Generalizing semidistributivity

A latticeL is called congruence normal if it can be generated by doubling of convex sets starting with the one-element lattice. In the special case of intervals, the lattice is called bounded. It has been proven thatL is bounded if and only ifL is congruence normal and semidistributive.In this paper we study the connection between certain classes of convex sets and generalized semidistributive laws. These so-called doubling classes are pseudovarieties which can be described by implications as well as by forbiden substructures. In the end, we examine the structure of the lattice of all doubling classes.     

Almost distributive sublattices and congruence heredity

A congruence lattice L of an algebra A is hereditary if every 0-1 sublattice of L is the congruence lattice of an algebra on A. Suppose that L is a finite lattice obtained from a distributive lattice by doubling a convex subset. We prove that every congruence lattice of a finite algebra isomorphic to L is hereditary. Presented by E. W. Kiss. Received July 18, 2005; accepted in final form April 2, 2006.     

Lattice ordered effect algebras

The purpose of this paper is to discuss some structural properties of lattice ordered effect algebras. We will use these structural properties to find certain lattices and classes of lattices that do not admit an effect algebra structure. Finally, using these structural properties, we will show that if L is the face lattice of a convex polytope in $ R^3 $ with more than 3 vertices, then L does not admit an effect algebra structure.Dedicated to the memory of Gian-Carlo Rota     

A note on essentially convex orders

In this note, we determine precisely which partially ordered sets (posets) have the property that, whenever they occur as subposets of a larger poset, they occur there convexly, i.e., as convex subposets. As a corollary, we also determine which lattices have the property that, if they occur as sublattices of a finite distributive lattice L, then they also occur as closed intervals in L. Throughout, all sets will be finite.Dedicated to the memory of Ivan RivalReceived May 5, 2003; accepted in final form October 3, 2004.This revised version was published online in August 2005 with a corrected cover date.     

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.     

Sublattices of product spaces: Hulls, representations and counting

The Cartesian product of lattices is a lattice, called a product space, with componentwise meet and join operations. A sublattice of a lattice L is a subset closed for the join and meet operations of L. The sublattice hullLQ of a subset Q of a lattice is the smallest sublattice containing Q. We consider two types of representations of sublattices and sublattice hulls in product spaces: representation by projections and representation with proper boundary epigraphs. We give sufficient conditions, on the dimension of the product space and/or on the sublattice hull of a subset Q, for LQ to be entirely defined by the sublattice hulls of the two-dimensional projections of Q. This extends results of Topkis (1978) and of Veinott [Representation of general and polyhedral subsemilattices and sublattices of product spaces, Linear Algebra Appl. 114/115 (1989) 681-704]. We give similar sufficient conditions for the sublattice hull LQ to be representable using the epigraphs of certain isotone (i.e., nondecreasing) functions defined on the one-dimensional projections of Q. This also extends results of Topkis and Veinott. Using this representation we show that LQ is convex when Q is a convex subset in a vector lattice (Riesz space), and is a polyhedron when Q is a polyhedron in Rⁿ.We consider in greater detail the case of a finite product of finite chains (i.e., totally ordered sets). We use the representation with proper boundary epigraphs and provide upper and lower bounds on the number of sublattices, giving a partial answer to a problem posed by Birkhoff in 1937. These bounds are close to each other in a logarithmic sense. We define a corner representation of isotone functions and use it in conjunction with the representation with proper boundary epigraphs to define an encoding of sublattices. We show that this encoding is optimal (up to a constant factor) in terms of memory space. We also consider the sublattice hull membership problem of deciding whether a given point is in the sublattice hull LQ of a given subset Q. We present a good characterization and a polynomial time algorithm for this sublattice hull membership problem. We construct in polynomial time a data structure for the representation with proper boundary epigraphs, such that sublattice hull membership queries may be answered in time logarithmic in the size |Q| of the given subset.     

Consistent lattices and Steinitz spaces

It is well-known that a finite lattice L is isomorphic to the lattice of flats of a matroid if and only if L is geometric. A result due to Edelman (see [1], Theorem 3.3) states that a lattice is meet-distributive if and only if it is isomorphic to the lattice of all closed sets of a convex geometry. In this note we prove that a finite lattice is the lattice of closed sets of a closure space with the Steinitz exchange property if and only if it is a consistent lattice. Received February 28, 1997; accepted in final form February 2, 1998.     

