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Papers by James Nation

Journal of the Australian Mathematical Society, 1997

We investigate the number and size of the maximal sublattices of a finite lattice. For any positi... more We investigate the number and size of the maximal sublattices of a finite lattice. For any positive integerk, there is a finite lattice L with more that ]L]ksublattices. On the other hand, there are arbitrary large finite lattices which contain a maximal sublattice with only 14 elements. It is shown that every bounded lattice is isomorphic to the Frattini sublattice (the intersection of all maximal sublattices) of a finite bounded lattice.

BioData mining, 2016

Technological advances enable the cost-effective acquisition of Multi-Modal Data Sets (MMDS) comp... more Technological advances enable the cost-effective acquisition of Multi-Modal Data Sets (MMDS) composed of measurements for multiple, high-dimensional data types obtained from a common set of bio-samples. The joint analysis of the data matrices associated with the different data types of a MMDS should provide a more focused view of the biology underlying complex diseases such as cancer that would not be apparent from the analysis of a single data type alone. As multi-modal data rapidly accumulate in research laboratories and public databases such as The Cancer Genome Atlas (TCGA), the translation of such data into clinically actionable knowledge has been slowed by the lack of computational tools capable of analyzing MMDSs. Here, we describe the Joint Analysis of Many Matrices by ITeration (JAMMIT) algorithm that jointly analyzes the data matrices of a MMDS using sparse matrix approximations of rank-1. The JAMMIT algorithm jointly approximates an arbitrary number of data matrices by ra...

Order, 2014

This note gives a complete characterization of when the ordinal sum of two lattices (the lattice ... more This note gives a complete characterization of when the ordinal sum of two lattices (the lattice obtained by placing the second lattice on top of the first) is projective. This characterization applies not only to the class of all lattices, but to any variety of lattices, and in particular, to the class of distributive lattices. Lattices L with the property that every epimorphism onto L has an isotone section are also characterized.

Order

This note gives a complete characterization of when the ordinal sum of two lattices (the lattice ... more This note gives a complete characterization of when the ordinal sum of two lattices (the lattice obtained by placing the second lattice on top of the first) is projective. This characterization applies not only to the class of all lattices, but to any variety of lattices, and in particular, to the class of distributive lattices. Lattices L with the property that every epimorphism onto L has an isotone section are also characterized.

Group codes that use certain complex matrix groups are analyzed. These groups are complex analogu... more Group codes that use certain complex matrix groups are analyzed. These groups are complex analogues of the real Coxeter re- flection groups. It is shown that some types of complex permutation groups have good properties for use in coding, while others do not.

Pacific Journal of Mathematics, 1978

This paper gives necessary and sufficient conditions for a lattice to be projective. The conditio... more This paper gives necessary and sufficient conditions for a lattice to be projective. The conditions are the Whitman condition, and a condition of Jόnsson, and two new conditions explained below.

Order, 1989

ABSTRACT A useful construction for lattices is doubling a convex subsetI of a latticeL, i.e., rep... more ABSTRACT A useful construction for lattices is doubling a convex subsetI of a latticeL, i.e., replacingI byI2. It is shown that this construction preserves a generalized semidistributivity condition (C). Varieties of lattices in which every lattice satisfies (C) are characterized equationally.

Journal of the Australian Mathematical Society, 1997

We investigate the number and size of the maximal sublattices of a finite lattice. For any positi... more We investigate the number and size of the maximal sublattices of a finite lattice. For any positive integerk, there is a finite lattice L with more that ]L]ksublattices. On the other hand, there are arbitrary large finite lattices which contain a maximal sublattice with only 14 elements. It is shown that every bounded lattice is isomorphic to the Frattini sublattice (the intersection of all maximal sublattices) of a finite bounded lattice.

Every lattice is the complete join of all its one-element sublattices. In this paper we address t... more Every lattice is the complete join of all its one-element sublattices. In this paper we address the question: Which lattices L have the property that L is finitely join reducible in Sub L? That is, when do there exist proper sublattices A, B such that L = A ∨ B? In particular, could it be that every nontrivial lattice has this property, in which case every element of Sub L would be finitely join reducible? The authors would like to thank David Wasserman and M. E. Adams for bringing this problem to our attention, along with some elementary observations and helpful discussion. Let us mention a related problem. Recall the following result of Tom Whaley [4].

