James B. Nation
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Research Interests
James B. Nation has a total of 33 co-authors. Here are the top co-authors by number of shared publications:
- Kira Adaricheva
Hofstra University
21 shared publications - Jennifer Hyndman
York University
13 shared publications - Joy N. Nishida
University of Hawaii at Manoa
11 shared publications - Gianluca Paolini
Department of Mathematics “Giuseppe Peano” University of Torino Torino Italy
4 shared publications - Ralph Freese
Department of Mathematics, University of Hawaii, Honolulu, USA
4 shared publications - George Grätzer
Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada
3 shared publications
Latest Publications
- 16 April 2026
- Published by Springer Nature in Selecta Mathematica
- Vol. 32 (2) , 41
Elementary properties of free lattices II: decidability of the universal theory
- 16 April 2026
- Published by Springer Nature in Selecta Mathematica
- Vol. 32 (2) , 41
Elementary properties of free lattices II: decidability of the universal theory
- Open Access
Elementary properties of free lattices III: Undecidability of the full theory
In [6] we proved that the universal theory of infinite free lattices is (algorithmically) decidable, leaving open the problem of decidability of the full theory of an (infinite) free lattice. We solve this problem by proving that, for every cardinal $κ\geq 3$, the first-order theory of the free lattice $\mathbf{F}_κ$ is undecidable.Open AccessElementary properties of free lattices III: Undecidability of the full theory
In [6] we proved that the universal theory of infinite free lattices is (algorithmically) decidable, leaving open the problem of decidability of the full theory of an (infinite) free lattice. We solve this problem by proving that, for every cardinal $κ\geq 3$, the first-order theory of the free lattice $\mathbf{F}_κ$ is undecidable. - Open Access
- 15 April 2025
- Published by Universidad de La Frontera in Cubo (Temuco)
- Vol. 27 (1) , 83-92
Congruences of infinite semidistributive lattices
Not every finite distributive lattice is isomorphic to the congruence lattice of a finite semidistributive lattice. This note provides a construction showing that many of these finite distributive lattices are isomorphic to congruence lattices of infinite semidistributive lattices.
- 15 April 2025
- Published by Universidad de La Frontera in Cubo (Temuco)
- Vol. 27 (1) , 83-92
Open AccessCongruences of infinite semidistributive lattices
Not every finite distributive lattice is isomorphic to the congruence lattice of a finite semidistributive lattice. This note provides a construction showing that many of these finite distributive lattices are isomorphic to congruence lattices of infinite semidistributive lattices.
- Open Access
- 1 October 2024
- Published by Universidad de La Frontera in Cubo (Temuco)
- Vol. 26 (3) , 443-473
Congruences of finite semidistributive lattices
We show that there are finite distributive lattices that are not the congruence lattice of any finite semidistributive lattice. For \(0 \leq k \leq 2\), the distributive lattice \((\mathbf{B}_k)_{++} = \mathbf{2} + \mathbf{B}_k\), where \(\mathbf{B}_k\) denotes the boolean lattice with \(k\) atoms, is not the congruence lattice of any finite semidistributive lattice. Neither can these lattices be a filter in the congruence lattice of a finite semidistributive lattice. However, each \((\mathbf{B}_k)_{++}\) with \(k \geq 3\) is the congruence lattice of a finite semidistributive lattice, say \(\mathbf{L}_k\). These lattices \(\mathbf{L}_k\) cannot be bounded (in the sense of McKenzie), as no \((\mathbf{B}_k)_{++}\) \((k \geq 0)\) is the congruence lattice of a finite bounded lattice. A companion paper shows that every \((\mathbf{B}_k)_{++}\) \((k \geq 0)\) can be represented as the congruence lattice of an infinite semidistributive lattice. We also find sufficient conditions for a finite distributive lattice to be representable as the congruence lattice of a finite bounded (and hence semidistributive) lattice.- 1 October 2024
- Published by Universidad de La Frontera in Cubo (Temuco)
- Vol. 26 (3) , 443-473
Open AccessCongruences of finite semidistributive lattices
We show that there are finite distributive lattices that are not the congruence lattice of any finite semidistributive lattice. For \(0 \leq k \leq 2\), the distributive lattice \((\mathbf{B}_k)_{++} = \mathbf{2} + \mathbf{B}_k\), where \(\mathbf{B}_k\) denotes the boolean lattice with \(k\) atoms, is not the congruence lattice of any finite semidistributive lattice. Neither can these lattices be a filter in the congruence lattice of a finite semidistributive lattice. However, each \((\mathbf{B}_k)_{++}\) with \(k \geq 3\) is the congruence lattice of a finite semidistributive lattice, say \(\mathbf{L}_k\). These lattices \(\mathbf{L}_k\) cannot be bounded (in the sense of McKenzie), as no \((\mathbf{B}_k)_{++}\) \((k \geq 0)\) is the congruence lattice of a finite bounded lattice. A companion paper shows that every \((\mathbf{B}_k)_{++}\) \((k \geq 0)\) can be represented as the congruence lattice of an infinite semidistributive lattice. We also find sufficient conditions for a finite distributive lattice to be representable as the congruence lattice of a finite bounded (and hence semidistributive) lattice. - 16 May 2024
- Published by Walter de Gruyter GmbH in Forum Mathematicum
- Vol. 37 (2) , 581-592
Elementary properties of free lattices
We start a systematic analysis of the first-order model theory of free lattices. Firstly, we prove that the free lattices of finite rank are not positively indistinguishable, as there is a positive -sentence true in- 16 May 2024
- Published by Walter de Gruyter GmbH in Forum Mathematicum
