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George Grätzer

Publications and source records attributed to George Grätzer.

At least 37 records · Page 2Linked to original sources
arXiv2015

Homomorphisms and principal congruences of bounded lattices

Two years ago, I characterized the order $\Princl L$ of principal congruences of a bounded lattice $L$ as a bounded order. If $K$ and $L$ are bounded lattices and $\gf$ is a \zo homomorphism of $K$ into~$L$, then there is a natural isotone \zo-map $\gf_{\Hom}$ from $\Princl K$ into $\Princl L$. We prove the converse: For bounded orders $P$ and $Q$ and an isotone \zo map $\gy$ of $P$ into $Q$, we represent $P$ and $Q$ as $\Princl K$ and $\Princl L$ for bounded lattices $K$ and $L$ with a \zo homomorphism $\gf$ of $K$ into $L$, so that $\gy$ is represented as $\gf_{\Hom}$.

math.RA↗
arXiv2015

Congruences in slim, planar, semimodular lattices: The Swing Lemma

In an earlier paper, to describe how a congruence spreads from a prime interval to another in a finite lattice, I introduced the concept of prime-perspectivity and its transitive extension, prime-projectivity and proved the Prime-projectivity Lemma. In this paper, I specialize the Prime-projectivity Lemma to slim, planar, semimodular lattices to obtain the Swing Lemma, a very powerful description of the congruence generated by a prime interval in this special class of lattices.

math.RA↗
arXiv2014

Congruences and trajectories in planar semimodular lattices

A 1955 result of J.~Jakubí k states that for the prime intervals $\fp$ and $\fq$ of a finite lattice, $\con{\fp} \geq \con{\fq}$ if{}f $\fp$ is congruence-projective to~$\fq$ (\emph{via} intervals of arbitrary size). The problem is how to determine whether $\con{\fp} \geq \con{\fq}$ involving only prime intervals. Two recent papers approached this problem in different ways. G. Czédli's used trajectories for slim rectangular lattices---a special subclass of slim, planar, semimodular lattices. I used the concept of prime-projectivity for arbitrary finite lattices. In this note I show how my approach can be used to generalize Czédli's result to arbitrary slim, planar, semimodular lattices.

math.RA↗
arXiv2014

Congruences of fork extensions of slim semimodular lattices

For a slim, planar, semimodular lattice $L$ and covering square~$S$, G.~Czédli and E.\,T.~Schmidt introduced the fork extension, $L[S]$, which is also a slim, planar, semimodular lattice. We investigate when a congruence of $L$ extends to $L[S]$. We introduce a join-irreducible congruence $\boldsymbolγ(S)$ of $L[S]$. We determine when it is new, in the sense that it is not generated by a join-irreducible congruence of $L$. When it is new, we describe the congruence $\boldsymbolγ(S)$ in great detail. The main result follows: \emph{In the order of join-irreducible congruences of a slim, planar, semimodular lattice $L$, the congruence $\boldsymbolγ(S)$ has \emph{at most two covers.}}

math.RA↗
arXiv2013

Representing finite distributive lattices as congruence lattices of lattices

Dilworth's theorem. Every finite distributive lattice $D$ can be represented as the congruence lattice of a finite lattice $L$. We want: Every finite distributive lattice $D$ can be represented as the congruence lattice of a nice finite lattice $L$. nice = sectionally complemented, uniform, semimodular, given automorphism group, regular, uniform, isoform

math.RA↗
arXiv2013

Congruences of the fork extensions. I. The Congruence Extension Property

For a slim, planar, semimodular lattice, G. Czédli and E.\,T. Schmidt introduced the fork extension in 2012. In this note we prove that the fork extension has the Congruence Extension Property. This paper has been merged with Part II, under the title Congruences of fork extensions of slim semimodular lattices, see arXiv: 1307.8404

math.RA↗
arXiv2013

Congruences of the fork extensions. II. The congruence gamma

G. Czédli and E.\,T. Schmidt introduced in 2012 the fork extension. Continuing from Part I, we investigate the congruences of a fork extension. This paper has been merged with Part I, under the title Congruences of fork extensions of slim semimodular lattices, see arXiv: 1307.8404

math.RA↗
arXiv2013

A technical lemma for congruences of finite lattices

The classical Technical Lemma for congruences is not difficult to prove but it is very efficient in its applications. We present here a Technical Lemma for congruences on \emph{finite lattices}. This is not difficult to prove either but it has already has proved its usefulness in some applications.

math.RA↗
arXiv2005

The M\_{3}[D] construction and n-modularity

In 1968, E. T. Schmidt introduced the M\_3[D] construction, an extension of the five-element nondistributive lattice M\_3 by a bounded distributive lattice D, defined as the lattice of all triples $(x, y, z) \in D^3$ satisfying $x \mm y = x \mm z = y \mm z$. 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 Un such that U1 is modularity and Un+1 is properly weaker than Un. Let Mn denote the variety defined by Un, 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, $IdM\_3[L] \cong M\_3[Id L]$. 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 R. W. Quackenbush: Is the tensor product $A\otimes B$ 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 $M\_4 \otimes L$ 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 R. P. 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 G. Grätzer, H. Lakser, and E. T. Schmidt yields a 3-modular lattice.

