Notes on sectionally complemented lattices. II

Summary G. Grätzer and H. Lakser proved in 1986 that for the finite distributive lattices D and E, with |D| > 1, and for the {0, 1}-homomorphism φ of D into E, there exists a finite lattice L and an ideal I of L such that D ≡ Con L, E ≡ Con I, and φ is represented by the restriction map. In their recent survey of finite congruence lattices, G. Grätzer and E. T. Schmidt ask whether this result can be improved by requiring that L be sectionally complemented. In this note, we provide an affirmative answer. The key to the solution is to generalize the 1960 sectional complement (see Part I) from finite orders to finite preorders.     

A Lower Bound for Congruence Representations

G. Grätzer, H. Lakser, and E. T. Schmidt proved that every finite distributive lattice D with n join-irreducible elements can be represented as the congruence lattice of a lattice L with O(n2) elements. G. Grätzer, I. Rival, and N. Zaguia gave kn, < 2, as a lower bound for the size of such a lattice L; a sharper form, 1/64(n/log2n)2, of this result was given by Y. Zhang.In this note, we apply a recent result of R. Freese, to obtain 1/16 n2/log2n as a lower bound. We also give a direct proof of Freese's result.     

A representation of m-algebraic lattices

In this paper, we give a short proof of the following result of G. Grätzer and E. T. Schmidt: every m-algebraic lattice can be represented as the lattice of m-complete congruence relations of some m-complete modular lattice.Dedicated to Bjarni Jonsson on his 70th birthdayThe research of the first author was supported by the NSERC of Canada.The research of the third author was supported by the Hungarian National Foundation for Scientific Research, under Grant No. 1903.     

Notes on sectionally complemented lattices. IV How far does the Atom Lemma go?

There are two results in the literature that prove that the ideal lattice of a finite, sectionally complemented, chopped lattice is again sectionally complemented. The first is in the 1962 paper of G. Grätzer and E. T. Schmidt, where the ideal lattice is viewed as a closure space to prove that it is sectionally complemented; we call the sectional complement constructed then the 1960 sectional complement. The second is the Atom Lemma from a 1999 paper of the same authors that states that if a finite, sectionally complemented, chopped lattice is made up of two lattices overlapping in an atom and a zero, then the ideal lattice is sectionally complemented. In this paper, we show that the method of proving the Atom Lemma also applies to the 1962 result. In fact, we get a stronger statement, in that we get many sectional complements and they are rather close to the componentwise sectional complement.     

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.     

Notes on sectionally complemented lattices. III

Summary In a recent survey article, G. Grätzer and E. T. Schmidt raise the problem when is the ideal lattice of a sectionally complemented chopped lattice sectionally complemented. The only general result is a 1999 lemma of theirs, stating that if the finite chopped lattice is the union of two ideals that intersect in a two-element ideal U, then the ideal lattice of M is sectionally complemented. In this paper, we present examples showing that in many ways their result is optimal. A typical result is the following: For any finite sectionally complemented lattice U with more than two elements, there exists a finite sectionally complemented chopped lattice M that is (i) the union of two ideals intersecting in the ideal U; (ii) the ideal lattice of M is not sectionally complemented.     

Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices

Let be a {0, 1}-homomorphism of a finite distributive lattice D into the congruence lattice Con L of a rectangular (whence finite, planar, and semimodular) lattice L. We prove that L is a filter of an appropriate rectangular lattice K such that ConK is isomorphic with D and is represented by the restriction map from Con K to Con L. The particular case where is an embedding was proved by E.T. Schmidt. Our result implies that each {0, 1}-lattice homomorphism between two finite distributive lattices can be represented by the restriction of congruences of an appropriate rectangular lattice to a rectangular filter.     

On complete congruence lattices of join-infinite distributive lattices

In [5] G. Gr?tzer and E. T. Schmidt raised the problem of characterizing the complete congruence lattices of complete lattices satisfying the Join-Infinite Distributive Identity (JID) and the Meet-Infinite Distributive Identity (MID) and proved the theorem: Any complete lattice with more than two elements and with a meet-irreducible zero cannot be represented as the lattice of complete congruence relations of a complete lattice satisfying the (JID) and (MID). In this note we generalize this result by showing that the complete congruence lattice of every complete lattice satisfying (JID) and (MID) is a zero-dimensional complete lattice satisfying (JID). Some consequences are discussed. Received March 6, 2000; accepted in final form September 12, 2000.     

