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

Defying Boundaries: Mary Musgrove in Early Colonial Georgia

Often referred to as the ‘Pocahontas of Georgia,’ Mary Musgrove played a very prominent role in facilitating peaceful relationships between Native Americans and English settlers. And, much like Pocahontas, recent scholarship on Mary Musgrove has slowly been chipping away at the mask designated to her by popular memory. Historian Michael D. Green argues that Mary Musgrove’s life “represented a distinct vision for the future of the English in America.” This vision was one in which Native American and English identities could be combined, which Mary intended not only for herself, but also for English colonists and the Creeks. Mary’s vision of a world where one could live in peace as both a Native American and English person was, however, severely shattered; Mary Musgrove’s rise to power as a businesswoman and landowner of ‘mixed blood’ threatened English colonists’ predominantly white, patriarchal society and ultimately led to the absolution of their relationship

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

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

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