math.GM↗
arXiv2005

Flat semilattices

Let $A$, $B$, and $S$ be (v,0)-semilattices and let $f: A\to B$ be a (v,0)-embedding. Then the canonical map, $f \otimes \id\_S$, of the tensor product $A \otimes S$ into the tensor product $B \otimes S$ is not necessarily an embedding. The (v,0)-semilattice $S$ is flat, if for every embedding $f : A\to B$, the canonical map $f\otimes\id$ is an embedding. We prove that a (v,0)-semilattice $S$ is flat if and only if it is distributive.

math.GM↗
arXiv2005

Tensor products of semilattices with zero, revisited

Let A and B be lattices with zero. The classical tensor product, $A\otimes B$, of A and B as join-semilattices with zero is a join-semilattice with zero; it is, in general, not a lattice. We define a very natural condition: $A \otimes B$ is capped (that is, every element is a finite union of pure tensors) under which the tensor product is always a lattice. Let Conc L denote the join-semilattice with zero of compact congruences of a lattice L. Our main result is that the following isomorphism holds for any capped tensor product: $Conc A\otimes Conc B \cong Conc(A \otimes B)$. This generalizes from finite lattices to arbitrary lattices the main result of a joint paper by the first author, H. Lakser, and R. W. Quackenbush.

math.GM↗
arXiv2005

A new lattice construction: the box product

In a recent paper, the authors have proved that for lattices A and B with zero, the isomorphism $Conc(A \otimes B)\cong Conc A \otimes Conc B$, holds, provided that the tensor product satisfies a very natural condition (of being capped) implying that $A\otimes B$ is a lattice. In general, $A \otimes B$ is not a lattice; for instance, we proved that $M\_3\otimes F(3)$ is not a lattice. In this paper, we introduce a new lattice construction, the box product for arbitrary lattices. The tensor product construction for complete lattices introduced by G. N. Raney in 1960 and by R. Wille in 1985 and the tensor product construction of A. Fraser in 1978 for semilattices bear some formal resemblance to the new construction. For lattices A and B, while their tensor product $A\otimes B$ (as semilattices) is not always a lattice, the box product, $A \bp B$, is always a lattice. Furthermore, the box product and some of its ideals behave like an improved tensor product. For example, if A and B are lattices with unit, then the isomorphism $Conc(A $\bp$ B) \cong Conc A \otimes Conc B$ holds. There are analogous results for lattices A and B with zero and for a bounded lattice A and an arbitrary lattice B. A join-semilattice S with zero is called {0}-representable, if there exists a lattice L with zero such that $Conc L \cong S$. The above isomorphism results yield the following consequence: The tensor product of two {0}-representable semilattices is {0}-representable.

math.GM↗
arXiv2005

Tensor products and transferability of semilattices

In general, the tensor product, $A\otimes B$, of the lattices A and B with zero is not a lattice (it is only a join-semilattice with zero). If $A \otimes B$ is a capped tensor product, then $A \otimes B$ is a lattice (the converse is not known). In this paper, we investigate lattices A with zero enjoying the property that $A \otimes B$ is a capped tensor product, for every lattice B with zero; we shall call such lattices amenable. The first author introduced in 1966 the concept of a sharply transferable lattice. In 1972, H. Gaskill [5] defined, similarly, sharply transferable semilattices, and characterized them by a very effective condition (T). We prove that a finite lattice A is amenable iff it is sharply transferable as a join-semilattice. For a general lattice A with zero, we obtain the result: A is amenable iff A is locally finite and every finite sublattice of A is transferable as a join-semilattice. This yields, for example, that a finite lattice A is amenable iff $A\otimes F(3)$ is a lattice iff A satisfies (T), with respect to $\jj$. In particular, $M3 \otimes F(3)$ is not a lattice. This solves a problem raised by R. W. Quackenbush in 1985 whether the tensor product of lattices with zero is always a lattice.

math.GM↗
arXiv2005

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

For a finite lattice L, let EL 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 EL, as follows: Theorem. Let E be a quasi-ordering on a finite set P. Then the following conditions are equivalent: (i) There exists a finite lattice L such that (J(L),EL) is isomorphic to the quasi-ordered set (P,E). (ii) There are not exactly two elements x in P such that p E x, for any p in P. For a finite lattice L, let je(L) = |J(L)|-|J(Con L)|, where Con L is the congruence lattice of L. It is well-known that the inequality je(L) $\ge$ 0 holds. For a finite distributive lattice D, let us define the join-excess function: JE(D) = min(je(L) | Con L isomorphic to D). We provide a formula for computing the join-excess function of a finite distributive lattice D. This formula implies that JE(D) $\le$ (2/3)| 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.

math.GM↗