arXiv subjects
Publications and source records attributed to G. Grätzer.
Let $A$ and $B$ be finite ordered sets. We show that if the ordered sets of isotone self-maps $A^A$ and $B^B$ (ordered pointwise) are isomorphic, then $A$ and $B$ are isomorphic. This resolves a question originating with D. Duffus in 1978, with related results by D. Duffus--R. Wille in 1979 and J. Farley in 2023.
We continue the study of exponent-cancellation for finite ordered sets. It is known that $A$ can be reconstructed from $A^{A}$, from $(A^{A})^{A}$, and from $A^{A^{A}}$. In this note we prove the next result in this hierarchy: the ordered set $(A^{A^{A}})^{A^{A^{A}}}$ determines $A$ up to isomorphism.
Let $A$ be a finite ordered set. Define the ordered set $A^A$ as the set of all maps from $A$ to $A$, ordered pointwise. Let ${}^{A} A$ be the dual of $A^A$. We prove results in the spirit of Parts~I--III, but now using both $A^A$ and ${}^{A}A$. For example, if \[ \Bigl({}^{{}^{ {}^{ {}^{A}A}A}A}A\Bigr)^{A^{A^{A}}} \] is isomorphic to \[ \Bigl({}^{ {}^{ {}^{ {}^{B}B}B}B}B\Bigr)^{B^{B^{B}}} \] for finite ordered sets $A$ and $B$, then $A$ is isomorphic to $B$.
In 1960, G. Grätzer and E.\,T. Schmidt proved that every finite distributive lattice can be represented as the congruence lattice of a sectionally complemented finite lattice $L$. For $u \leq v$ in $L$, they constructed a sectional complement, which is now called the \emph{1960 sectional complement}. In 1999, G. Grätzer and E.\,T. Schmidt discovered a very simple way of constructing a sectional complement in the ideal lattice of a chopped lattice made up of two sectionally complemented finite lattices overlapping in only two elements -- the Atom Lemma. The question was raised whether this simple process can be generalized to an algorithm that finds the 1960 sectional complement. In 2006, G.~Grätzer and M. Roddy discovered such an algorithm -- allowing a wide latitude how it is carried out. In this paper we prove that the wide latitude apparent in the algorithm is deceptive: whichever way the algorithm is carried out, it~produces the same sectional complement. This solves, in fact, Problems 2 and 3 of the Grätzer-Roddy paper. Surprisingly, the unique sectional complement provided by the algorithm is the 1960 sectional complement, solving Problem 1 of the same paper.
If $A$ is an algebra and \bgt is a tolerance on $A$, then $A/\bgt$ is a multi-algebra in a natural way. We give an example to show that not every multi-algebra arises in this manner. We slightly generalize the construction of $A/\bgt$ and prove that every multi-algebra arises from this modified construction.
In a recent paper, G. Czédli and E.\,T. Schmidt present a structure theorem for planar semimodular lattices. In this note, we present an alternative proof.
A new result of G. Czédli states that for an ordered set $P$ with at least two elements and a group $G$, there exists a bounded lattice $L$ such that the ordered set of principal congruences of $L$ is isomorphic to $P$ and the automorphism group of $L$ is isomorphic to $G$. I provide an alternative proof utilizing a result of mine with J. Sichler from the late 1960-s.
A recent result of G. Czédli relates the ordered set of principal congruences of a bounded lattice $L$ with the ordered set of principal congruences of a~bounded sublattice $K$ of $L$. In this note, I sketch a new proof.
Let $L$ be a slim, planar, semimodular lattice (slim means that it does not contain ${\mathsf M}_3$-sublattices). We call the interval $I = [o, i]$ of $L$ \emph{rectangular}, if there are $u_l, u_r \in [o, i] - \{o,i\}$ such that $o = u_l \wedge u_r$ and $i = u_l \vee u_r$, where $u_l$ is to the left of $u_r$. We prove that a rectangular interval of a rectangular lattice is a rectangular lattice. As an application, we get a recent result of G. Czédli.
