n-distributivity,dimension and Carathéodory's theorem

A. Huhn proved that the dimension of Euclidean spaces can be characterized through algebraic properties of the lattices of convex sets. In fact, the lattice of convex sets of isn+1-distributive but notn-distributive. In this paper his result is generalized for a class of algebraic lattices generated by their completely join-irreducible elements. The lattice theoretic form of Carathéodory's theorem characterizesn-distributivity in such lattices. Several consequences of this result are studied. First, it is shown how infiniten-distributivity and Carathéodory's theorem are related. Then the main result is applied to prove that for a large class of lattices beingn-distributive means being in the variety generated by the finiten-distributive lattices. Finally,n-distributivity is studied for various classes of lattices, with particular attention being paid to convexity lattices of Birkhoff and Bennett for which a Helly type result is also proved.Presented by J. Sichler.     

Some Structure Theorems for Atomistic Algebraic Lattices

We prove that any atomistic algebraic lattice is a direct product of subdirectly irreducible lattices iff its congruence lattice is an atomic Stone lattice. We define on the set A(L) of all atoms of an atomistic algebraic lattice L a relation R as follows: for a, b A(L), (a, b) R ? θ(0, a) ∧ θ(0, b) ≠ ?_{Con L} . We prove that Con L is a Stone lattice iff R is transitive and we give a characterization of Cen (L) using R. We also give a characterization of weakly modular atomistic algebraic lattices.     

Matchings and radon transforms in lattices. I. Consistent lattices

An element in a lattice is join-irreducible if x=ab implies x=a or x=b. A meet-irreducible is a join-irreducible in the order dual. A lattice is consistent if for every element x and every join-irreducible j, the element xj is a join-irreducible in the upper interval [x, î]. We prove that in a finite consistent lattice, the incidence matrix of meet-irreducibles versus join-irreducibles has rank the number of join-irreducibles. Since modular lattices and their order duals are consistent, this settles a conjecture of Rival on matchings in modular lattices.     

Join-semidistributive lattices and convex geometries

We introduce the notion of a convex geometry extending the notion of a finite closure system with the anti-exchange property known in combinatorics. This notion becomes essential for the different embedding results in the class of join-semidistributive lattices. In particular, we prove that every finite join-semidistributive lattice can be embedded into a lattice S_P(A) of algebraic subsets of a suitable algebraic lattice A. This latter construction, S_P(A), is a key example of a convex geometry that plays an analogous role in hierarchy of join-semidistributive lattices as a lattice of equivalence relations does in the class of modular lattices. We give numerous examples of convex geometries that emerge in different branches of mathematics from geometry to graph theory. We also discuss the introduced notion of a strong convex geometry that might promise the development of rich structural theory of convex geometries.     

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.     

The Asymptotic Number of Planar,Slim, Semimodular Lattice Diagrams

A lattice L is slim if it is finite and the set of its join-irreducible elements contains no three-element antichain. We prove that there exists a positive constant C such that, up to similarity, the number of planar diagrams of slim semimodular lattices of size n is asymptotically C · 2 ⁿ .     

Coordinatization of finite join-distributive lattices

Join-distributive lattices are finite, meet-semidistributive, and semimodular lattices. They are the same as lattices with unique irreducible decompositions, introduced by R.P. Dilworth in 1940, and many alternative definitions and equivalent concepts have been discovered or rediscovered since then. Let L be a join-distributive lattice of length n, and let k denote the width of the set of join-irreducible elements of L. We prove that there exist k ? 1 permutations acting on {1, . . . , n} such that the elements of L are coordinatized by k-tuples over {0, . . . , n}, and the permutations determine which k-tuples are allowed. Since the concept of join-distributive lattices is equivalent to that of antimatroids and convex geometries, our result offers a coordinatization for these combinatorial structures.     

The M3[D] construction and n-modularity

In 1968, Schmidt introduced the M ₃[D] construction, an extension of the five-element modular nondistributive lattice M ₃ by a bounded distributive lattice D, defined as the lattice of all triples satisfying . The lattice M ₃[D] is a modular congruence-preserving extension of D.? In this paper, we investigate this construction for an arbitrary lattice L. For every n > 0, we exhibit an identity such that is modularity and is properly weaker than . Let M _n denote the variety defined by , the variety of n-modular lattices. If L is n-modular, then M ₃[L] is a lattice, in fact, a congruence-preserving extension of L; we also prove that, in this case, Id M ₃[L] M ₃[Id L]. ? We provide an example of a lattice L such that M ₃[L] is not a lattice. This example also provides a negative solution to a problem of Quackenbush: Is the tensor product of two lattices A and B with zero always a lattice. We complement this result by generalizing the M ₃[L] construction to an M ₄[L] construction. This yields, in particular, a bounded modular lattice L such that M ₄ L is not a lattice, thus providing a negative solution to Quackenbush’s problem in the variety M of modular lattices.? Finally, we sharpen a result of Dilworth: Every finite distributive lattice can be represented as the congruence lattice of a finite 3-modular lattice. We do this by verifying that a construction of Gr?tzer, Lakser, and Schmidt yields a 3-modular lattice. Received May 26, 1998; accepted in final form October 7, 1998.     

Involutive Residuated Lattices Based on Modular and Distributive Lattices

An involutive residuated lattice (IRL) is a lattice-ordered monoid possessing residual operations and a dualizing element. We show that a large class of self-dual lattices may be endowed with an IRL structure, and give examples of lattices which fail to admit IRLs with natural algebraic conditions. A classification of all IRLs based on the modular lattices M _n is provided.     

