Congruence lattices of semilattices

The main result of this paper is that the class of congruence lattices of semilattices satisfies no nontrivial lattice identities. It is also shown that the class of subalgebra lattices of semilattices satisfies no nontrivial lattice identities. As a consequence it is shown that if V is a semigroup variety all of whose congruence lattices satisfy some fixed nontrivial lattice identity, then all the members of V are groups with exponent dividing a fixed finite number

A class of infinite convex geometries

Various characterizations of finite convex geometries are well known. This note provides similar characterizations for possibly infinite convex geometries whose lattice of closed sets is strongly coatomic and lower continuous. Some classes of examples of such convex geometries are given.Comment: 10 page

Lattices of quasi-equational theories as congruence lattices of semilattices with operators, part I

We show that for every quasivariety K of structures (where both functions and relations are allowed) there is a semilattice S with operators such that the lattice of quasi-equational theories of K (the dual of the lattice of sub-quasivarieties of K) is isomorphic to Con(S;+; 0; F). As a consequence, new restrictions on the natural quasi-interior operator on lattices of quasi-equational theories are found

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

The Bases of Association Rules of High Confidence

We develop a new approach for distributed computing of the association rules of high confidence in a binary table. It is derived from the D-basis algorithm in K. Adaricheva and J.B. Nation (TCS 2017), which is performed on multiple sub-tables of a table given by removing several rows at a time. The set of rules is then aggregated using the same approach as the D-basis is retrieved from a larger set of implications. This allows to obtain a basis of association rules of high confidence, which can be used for ranking all attributes of the table with respect to a given fixed attribute using the relevance parameter introduced in K. Adaricheva et al. (Proceedings of ICFCA-2015). This paper focuses on the technical implementation of the new algorithm. Some testing results are performed on transaction data and medical data.Comment: Presented at DTMN, Sydney, Australia, July 28, 201

Primitive Lattice Varieties

A variety is primitive if every subquasivariety is equational, i.e. a subvariety. In this paper, we explore the connection between primitive lattice varieties and Whitman’s condition (W). For example, if every finite subdirectly irreducible lattice in a locally finite variety V satisfies Whitman’s condition (W), then V is primitive. This allows us to construct infinitely many sequences of primitive lattice varieties, and to show that there are 2ℵ0 such varieties. Some lattices that fail (W) also generate primitive varieties. But if I is a (W)-failure interval in a finite subdirectly irreducible lattice L, and L[I] denotes the lattice with I doubled, then V(L[I]) is never primitive

Lattices of quasi-equational theories as congruence lattices of semilattices with operators, Part I

We show that for every quasivariety K of structures (where both functions and relations are allowed) there is a semilattice S with operators such that the lattice of quasi-equational theories of K (the dual of the lattice of sub-quasivarieties of K) is isomorphic to Con(S,+,0,F). As a consequence, new restrictions on the natural quasi-interior operator on lattices of quasi-equational theories are found.Comment: Presented on International conference "Order, Algebra and Logics", Vanderbilt University, 12-16 June, 2007 25 pages, 2 figure

A metapopulation model for whale-fall specialists: The largest whales are essential to prevent species extinctions

The sunken carcasses of great whales (i.e., whale falls) provide an important deep-sea habitat for more than 100 species that may be considered whale-fall specialists. Commercial whaling has reduced the abundance and size of whales, and thus whale-fall habitats, as great whales were hunted and removed from the oceans, often to near extinction. In this article, we use a metapopulation modeling approach to explore the consequences of whaling to the abundance and persistence of whale-fall habitats in the deep sea and to the potential for extinction of whale-fall specialists. Our modeling indicates that the persistence of metapopulations of whale-fall specialists is linearly related to the abundance of whales, and extremely sensitive (to the fourth power) to the mean size of whales. Thus, whaling-induced declines in the mean size of whales are likely to have been as important as declines in whale abundance to extinction pressure on whale-fall specialists. Our modeling also indicates that commercial whaling, even under proposed sustainable yield scenarios, has the potential to yield substantial extinction of whale-fall specialists. The loss of whale-fall habitat is likely to have had the greatest impact on the diversity of whale-fall specialists in areas where whales have been hunted for centuries, allowing extinctions to proceed to completion. The North Atlantic experienced dramatic declines, and even extirpation, of many whale species before the 20th century; thus, extinctions of whale-fall specialists are likely to have already occurred in this region. Whale depletions have occurred more recently in the Southern Hemisphere and across most of the North Pacific; thus, these regions may still have substantial extinction debts, and many extant whale-fall specialists may be destined for extinction if whale populations do not recover in abundance and mean size over the next few decades. Prior to the resumption of commercial whaling, or the loosening of protections to reduce incidental take, the impacts of hunting on deep-sea whale-fall ecosystems, as well as differential protection of the largest whales within and across species, should be carefully considered

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