Let the finite distributive lattice $D$ be isomorphic to the congruence lattice of a finite lattice $L$. Let $Q$ denote those elements of $D$ that correspond to principal congruences under this isomorphism. Then $Q$ contains $0,1 \in D$ and all the join-irreducible elements of $D$. If $Q$ contains exactly these elements, we say that $L$ is a minimal representations of $D$ by principal congruences of the lattice $L$. We characterize finite distributive lattices $D$ with a minimal representation by principal congruences with the property that $D$ has at most two dual atoms.
In this note, we discuss planar lattices generated by their atoms. We prove that if $L$ is a planar lattice generated by $n$ atoms, then both the left and the right boundaries of $L$ have at most $n+1$ elements. On the other hand, $L$ can be arbitrarily large. For every $k > 1$, we construct a planar lattice $L$ generated by $4$ atoms such that $L$ has more than $k$ elements.
A classical result of R.\,P. Dilworth states that every finite distributive lattice $D$ can be represented as the congruence lattice of a finite lattice~$L$. A~sharper form was published in G.~Grätzer and E.\,T. Schmidt in 1962, adding the requirement that all congruences in $L$ be principal. Another variant, published in 1998 by the authors and E.\,T. Schmidt, constructs a planar semimodular lattice $L$. In this paper, we merge these two results: we construct $L$ as a planar semimodular lattice in which all congruences are principal. This paper relies on the techniques developed by the authors and E.\,T. Schmidt in the 1998 paper.
In this note, I find a new property of the congruence lattice, Con$L$, of an SPS lattice $L$ (slim, planar, semimodular, where "slim" is the absence of~$\mathsf M_3$ sublattices) with more than $2$ elements: \emph{there are at least two dual atoms in Con$L$}. So the three-element chain cannot be represented as the congruence lattice of an SPS lattice, supplementing a~result of G. Czédli.
Following G.~Grätzer and E.~Knapp, 2009, a planar semimodular lattice $L$ is \emph{rectangular}, if~the left boundary chain has exactly one doubly-irreducible element, $c_l$, and the right boundary chain has exactly one doubly-irreducible element, $c_r$, and these elements are complementary. The Czédli-Schmidt Sequences, introduced in 2012, construct rectangular lattices. We use them to prove some structure theorems. In particular, we prove that for a slim (no $\mathsf{M}_3$ sublattice) rectangular lattice~$L$, the congruence lattice $\Con L$ has exactly $\length[c_l,1] + \length[c_r,1]$ dual atoms and a dual atom in $\Con L$ is a congruence with exactly two classes. We also describe the prime ideals in a slim rectangular lattice.
In the second edition of the congruence lattice book, Problem 22.1 asks for a characterization of subsets $Q$ of a finite distributive lattice $D$ such that there is a finite lattice $L$ whose congruence lattice is isomorphic to $D$ and under this isomorphism $Q$ corresponds the the principal congruences of $L$. In this note, we prove some preliminary results.
IIn a finite lattice, a congruence spreads from a prime interval to another by a sequence of congruence-perspectivities through \emph{intervals of arbitrary size}, by a 1955 result of J. Jakubík. In this note, I introduce the concept of \emph{prime-perspectivity} and prove the Prime-projectivity Lemma: a congruence spreads from a prime interval to another by a sequence of prime-perspectivities through \emph{prime ntervals}. A planar semimodular lattice is \emph{slim} if it contains no $\mathsf{M}_3$ sublattice. I introduce the Swing Lemma, a very strong version of the Prime-projectivity Lemma for slim, planar, semimodular lattices.
Let $L$ be a lattice. We call a congruence relation $\gQ$ of $L$ isoform, if any two congruence classes of $\gQ$ are isomorphic (as lattices). Let us call the lattice $L$ isoform, if all congruences of $L$ are isoform. G. Grätzer and E.\,T. Schmidt proved that every finite distributive lattice $D$ can be represented as the congruence lattice of a finite isoform lattice $L$. We now prove that every finite lattice $K$ has a congruence-preserving extension to a finite isoform lattice $L$.
We characterize the order of principal congruences of a bounded lattice as a bounded ordered set. We also state a number of open problems in this new field.