Congruence lattices of function lattices

Thefunction lattice L ^P is the lattice of all isotone maps from a posetP into a latticeL.D. Duffus, B. Jónsson, and I. Rival proved in 1978 that for afinite poset P, the congruence lattice ofL ^P is a direct power of the congruence lattice ofL; the exponent is |P|.This result fails for infiniteP. However, utilizing a generalization of theL ^P construction, theL[D] construction (the extension ofL byD, whereD is a bounded distributive lattice), the second author proved in 1979 that ConL[D] is isomorphic to (ConL) [ConD] for afinite lattice L.In this paper we prove that the isomorphism ConL[D](ConL)[ConD] holds for a latticeL and a bounded distributive latticeD iff either ConL orD is finite.The research of the first author was supported by the NSERC of Canada.The research of the second author was supported by the Hungarian National Foundation for Scientific Research, under Grant No. 1903.     

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.     

Counting formulas for glued lattices

A tolerance relation of a lattice L, i.e., a reflexive and symmetric relation of L which is compatible with join and meet, is called glued if covering blocks of have nonempty intersection. For a lattice L with a glued tolerance relation we prove a formula counting the number of elements of L with exactly k lower (upper) covers. Moreover, we prove similar formulas for incidence structures and graphs and we give a new proof of Dilworth's covering theorem.     

On the lattice formed by all normal subloops of a finite loop

All normal subloops of a loopG form a modular latticeL _n (G). It is shown that a finite loopG has a complemented normal subloop lattice if and only ifG is a direct product of simple subloops. In particular,L _n (G) is a Boolean algebra if and only if no two isomorphic factors occurring in a decomposition ofG are abelian groups. The normal subloop lattice of a finite loop is a projective geometry if and only ifG is an elementary abelianp-group for some primep.     

Identities of cofinal sublattices

If V is a variety of lattices and L a free lattice in V on uncountably many generators, then any cofinal sublattice of L generates all of V. On the other hand, any modular lattice without chains of order-type +1 has a cofinal distributive sublattice. More generally, if a modular lattice L has a distributive sublattice which is cofinal modulo intervals with ACC, this may be enlarged to a cofinal distributive sublattice. Examples are given showing that these existence results are sharp in several ways. Some similar results and questions on existence of cofinal sublattices with DCC are noted.This work was done while the first author was partly supported by NSF contract MCS 82-02632, and the second author by an NSF Graduate Fellowship.     

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.     

Intervals in lattices of quasiorders

We investigate the structure of intervals in the lattice of all closed quasiorders on a compact or discrete space. As a first step, we show that if the intervalI has no infinite chains then the underlying space may be assumed to be finite, and in particular,I must be finite, too. We compute several upper bounds for its size in terms of its heighth, which in turn can be computed easily by means of the least and the greatest element ofI. The cover degreec of the interval (i.e. the maximal number of atoms in a subinterval) is less than 4h. Moreover, ifc4(n–1) thenI contains a Boolean subinterval of size 2 ⁿ , and ifI is geometric then it is already a finite Boolean lattice. While every finite distributive lattice is isomorphic to some interval of quasiorders, we show that a nondistributive finite interval of quasiorders is neither a vertical sum nor a horizontal sum of two lattices, with exception of the pentagon. Many further lattices are excluded from the class of intervals of quasiorders by the fact that no join-irreducible element of such an interval can have two incomparable join-irreducible complements. Up to isomorphism, we determine all quasiorder intervals with less than 9 elements and all quasiorder intervals with two complementary atoms or coatoms.     

Sublattices of complete lattices with continuity conditions

Various embedding problems of lattices into complete lattices are solved. We prove that for any join-semilattice S with the minimal join-cover refinement property, the ideal lattice Id S of S is both algebraic and dually algebraic. Furthermore, if there are no infinite D-sequences in J(S), then Id S can be embedded into a direct product of finite lower bounded lattices. We also find a system of infinitary identities that characterize sublattices of complete, lower continuous, and join-semidistributive lattices. These conditions are satisfied by any (not necessarily finitely generated) lower bounded lattice and by any locally finite, join-semidistributive lattice. Furthermore, they imply M. Erné’s dual staircase distributivity.On the other hand, we prove that the subspace lattice of any infinite-dimensional vector space cannot be embedded into any ℵ₀-complete, ℵ₀-upper continuous, and ℵ₀-lower continuous lattice. A similar result holds for the lattice of all order-convex subsets of any infinite chain.Dedicated to the memory of Ivan RivalReceived April 4, 2003; accepted in final form June 16, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Primes,irreducibles and extremal lattices

