Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2208.03745

On the algorithmic construction of the 1960 sectional complement

G. Grätzer·G. Klus·A. Nguyen
View original source ↗Read source document ↗Export metadata ↓

Abstract

In 1960, G. Gr\"atzer 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. Gr\"atzer 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.~Gr\"atzer 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 Gr\"atzer-Roddy paper. Surprisingly, the unique sectional complement provided by the algorithm is the 1960 sectional complement, solving Problem 1 of the same paper.

Explore related subjects

math.RA

Keep this discovery

BibTeXRIS

G. Grätzer, G. Klus, A. Nguyen. 2022-08-07. On the algorithmic construction of the 1960 sectional complement. https://arxiv.org/abs/2208.03745

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Invariants of Nilpotent Lie Algebras via Geometry and Algebra with a Focus on Computation

We consider the problem of computing rational invariants of nilpotent Lie algebras. We compare two methods that are commonly used for this task: the method of integral curves and the Dixmier map. Given a derivation of a rational function field with polynomial coefficients, we formulate a condition under which the kernel can be recovered from a family of rational integral curves, and we show that triangular derivations satisfy this hypothesis. This yields an explicit description of the kernel as a purely transcendental extension and produces algebraically independent generators. We also show that, in the triangular case, the resulting generators agree with those obtained from the Dixmier map via a local slice. A careful analysis of the generating set obtained from this method leads to an algorithm for computing generators of the rational invariant field of a nilpotent Lie algebra. An implementation of the methods is available in the SageMath system.

math.RA↗
arXiv2026

Quasilinear multiplication in the real Cayley--Dickson tower

Direct evaluation of the defining product in the real Cayley--Dickson algebra $A_n$, of dimension $N=2^n$, has quadratic arithmetic complexity. This paper gives a uniform algorithm for multiplication using $O(N\log N)$ real arithmetic operations and $O(N)$ auxiliary storage. The algorithm reduces multiplication to the alternating product on the imaginary subspace, then evaluates that product by a two-call recursion over one fixed quadratic coefficient extension. For $n\ge1$, the resulting bilinear algorithm uses at most $(9n-15)2^{n-1}+10$ input-dependent real multiplications, and for $n\ge3$, the specified arithmetic schedule uses $(34n-83)2^{n-1}+50$ real operations in total. Under this counting convention, the quasilinear schedule uses fewer operations than direct multiplication for $N\ge16$ and than the uniform Cariow--Cariowa method for $N\ge32$. The algorithm is implemented in the MIT-licensed C11 library fastCD, with a NumPy-backed Python interface, and its results are checked against an independent implementation of the defining recursion. In single-core benchmarks against direct multiplication and the uniform Cariow--Cariowa method, the quasilinear implementation had the lowest mean time of the three at every tested dimension $N\ge32$, for both single and batched products, and was roughly $16$ times faster than direct multiplication at $N=1024$.

math.RA↗
arXiv2026

Graded classification of Leavitt path algebras in terms of strong shift equivalence

Given two finite essential adjacency matrices $A$ and $B$, Hazrat's graded classification conjectures posit that an order preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism of $K_0$ groups implies graded Morita equivalence of the Leavitt path algebras of $A$ and $B$, while the pointed version predicts a graded isomorphism of the Leavitt path algebras when the $K_0$ group isomorphism additionally preserves the class of the regular module. For any field $k$, we show that the Leavitt path algebras over $k$ of $A$ and $B$ are graded Morita equivalent if and only if $A$ and $B$ are strong shift equivalent. By appealing to counterexamples of Kim and Roush from symbolic dynamics, this shows that Hazrat's graded classification conjectures are false.

math.RA↗