Matching in modular lattices

Let L be a finite lattice. A map f of the join irreducible elements of L to the meet irreducible elements of L is called a matching of L if f is one-to-one and x?f(x) for each join irreducible x. We investigate this conjecture: every finite modular lattice has a matching. The conjecture is verified for certain classes of modular lattices.     

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.     

Dimension and decomposition in modular upper-continuous lattices

Slim rectangular lattices are special planar semimodular lattices introduced by G. Grätzer and E. Knapp in 2009. They are finite semimodular lattices L such that the ordered set Ji L of join-irreducible elements of L is the cardinal sum of two nontrivial chains. After describing these lattices of a given length n by permutations, we determine their number, |SRectL(n)|. Besides giving recursive formulas, which are effective up to about n = 1000, we also prove that |SRectL(n)| is asymptotically (n - 2)! · \({e^{2}/2}\). Similar results for patch lattices, which are special rectangular lattices introduced by G. Czédli and E. T. Schmidt in 2013, and for slim rectangular lattice diagrams are also given.     

Embedding and unsolvability theorems for modular lattices

LetR be a nontrivial ring with 1 and δ a cardinal. Let,L(R, δ) denote the lattice of submodules of a free unitaryR-module on δ generators. Let ? be the variety of modular lattices. A lattice isR-representable if embeddable in the lattice of submodules of someR-module; ?(R) denotes the quasivariety of allR-representable lattices. Let ω denote aleph-null, and let a (m, n) presentation havem generators andn relations,m, n≤ω. THEOREM. There exists a (5, 1) modular lattice presentation having a recursively unsolvable word problem for any quasivarietyV,V ? ?, such thatL(R, ω) is inV. THEOREM. IfL is a denumerable sublattice ofL(R, δ), then it is embeddable in some sublatticeK ofL(R,δ^*) having five generators, where δ^*=δ for infinite δ and δ^*=4δ(m+1) if δ is finite andL has a set ofm generators. THEOREM. The free ?(R)-lattice on ω generators is embeddable in the free ?(R)-lattice on five generators. THEOREM. IfL has an (m, n), ?(R)-presentation for denumerablem and finiten, thenL is embeddable in someK having a (5, 1) ?(R)-presentation.     

Extension theorems for linear lattices with positive algebraic basis

An ordered linear spaceV with positive wedgeK is said to satisfy extension property (E1) if for every subspaceL ₀ ofV such thatL ₀ ∩K is reproducing inL ₀, and every monotone linear functionalf ₀ defined onL ₀,f ₀ has a monotone linear extension to all ofV. A linear latticeX is said to satisfy extension property (E2) if for every sublatticeL ofX, and every linear functionalf defined onL which is a lattice homomorphism,f has an extensionf′ to all ofX which is also a linear functional and a lattice homomorphism. In this paper it is shown that a linear lattice with a positive algebraic basis has both extension property (E1) and (E2). In obtaining this result it is shown that the linear span of a lattice idealL and an extremal element not inL is again a lattice ideal. (HereX does not have to have a positive algebraic basis.) It is also shown that a linear lattice which possesses property (E2) must be linearly and lattice isomorphic to a functional lattice. An example is given of a function lattice which has property (E2) but does not have a positive algebraic basis. Yudin [12] has shown a reproducing cone in ann-dimensional linear lattice to be the intersection of exactlyn half-spaces. Here it is shown that the positive wedge in ann-dimensional archimedean ordered linear space satisfying the Riesz decomposition property must be the intersection ofn half-spaces, and hence the space must be a linear lattice with a positive algebraic basis. The proof differs from those given for the linear lattice case in that it uses no special techniques, only well known results from the theory of ordered linear space.     

A condition for modular lattices

This paper proves that a geometric lattice of rank n is a modular lattice if its every maximal chain contains a modular element of rank greater than 1 and less than n. This result is generalized to a more general lattices of finite rank. The first author is partially supported by the National Natural Science Foundation of China (Grant No. 10471016).     

A cover-preserving embedding of semimodular lattices into geometric lattices

Extending former results by G. Grätzer and E.W. Kiss (1986) [5] and M. Wild (1993) [9] on finite (upper) semimodular lattices, we prove that each semimodular lattice L of finite length has a cover-preserving embedding into a geometric lattice G of the same length. The number of atoms of our G equals the number of join-irreducible elements of L.     

Finite modular congruence lattices

We consider the variety of modular lattices generated by all finite lattices obtained by gluing together some M₃’s. We prove that every finite lattice in this variety is the congruence lattice of a suitable finite algebra (in fact, of an operator group). Received February 26, 2004; accepted in final form December 16, 2004.     

Graphical algebras — a new approach to congruence lattices

In 1970, H. Werner considered the question of which sublattices of partition lattices are congruence lattices for an algebra on the underlying set of the partition lattices. He showed that a complete sublattice of a partition lattice is a congruence lattice if and only if it is closed under a new operation called graphical composition. We study the properties of this new operation, viewed as an operation on an abstract lattice. We obtain some necessary properties, and we also obtain some sufficient conditions for an operation on an abstract lattice L to be this operation on a congruence lattice isomorphic to L. We use this result to give a new proof of Grätzer and Schmidt’s result that any algebraic lattice occurs as a congruence lattice.