Algebra Universalis, 1977

Algebra Universalis, 1995

Alan once told me that he really liked elegant mathematics: simple ideas that give profound insig... more Alan once told me that he really liked elegant mathematics: simple ideas that give profound insights. Of course, everyone does, and it was for that reason that Alan was particularly proud of his doubling construction. It is a method which is simultaneously powerful and simple, with subtleties that go beyond the surface. It is a subject to which Alan kept returning till the end. Here I want to survey some areas of lattice theory and algebra which are closely tied to the doubling construction. Because so many concepts are interrelated, the construction will be sometimes on the surface and sometimes hidden, but I can assure you from experience that it is always used as a tool in the research stage. Let us begin at the beginning, around 1969. Ralph McKenzie had shown that splitting lattices are projective [26], and Alan and Steve Comer were discussing whether the same might be true for other lattice varieties. No, said Steve, and showed him how M33, which is a splitting modular lattice, is a homomorphic image of M;~ 3 (see Figure 1). Probably Alan was also aware that George Gr/itzer had doubled points in his work on ideal lattices [23], [24]. Anyway, this struck a responsive chord with Alan, and he soon had the construction written down in general form, and in its proper context, and used it to obtain a simple proof of Whitman's solution to the word problem for free lattices [4].

Algebra Universalis, 2010

We show that in a semimodular lattice L of finite length, from any prime interval we can reach an... more We show that in a semimodular lattice L of finite length, from any prime interval we can reach any maximal chain C by an up- and a down-perspectivity. Therefore, C is a congruence-determining sublattice of L.

Algebra universalis, 2006

Transactions of the American Mathematical Society, 1982

Every finite semidistributive lattice satisfying Whitman's condition is isomorphic to a sublattic... more Every finite semidistributive lattice satisfying Whitman's condition is isomorphic to a sublattice of a free lattice. Introduction. The aim of this paper is to show that a finite semidistributive lattice satisfying Whitman's condition can be embedded in a free lattice. This confirms a conjecture of Bjarni Jónsson, and indeed our proof will follow the line of approach originally suggested by him in unpubhshed notes around 1960. This approach was later described in Jónsson and Nation [15], to which the reader is referred for a more complete discussion of the background material and related work than will be given here. Let us recall some relevant definitions and results. A finite sublattice of a free lattice satisfies Whitman's condition [23] (W) ab < c + d iff a < c + d or b < c + d or ab < c or ab < d and the semidistributive laws introduced by Jónsson [12] (SDV) u = a + b = a + c implies u = a + be, (SDA) u = ab = ac implies u = a(b + c). As in [15], we shall refer to a finite lattice satisfying these three conditions as an S-lattice. We will often use the following (equivalent) form of the semidistributive laws [14]. (SDV) u = 2 a,,-2 bj implies u = 2,-2, a,bp (SDA) u = n a,: = LI bj implies u = II, II, (a,. + bj). Let J(L) denote the set of nonzero join-irreducible elements in a finite lattice L. Every element p G J(L) has a unique lower cover, which we will denote by p^. If />" G J(L), letpt<1 = (pf)+. Dually, M(L) denotes the set of nonunit meet-irreducible elements of L, and for y G M(L), y* >y. In a finite semidistributive lattice there is a bijection between J(L) and M(L), p <-» k(p) = 2 {x G L: x > pt andx £p}. (In fact, A. Day has shown that this characterizes finite semidistributive lattices [4].) Now px = p, iff x > p" and x %p, and, by (SDA), p/c(p) = p"; thus k(p) is the largest element in L with this property. Repeatedly we will use the following observations.

Proceedings of the American Mathematical Society, 1979

It is shown that the generalized word problem for lattices is solvable. Moreover, one can recursi... more It is shown that the generalized word problem for lattices is solvable. Moreover, one can recursively decide if two finitely presented lattices are isomorphic. It is also shown that the automorphism group of a finitely presented lattice is finite.

Order, 2004

For closure operators Γ and ∆ on the same set X, we say that ∆ is a weak (resp. strong) extension... more For closure operators Γ and ∆ on the same set X, we say that ∆ is a weak (resp. strong) extension of Γ if Cl(X, Γ) is a complete meet-subsemilattice (resp. complete sublattice) of Cl(X, ∆). This context is used to describe describe the extensions of a finite lattice that preserve various properties.

