arXiv subjects

Joseph Van Name

Publications and source records attributed to Joseph Van Name.

6 recordsLinked to original sources

arXiv2018

Generalizations of Laver tables

We shall generalize the notion of a Laver table to algebras which may have many generators, several fundamental operations, fundamental operations of arity higher than 2, and to algebras where only some of the operations are self-distributive or where the operations satisfy a generalized version of self-distributivity. These algebras mimic the algebras of rank-into-rank embeddings $\mathcal{E}_λ/\equiv^γ$ in the sense that composition and the notion of a critical point make sense for these sorts of algebras.

math.LO↗

arXiv2013

Ultraparacompactness and Ultranormality

In this note, we shall overview some results related to ultraparacompactness and ultranormality in the general topological and point-free contexts. This note contains some standard results and counterexamples along with some results which are not that well known and even of my new results.

math.GN↗

arXiv2012

A Generalization of the notion of a $P$-space to proximity spaces

In this note, we shall generalize the notion of a $P$-space to proximity spaces and investigate the basic properties of these proximities. We therefore define a $P_{\aleph_{1}}$-proximity to be a proximity where if $A_{n}\prec B$ for all $n\in\mathbb{N}$, then $\bigcup_{n}A_{n}\prec B$. It turns out that the class of $P_{\aleph_{1}}$-proximities is equivalent to the class of $σ$-algebras. Furthermore, the $P_{\aleph_{1}}$-proximity coreflection of a proximity space is the $σ$-algebra of proximally Baire sets.

math.GN↗

arXiv2012

Constructing Ultrapowers from Elementary Extensions of Full Clones

Let $A$ be an infinite set. Let $Ω(A)$ be the algebra over $A$ where every constant is a fundamental constant and every finitary function is a fundamental operation. We shall give a method of representing any algebra $\mathcal{L}$ in the variety generated by $Ω(A)$ as limit reduced powers and even direct limits of limit reduced powers of $\mathcal{L}$. If the algebra $\mathcal{L}$ is elementarily equivalent to $Ω(A)$, then this construction represents $Ω(A)$ as a limit ultrapower and also as direct limits of limit ultrapowers of $Ω(A)$. This method therefore gives a method of representing Boolean ultrapowers and other generalizations of the ultrapower construction as limit ultrapowers and direct limits of limit ultrapowers.

math.LO↗

arXiv2012

A Duality Between Non-Archimedean Uniform Spaces and Subdirect Powers of Full Clones

A uniform space is said to be non-Archimedean if it is generated by equivalence relations. If $λ$ is a cardinal, then a non-Archimedean uniform space $(X,\mathcal{U})$ is $λ$-totally bounded if each equivalence relation in $\mathcal{U}$ partitions $X$ into less than $λ$ blocks. If $A$ is an infinite set, then let $Ω(A)$ be the algebra with universe $A$ and where each $a\in A$ is a fundamental constant and every finitary function is a fundamental operation. We shall give a duality between complete non-Archimedean $|A|^{+}$-totally bounded uniform spaces and subdirect powers of $Ω(A)$. We shall apply this duality to characterize the algebras dual to supercomplete non-Archimedean uniform spaces.

math.GN↗

arXiv2011

Duality Between Uniform Spaces and Boolean Algebras

In this note we shall generalize the Stone duality between compact totally disconnected spaces and Boolean algebras to a duality between all complete non-Archimedean uniform spaces and Boolean algebras.

math.GN↗