x) E g(y) holds if and only if x < y, then e is called sharply transferable. It is shown that every finite transferable lattice is sharply transferable. Introduction. A lattice e, is called transferable if and only if, whenever 15 can be embedded into the lattice I(f3) of all ideals of a lattice X3C, C can be embedded into i; itself. If for every lattice embeddingf of e& into I(9C), there exists an embedding g of e into SC satisfying the further condition that for x and y in L, f(x) e g(y) holds if and only if x < y, then C is called sharply transferable. The concept of a transferable lattice was introduced in [5], where it was called Property (P). Sharp transferability was later introduced as a related property whose characterization might be more tractable and might shed light on transferability itself. In [31 finite sharply transferable lattices were characterized and in [4] they were shown to be the same as finite sublattices of free lattices. This paper shows that the finite transferable lattices coincide with the finite sharply transferable lattices, thus answering the question raised in [5]. To fix the notation, we recall here some of the definitions from [7]. If X and Y are subsets of a poset, we say that Y dominates X iff for every x E X there exists y E Y such that x 6 y. This is here denoted X < Y. If C is a lattice, x E L, and U is a subset of L, then U is called an C-cover of x, if x < V U. If, furthermore, x + u for each u E U, then U is a nontrivial et-cover of x. If U is a nontrivial ,-cover of x with the property that whenever U' is an e-cover of x and U' < U then U' contains U, then U is called a minimal C-cover of x. A lattice C is said to satisfy condition (Rv) iff there exists a mapping p: C e such that, if U is a minimal Ce-cover of x and u E U, then p(x) < p(u) for each u E U (here X denotes the natural numbers). For a finite lattice, condition (RA) can be defined as the dual of condition (Rv). The lattice C satisfies condition (W) iff for all a, b, c, d E L, if a A b 6 c V d, then either a < c V d, b < c V d, a A b < c, or a A b < d. The result in [3] shows that a finite lattice is sharply transferable if and only if it Received by the editors November 13, 1979. 1980 Mathematics Subject Classification Primary 06A20.

Distributive lattices with a dual homomorphic operation. II

The lattices of the title generalize the concept of a De Morgan lattice. A representation in terms of ordered topological spaces is described. This topological duality is applied to describe homomorphisms, congruences, and subdirectly irreducible and free lattices in the category. In addition, certain equational subclasses are described in detail.

Free lattices in some small varieties

Free lattices in some small varieties

The level polynomials of the free distributive lattices

We show that there exist a set of polynomials { L k ⋎ k = 0, 1⋯} such that L k ( n) is the number of elements of rank k in the free distributive lattice on n generators. L 0( n) = L 1( n) = 1 for all n and the degree of L k is k−1 for k⩾1. We show that the coefficients of the L k can be calculated using another family of polynomials, P j . We show how to calculate L k for k = 1,…,16 and P j for j = 0,…,10. These calculations are enough to determine the number of elements of each rank in the free distributive lattice on 5 generators a result first obtained by Church [2]. We also calculate the asymptotic behavior of the L k 's and P j 's.