- Vol. 37 (2) , 581-592
Elementary properties of free lattices
We start a systematic analysis of the first-order model theory of free lattices. Firstly, we prove that the free lattices of finite rank are not positively indistinguishable, as there is a positive -sentence true in- Open Access
Elementary Properties of Free Lattices
We start a systematic analysis of the first-order model theory of free lattices. Firstly, we prove that the free lattices of finite rank are not positively indistinguishable, as there is a positive $\exists \forall$-sentence true in $\mathbf F_3$ and false in $\mathbf F_4$. Secondly, we show that every model of $\mathrm{Th}(\mathbf F_n)$ admits a canonical homomorphism into the profinite-bounded completion $\mathbf H_n$ of $\mathbf F_n$. Thirdly, we show that $\mathbf H_n$ is isomorphic to the Dedekind-MacNeille completion of $\mathbf F_n$, and that $\mathbf H_n$ is not positively elementarily equivalent to $\mathbf F_n$, as there is a positive $\forall\exists$-sentence true in $\mathbf H_n$ and false in $\mathbf F_n$. Finally, we show that $\mathrm{DM}(\mathbf F_n)$ is a retract of $\mathrm{Id}(\mathbf F_n)$ and that for any lattice $\mathbf K$ which satisfies Whitman's condition $\mathrm{(W)}$ and which is generated by join prime elements, the three lattices $\mathbf K$, $\mathrm{DM}(\mathbf K)$, and $\mathrm{Id}(\mathbf K)$ all share the same positive universal first-order theory.Open AccessElementary Properties of Free Lattices
We start a systematic analysis of the first-order model theory of free lattices. Firstly, we prove that the free lattices of finite rank are not positively indistinguishable, as there is a positive $\exists \forall$-sentence true in $\mathbf F_3$ and false in $\mathbf F_4$. Secondly, we show that every model of $\mathrm{Th}(\mathbf F_n)$ admits a canonical homomorphism into the profinite-bounded completion $\mathbf H_n$ of $\mathbf F_n$. Thirdly, we show that $\mathbf H_n$ is isomorphic to the Dedekind-MacNeille completion of $\mathbf F_n$, and that $\mathbf H_n$ is not positively elementarily equivalent to $\mathbf F_n$, as there is a positive $\forall\exists$-sentence true in $\mathbf H_n$ and false in $\mathbf F_n$. Finally, we show that $\mathrm{DM}(\mathbf F_n)$ is a retract of $\mathrm{Id}(\mathbf F_n)$ and that for any lattice $\mathbf K$ which satisfies Whitman's condition $\mathrm{(W)}$ and which is generated by join prime elements, the three lattices $\mathbf K$, $\mathrm{DM}(\mathbf K)$, and $\mathrm{Id}(\mathbf K)$ all share the same positive universal first-order theory. - Open Access
The Tight Upper Bound for the Size of Single Deletion Error Correcting Codes in Dimension 11
A single deletion error correcting code (SDECC) is a set of fixed-length sequences consisting of two types of symbols, 0 and 1, such that the original sequence can be recovered for at most one deletion error. The upper bound for the size of SDECC is expected to be equal to the size of Varshamov-Tenengolts (VT) code, and this conjecture had been shown to be true when the code length is ten or less. In this paper, we discuss a method for calculating this upper bound by providing an integer linear programming solver with several linear constraints. As a new result, we obtained that the tight upper bound for the size of a single deletion error correcting code in dimension 11 is 172.
Open AccessThe Tight Upper Bound for the Size of Single Deletion Error Correcting Codes in Dimension 11
A single deletion error correcting code (SDECC) is a set of fixed-length sequences consisting of two types of symbols, 0 and 1, such that the original sequence can be recovered for at most one deletion error. The upper bound for the size of SDECC is expected to be equal to the size of Varshamov-Tenengolts (VT) code, and this conjecture had been shown to be true when the code length is ten or less. In this paper, we discuss a method for calculating this upper bound by providing an integer linear programming solver with several linear constraints. As a new result, we obtained that the tight upper bound for the size of a single deletion error correcting code in dimension 11 is 172.
- Open Access
Planar, infinite, semidistributive lattices
An FN lattice $F$ is a simple, infinite, semidistributive lattice. Its existence was recently proved by R. Freese and J.\,B. Nation. Let $\mathsf{B}_n$ denote the Boolean lattice with $n$ atoms. For a lattice $K$, let $K^+$ denote $K$ with a new unit adjoined. We prove that the finite distributive lattices: $\mathsf{B}_0^+, \mathsf{B}_1^+,\mathsf{B}_2^+, \dots$ can be represented as congruence lattices of infinite semidistributive lattices. The case $n = 0$ is the Freese-Nation result, which is utilized in the proof. We also prove some related representation theorems.Open AccessPlanar, infinite, semidistributive lattices
An FN lattice $F$ is a simple, infinite, semidistributive lattice. Its existence was recently proved by R. Freese and J.\,B. Nation. Let $\mathsf{B}_n$ denote the Boolean lattice with $n$ atoms. For a lattice $K$, let $K^+$ denote $K$ with a new unit adjoined. We prove that the finite distributive lattices: $\mathsf{B}_0^+, \mathsf{B}_1^+,\mathsf{B}_2^+, \dots$ can be represented as congruence lattices of infinite semidistributive lattices. The case $n = 0$ is the Freese-Nation result, which is utilized in the proof. We also prove some related representation theorems. - 10 May 2022
- Published by Springer Nature
- p. 69-91
Equaclosure Operators
- 10 May 2022
- Published by Springer Nature
- p. 161-216
The Six-Step Program: From (L, γ) to $(\text{L}_{\text{q}}(\mathcal K),\varGamma )$- 10 May 2022
- Published by Springer Nature
- p. 161-216
The Six-Step Program: From (L, γ) to $(\text{L}_{\text{q}}(\mathcal K),\varGamma )$