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Jonathan David Farley

Publications and source records attributed to Jonathan David Farley.

10 recordsLinked to original sources
arXiv2022

A Conjecture of Kozlov from the 1998 Proceedings of the American Mathematical Society: Non-Evasive Order Complexes and Generalizations of Non-Complemented Lattices

Let $P$ be a finite poset with an element $s$ such that (1) for all $x\in P$, either $s\vee x$ or $s\wedge x$ exists; and (2) for all $x,y\in P$ such that $x<y$, if $s\wedge x$ does not exist but $s\wedge y$ does exist, then $(s\wedge y)\vee x$ exists. Kozlov, the winner of the 2005 European Prize in Combinatorics ("for deep combinatorial results obtained by algebraic topology and particularly for the solution of a conjecture of Lovász"), conjectured in the 1998 Proceedings of the American Mathematical Society that the order complex of $P$ is non-evasive. We prove this conjecture.

math.CO↗
arXiv2021

On a Question of Grätzer and Lakser from the 1971 {\sl Transactions of the American Mathematical Society}

Grätzer and Lakser asked in the 1971 {\sl Transactions of the American Mathematical Society} if the pseudocomplemented distributive lattices in the amalgamation class of the subvariety generated by ${\bf 2}^n\oplus{\bf 1}$ can be characterized by the property of not having a $*$-homomorphism onto ${\bf 2}^i\oplus{\bf 1}$ for $1<i<n$. In this article, this question is answered. If you want to know the answer, you will have to read it (or skip to the last section).

math.CO↗
arXiv2020

Another Problem of Jónsson and McKenzie from 1982: Refinement Properties for Connected Powers of Posets

In 1982, Jónsson and McKenzie posed the following problem: "Find counter examples (or prove that none exist) to the refinement of $A^C\cong B^D$ [$A$, $B$, $C$, and $D$ non-empty posets] under" the condition "$C$, $D$, and $A^C$ are finite and connected." That is, in this situation, are there posets $E$, $X$, $Y$, and $Z$ such that $A\cong E^X$, $B\cong E^Y$, $C\cong Y\times Z$, and $D\cong X\times Z$? In this note, this problem is solved.

math.CO↗
arXiv2020

An Issue Raised in 1978 by a Then-Future Editor-in-Chief of the Journal "Order": Does the Endomorphism Poset of a Finite Connected Poset Tell Us That the Poset Is Connected?

In 1978, Dwight Duffus---editor-in-chief of the journal "Order" from 2010 to 2018 and chair of the Mathematics Department at Emory University from 1991 to 2005---wrote that "it is not obvious that $P$ is connected and $P^P$ isomorphic to $Q^Q$ implies that $Q$ is connected," where $P$ and $Q$ are finite non-empty posets. We show that, indeed, under these hypotheses $Q$ is connected and $P\cong Q$.

math.CO↗
arXiv2015

On incomplete lattice homomorphisms in subspaces of geometries: "half" a problem of Hartmanis from 1959

Turing Award winner Juris Hartmanis introduced in 1959 lattices of subspaces of generalized partitions ("partitions of type n"; "geometries" if $n = 2$). Hartmanis states it is "an unsolved problem whether there are any incomplete lattice homomorphisms in" lattices of subspaces of geometries. (He continues, "[I]f so how can these geometries be characterized.") We give a positive answer to this question.

math.LO↗
arXiv2004

Chain polynomials of distributive lattices are 75 % unimodal

It is shown that the numbers $c_i$ of chains of length $i$ in the proper part $L\setminus\{0,1\}$ of a distributive lattice $L$ of length $\ell +2$ satisfy the inequalities $$c_0<... ...>c_{\ell}.$$ This proves 75 % of the inequalities implied by the Neggers unimodality conjecture.

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arXiv1998

The Fixed Point Property for Posets of Small Width

The fixed point property for finite posets of width 3 and 4 is studied in terms of forbidden retracts. The ranked forbidden retracts for width 3 and 4 are determined explicitly. The ranked forbidden retracts for the width 3 case that are linearly indecomposable are examined to see which are minimal automorphic. Part of a problem of Niederle from 1989 is thus solved.

math.CO↗
arXiv1997

Chain Decomposition Theorems for Ordered Sets (and Other Musings)

A brief introduction to the theory of ordered sets and lattice theory is given. To illustrate proof techniques in the theory of ordered sets, a generalization of a conjecture of Daykin and Daykin, concerning the structure of posets that can be partitioned into chains in a ``strong'' way, is proved. The result is motivated by a conjecture of Graham's concerning probability correlation inequalities for linear extensions of finite posets.

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