Characterizing fully principal congruence representable distributive lattices

Motivated by a recent paper of G. Grätzer, a finite distributive lattice D is called fully principal congruence representable if for every subset Q of D containing 0, 1, and the set J(D) of nonzero join-irreducible elements of D, there exists a finite lattice L and an isomorphism from the congruence lattice of L onto D such that Q corresponds to the set of principal congruences of L under this isomorphism. A separate paper of the present author contains a necessary condition of full principal congruence representability: D should be planar with at most one join-reducible coatom. Here we prove that this condition is sufficient. Furthermore, even the automorphism group of L can arbitrarily be stipulated in this case. Also, we generalize a recent result of G. Grätzer on principal congruence representable subsets of a distributive lattice whose top element is join-irreducible by proving that the automorphism group of the lattice we construct can be arbitrary.     

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.     

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.     

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.     

Lattice Tensor Products i. Coordinatization

G. Grätzer and F. Wehrung introduced the lattice tensor product, A B, of the lattices A and B. One of the most important properties is that for a simple and bounded lattice A, the lattice A B is a congruence-preserving extension of B. The lattice A B is defined as the set of certain subsets of A B; there is no easy test when a subset belongs to A B. A special case, M ₃B, was earlier defined by G. Gräatzer and F. Wehrung as M ₃, the it Boolean triple construct, defined as a subset of B ³, with a simple criterion when a triple belongs. A~recent paper of G. Grätzer and E. T. Schmidt illustrates the importance of this Boolean triple arithmetic. In this paper we show that for any finite lattice A, we can ``coordinatize" A B, that is, represent A B as a subset of B ⁿ (where n is the number of join-irreducible elements of A), and provide an effective criteria to recognize the n-tuples of elements of B that occur in this representation. To show the utility of this coordinatization, we reprove a special case of the above result: for a finite simple lattice A, the lattice A B is a congruence-preserving extension of B.     

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.     

Combinatorial partitions of finite posets and lattices —Ramsey lattices

It is proved that for every finite latticeL there exists a finite latticeL such that for every partition of the points ofL into two classes there exists a lattice embeddingf:LL such that the points off(L) are in one of the classes.This property is called point-Ramsey property of the class of all finite lattices. In fact a stronger theorem is proved which implies the following: for everyn there exists a finite latticeL such that the Hasse-diagram (=covering relation) has chromatic number >n. We discuss the validity of Ramseytype theorems in the classes of finite posets (where a full discussion is given) and finite distributive lattices. Finally we prove theorems which deal with partitions of lattices into an unbounded number of classes.Presented by G. Grätzer.     

Lattice generation of small equivalences of a countable set

Given a countable set A, let Equ(A) denote the lattice of equivalences of A. We prove the existence of a four-generated sublattice Q of Equ(A) such that Q contains all atoms of Equ(A). Moreover, Q can be generated by four equivalences such that two of them are comparable. Our result is a reasonable generalization of Strietz [5, 6] from the finite case to the countable one; and in spite of its essentially simpler proof it asserts more for the countable case than [2, 3].Dedicated to George Grätzer on his 60th birthdayThis research was supported by the NFSR of Hungary (OTKA), grant no. T7442.     

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.     

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     

Freely adjoining a relative complement to a lattice

Let K be a lattice, and let a < b < c be elements of K. We adjoin freely a relative complement u of b in [a, c] to K to form the lattice L. For two polynomials A and B over K ∪ {u}, we find a very simple set of conditions under which A and B represent the same element in L, so that in L all pairs of relative complements in [a, c] can be described. Our major result easily follows: Let [a, c] be an interval of a lattice K; let us assume that every element in [a, c] has at most one relative complement. Then K has an extension L such that [a, c] in L, as a lattice, is uniquely complemented.As an immediate consequence, we get the classical result of R. P. Dilworth: Every lattice can be embedded into a uniquely complemented lattice. We also get the stronger form due to C. C. Chen and G. Grätzer: Every at most uniquely complemented bounded lattice has a {0, 1}-embedding into a uniquely complemented lattice. Some stronger forms of these results are also presented.A polynomial A over K ∪ {u} naturally represents an element 〈A 〉 of L. Let us call a polynomial A minimal, if it is of minimal length representing x. We characterize minimal polynomials.Dedicated to the memory of Ivan RivalReceived February 12, 2003; accepted in final form June 18, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Interpolation of monotone functions in lattices

We show that for every latticeL ₀ and for every cardinal there is a lattice on which every monotone function can be interpolated by a polynomial on any set of size .Presented by G. Grätzer.Dedicated to '"N