Related papers: Neat atom structures

(1) Let 1\leq k\leq \omega. Call an atom structure \alpha weakly k neat representable, the term algebra is in \RCA_n\cap \Nr_n\CA_{n+k}, but the complex algebra is not representable. Call an atom structure neat if there is an atomic algebra…

In this paper we give an alternative construction using Monk like algebras that are binary generated to show that the class of strongly representable atom structures is not elementary. The atom structures of such algebras are cylindric…

This paper introduces the notions of atoms and atomicity in $C$-algebras and obtains a characterisation of atoms in the $C$-algebra of transformations. Further, this work presents some necessary conditions and sufficient conditions for the…

We exhibit two relation algebra atom structures such that they are elementarily equivalent but their term algebras are not. This answers Problem 14.19 in the book Hirsch, R. and Hodkinson, I., "Relation Algebras by Games", North-Holland,…

(1) If $R$ is an affine algebra of dimension $d\geq 4$ over $\overline{\mathbb{F}}_{p}$ with $p>3$, then the group structure on ${\rm Um}_d(R)/{\rm E}_d(R)$ is nice. (2) If $R$ is a commutative noetherian ring of dimension $d\geq 2$ such…

Algebraic hyperstructures represent a natural extension of classical algebraic structures. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two…

We classify primitive, rank 1, omega-categorical structures having polynomially many types over finite sets. For a fixed number of 4-types, we show that there are only finitely many such structures and that all are built out of finitely…

$\omega$-clones are multi-sorted structures that naturally emerge as algebras for infinite trees, just as $\omega$-semigroups are convenient algebras for infinite words. In the algebraic theory of languages, one hopes that a language is…

We establish an Excision type theorem for niceness of group structure on the orbit space of unimodular rows of length $n$ modulo elementary action. This permits us to establish niceness for relative versions of results for the cases when $n…

I give an overview of recent developments in the structure and classification theory of separable, simple, nuclear C*-algebras. I will in particular focus on the role of quasidiagonality and amenability for classification, and on the…

We prove two results intended to streamline proofs about cellularity that pass through mutual algebraicity. First, we show that a countable structure $M$ is cellular if and only if $M$ is $\omega$-categorical and mutually algebraic. Second,…

To constitute atoms of a $\sigma$ algebra is not a easy task due to the large number of its elements. However, determining them via generators seems a feasible and simple way since most $\sigma$ algebras are generated by their smaller…

We show that for 1<n<m, the class Nr_nCA_m known to be non-elementary is pseudo elementary. When n and m are finite we use a two sorted theory, when n is finite and m infinite we use a three sorted one, and finally when both are infinite we…

Effect algebras are a generalization of many structures which arise in quantum physics and in mathematical economics. We show that, in every modular Archimedean atomic lattice effect algebra $E$ that is not an orthomodular lattice there…

Classification theory of elementary classes deals with first order (elementary) classes of structures (i.e. fixing a set T of first order sentences, we investigate the class of models of T with the elementary submodel notion). It tries to…

Using constructions of Hirsch and Hodkinson, we show that the class of strongly atom structures for various cylindric-like algebras is not elementary. This applies to diagonal free reducts and polyadic algebras with and without equality.…

A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…

We demonstrate that pure C*-algebras form a robust class by proving that pureness follows from very weak comparison and divisibility properties. Using this, we show that every simple, non-elementary C*-algebra with a unique quasitrace and…

Using model theoretic techniques that proved that the class of $n$ neat reducts of $m$ dimensional cylindric algebras, $\Nr_n\CA_m$, is not elementary, we prove the same result for $\Ra\CA_k$, $k\geq 5$, and we show that $\Ra\CA_k\subset…

We show that the nerve of a strict omega-category can be described algebraically as a simplicial set with additional operations subject to certain identities. The resulting structures are called sets with complicial identities. We also…