Journal of the Australian Mathematical Society, 1997

We investigate the number and size of the maximal sublattices of a finite lattice. For any positi... more We investigate the number and size of the maximal sublattices of a finite lattice. For any positive integerk, there is a finite lattice L with more that ]L]ksublattices. On the other hand, there are arbitrary large finite lattices which contain a maximal sublattice with only 14 elements. It is shown that every bounded lattice is isomorphic to the Frattini sublattice (the intersection of all maximal sublattices) of a finite bounded lattice.

BioData mining, 2016

Technological advances enable the cost-effective acquisition of Multi-Modal Data Sets (MMDS) comp... more Technological advances enable the cost-effective acquisition of Multi-Modal Data Sets (MMDS) composed of measurements for multiple, high-dimensional data types obtained from a common set of bio-samples. The joint analysis of the data matrices associated with the different data types of a MMDS should provide a more focused view of the biology underlying complex diseases such as cancer that would not be apparent from the analysis of a single data type alone. As multi-modal data rapidly accumulate in research laboratories and public databases such as The Cancer Genome Atlas (TCGA), the translation of such data into clinically actionable knowledge has been slowed by the lack of computational tools capable of analyzing MMDSs. Here, we describe the Joint Analysis of Many Matrices by ITeration (JAMMIT) algorithm that jointly analyzes the data matrices of a MMDS using sparse matrix approximations of rank-1. The JAMMIT algorithm jointly approximates an arbitrary number of data matrices by ra...

Order, 2014

This note gives a complete characterization of when the ordinal sum of two lattices (the lattice ... more This note gives a complete characterization of when the ordinal sum of two lattices (the lattice obtained by placing the second lattice on top of the first) is projective. This characterization applies not only to the class of all lattices, but to any variety of lattices, and in particular, to the class of distributive lattices. Lattices L with the property that every epimorphism onto L has an isotone section are also characterized.

Order

This note gives a complete characterization of when the ordinal sum of two lattices (the lattice ... more This note gives a complete characterization of when the ordinal sum of two lattices (the lattice obtained by placing the second lattice on top of the first) is projective. This characterization applies not only to the class of all lattices, but to any variety of lattices, and in particular, to the class of distributive lattices. Lattices L with the property that every epimorphism onto L has an isotone section are also characterized.

Group codes that use certain complex matrix groups are analyzed. These groups are complex analogu... more Group codes that use certain complex matrix groups are analyzed. These groups are complex analogues of the real Coxeter re- flection groups. It is shown that some types of complex permutation groups have good properties for use in coding, while others do not.

Pacific Journal of Mathematics, 1978

This paper gives necessary and sufficient conditions for a lattice to be projective. The conditio... more This paper gives necessary and sufficient conditions for a lattice to be projective. The conditions are the Whitman condition, and a condition of Jόnsson, and two new conditions explained below.

Order, 1989

ABSTRACT A useful construction for lattices is doubling a convex subsetI of a latticeL, i.e., rep... more ABSTRACT A useful construction for lattices is doubling a convex subsetI of a latticeL, i.e., replacingI byI2. It is shown that this construction preserves a generalized semidistributivity condition (C). Varieties of lattices in which every lattice satisfies (C) are characterized equationally.

Journal of the Australian Mathematical Society, 1997

We investigate the number and size of the maximal sublattices of a finite lattice. For any positi... more We investigate the number and size of the maximal sublattices of a finite lattice. For any positive integerk, there is a finite lattice L with more that ]L]ksublattices. On the other hand, there are arbitrary large finite lattices which contain a maximal sublattice with only 14 elements. It is shown that every bounded lattice is isomorphic to the Frattini sublattice (the intersection of all maximal sublattices) of a finite bounded lattice.

Every lattice is the complete join of all its one-element sublattices. In this paper we address t... more Every lattice is the complete join of all its one-element sublattices. In this paper we address the question: Which lattices L have the property that L is finitely join reducible in Sub L? That is, when do there exist proper sublattices A, B such that L = A ∨ B? In particular, could it be that every nontrivial lattice has this property, in which case every element of Sub L would be finitely join reducible? The authors would like to thank David Wasserman and M. E. Adams for bringing this problem to our attention, along with some elementary observations and helpful discussion. Let us mention a related problem. Recall the following result of Tom Whaley [4].