This paper studies certain types of join and meet-irreducibles called coprimes and primes. These elements can be used to characterize certain types of lattices. For example, a lattice is distributive if and only if every join-irreducible is coprime. Similarly, a lattice is meet-pseudocomplemented if and only if each atom is coprime. Furthermore, these elements naturally decompose lattices into sublattices so that often properties of the original lattice can be deduced from properties of the sublattice. Not every lattice has primes and coprimes. This paper shows that lattices which are long enough must have primes and coprimes and that these elements and the resulting decompositions can be used to study such lattices.The length of every finite lattice is bounded above by the minimum of the number of meet-irreducibles (meet-rank) and the number of join-irreducibles (join-rank) that it has. This paper studies lattices for which length=join-rank or length=meet-rank. These are called p-extremal lattices and they have interesting decompositions and properties. For example, ranked, p-extremal lattices are either lower locally distributive (join-rank=length), upper locally distributive (meet-rank=length) or distributive (join-rank=meet-rank=length). In the absence of the Jordan-Dedekind chain condition, p-extremal lattices still have many interesting properties. Of special interest are the lattices that satisfy both equalities. Such lattices are called extremal; this class includes distributive lattices and the associativity lattices of Tamari. Even though they have interesting decompositions, extremal lattices cannot be characterized algebraically since any finite lattice can be embedded as a subinterval into an extremal lattice. This paper shows how prime and coprime elements, and the poset of irreducibles can be used to analyze p-extremal and other types of lattices.The results presented in this paper are used to deduce many key properties of the Tamari lattices. These lattices behave much like distributive lattices even though they violate the Jordan-Dedekind chain condition very strongly having maximal chains that vary in length from N-1 to N(N-1)/2 where N is a parameter used in the construction of these lattices.     

Bigeneration in complete lattices and principal separation in ordered sets

By a recent observation of Monjardet and Wille, a finite distributive lattice is generated by its doubly irreducible elements iff the poset of all join-irreducible elements has a distributive MacNeille completion. This fact is generalized in several directions, by dropping the finiteness condition and considering various types of bigeneration via arbitrary meets and certain distinguished joins. This leads to a deeper investigation of so-called L-generators resp. C-subbases, translating well-known notions of topology to order theory. A strong relationship is established between bigeneration by (minimal) L-generators and so-called principal separation, which is defined in order-theoretical terms but may be regarded as a strong topological separation axiom. For suitable L, the complete lattices with a smallest join-dense L-subbasis consisting of L-primes are the L-completions of principally separated posets.     

On the number of join-irreducibles in a congruence representation of a finite distributive lattice

For a finite lattice L, let $ \trianglelefteq_L $ denote the reflexive and transitive closure of the join-dependency relation on L, defined on the set J(L) of all join-irreducible elements of L. We characterize the relations of the form $ \trianglelefteq_L $, as follows: Theorem. Let $ \trianglelefteq $ be a quasi-ordering on a finite set P. Then the following conditions are equivalent:(i) There exists a finite lattice L such that $ \langle J(L), \trianglelefteq_L $ is isomorphic to the quasi-ordered set $ \langle P, \trianglelefteq \rangle $.(ii) $ |\{x\in P|p \trianglelefteq x\}| \neq 2 $, for any $ p \in P $.For a finite lattice L, let $ \mathrm{je}(L) = |J(L)|-|J(\mathrm{Con} L)| $ where Con L is the congruence lattice of L. It is well-known that the inequality $ \mathrm{je}(L) \geq 0 $ holds. For a finite distributive lattice D, let us define the join- excess function:$ \mathrm{JE}(D) =\mathrm{min(je} (L) | \mathrm{Con} L \cong D). $We provide a formula for computing the join-excess function of a finite distributive lattice D. This formula implies that $ \mathrm{JE}(D) \leq (2/3)| \mathrm{J}(D)|$ , for any finite distributive lattice D; the constant 2/3 is best possible.A special case of this formula gives a characterization of congruence lattices of finite lower bounded lattices.Dedicated to the memory of Gian-Carlo Rota     

A note on “small representations of finite distributive lattices as congruence lattices”

G. Grätzer, H. Lakser, and E. T. Schmidt proved that every distributive lattice with n join-irreducible elements can be represented as the congruence lattice of a small lattice L, that is, a lattice L with O(n ² ) elements. G. Grätzer, I. Rival, and N. Zaguia proved that, for any <2, O(n ² ) can not be improved to O(n ). In this note we show that the theorem about small representation can be improved further to get a more delicate result.