Algebra Universalis, 1977

Algebra Universalis, 1995

Alan once told me that he really liked elegant mathematics: simple ideas that give profound insig... more Alan once told me that he really liked elegant mathematics: simple ideas that give profound insights. Of course, everyone does, and it was for that reason that Alan was particularly proud of his doubling construction. It is a method which is simultaneously powerful and simple, with subtleties that go beyond the surface. It is a subject to which Alan kept returning till the end. Here I want to survey some areas of lattice theory and algebra which are closely tied to the doubling construction. Because so many concepts are interrelated, the construction will be sometimes on the surface and sometimes hidden, but I can assure you from experience that it is always used as a tool in the research stage. Let us begin at the beginning, around 1969. Ralph McKenzie had shown that splitting lattices are projective [26], and Alan and Steve Comer were discussing whether the same might be true for other lattice varieties. No, said Steve, and showed him how M33, which is a splitting modular lattice, is a homomorphic image of M;~ 3 (see Figure 1). Probably Alan was also aware that George Gr/itzer had doubled points in his work on ideal lattices [23], [24]. Anyway, this struck a responsive chord with Alan, and he soon had the construction written down in general form, and in its proper context, and used it to obtain a simple proof of Whitman's solution to the word problem for free lattices [4].

Algebra Universalis, 2010

We show that in a semimodular lattice L of finite length, from any prime interval we can reach an... more We show that in a semimodular lattice L of finite length, from any prime interval we can reach any maximal chain C by an up- and a down-perspectivity. Therefore, C is a congruence-determining sublattice of L.

Algebra universalis, 2006

Transactions of the American Mathematical Society, 1982

Every finite semidistributive lattice satisfying Whitman's condition is isomorphic to a sublattic... more Every finite semidistributive lattice satisfying Whitman's condition is isomorphic to a sublattice of a free lattice. Introduction. The aim of this paper is to show that a finite semidistributive lattice satisfying Whitman's condition can be embedded in a free lattice. This confirms a conjecture of Bjarni Jónsson, and indeed our proof will follow the line of approach originally suggested by him in unpubhshed notes around 1960. This approach was later described in Jónsson and Nation [15], to which the reader is referred for a more complete discussion of the background material and related work than will be given here. Let us recall some relevant definitions and results. A finite sublattice of a free lattice satisfies Whitman's condition [23] (W) ab < c + d iff a < c + d or b < c + d or ab < c or ab < d and the semidistributive laws introduced by Jónsson [12] (SDV) u = a + b = a + c implies u = a + be, (SDA) u = ab = ac implies u = a(b + c). As in [15], we shall refer to a finite lattice satisfying these three conditions as an S-lattice. We will often use the following (equivalent) form of the semidistributive laws [14]. (SDV) u = 2 a,,-2 bj implies u = 2,-2, a,bp (SDA) u = n a,: = LI bj implies u = II, II, (a,. + bj). Let J(L) denote the set of nonzero join-irreducible elements in a finite lattice L. Every element p G J(L) has a unique lower cover, which we will denote by p^. If />" G J(L), letpt<1 = (pf)+. Dually, M(L) denotes the set of nonunit meet-irreducible elements of L, and for y G M(L), y* >y. In a finite semidistributive lattice there is a bijection between J(L) and M(L), p <-» k(p) = 2 {x G L: x > pt andx £p}. (In fact, A. Day has shown that this characterizes finite semidistributive lattices [4].) Now px = p, iff x > p" and x %p, and, by (SDA), p/c(p) = p"; thus k(p) is the largest element in L with this property. Repeatedly we will use the following observations.

Proceedings of the American Mathematical Society, 1979

It is shown that the generalized word problem for lattices is solvable. Moreover, one can recursi... more It is shown that the generalized word problem for lattices is solvable. Moreover, one can recursively decide if two finitely presented lattices are isomorphic. It is also shown that the automorphism group of a finitely presented lattice is finite.

Order, 2004

For closure operators Γ and ∆ on the same set X, we say that ∆ is a weak (resp. strong) extension... more For closure operators Γ and ∆ on the same set X, we say that ∆ is a weak (resp. strong) extension of Γ if Cl(X, Γ) is a complete meet-subsemilattice (resp. complete sublattice) of Cl(X, ∆). This context is used to describe describe the extensions of a finite lattice that preserve various properties.