Fully representable and *-semisimple topological partial *-algebras

We continue our study of topological partial *-algebras, focusing our attention to *-semisimple partial *-algebras, that is, those that possess a {multiplication core} and sufficiently many *-representations. We discuss the respective roles of invariant positive sesquilinear (ips) forms and representable continuous linear functionals and focus on the case where the two notions are completely interchangeable (fully representable partial *-algebras) with the scope of characterizing a *-semisimple partial *-algebra. Finally we describe various notions of bounded elements in such a partial *-algebra, in particular, those defined in terms of a positive cone (order bounded elements). The outcome is that, for an appropriate order relation, one recovers the \M-bounded elements introduced in previous works.Comment: 26 pages, Studia Mathematica (2012) to appea

On Complete Representations and Minimal Completions in Algebraic Logic, Both Positive and Negative Results

Fix a finite ordinal $n\geq 3$ and let $\alpha$ be an arbitrary ordinal. Let $\mathsf{CA}_n$ denote the class of cylindric algebras of dimension $n$ and $\sf RA$ denote the class of relation algebras. Let $\mathbf{PA}_{\alpha}(\mathsf{PEA}_{\alpha})$ stand for the class of polyadic (equality) algebras of dimension $\alpha$. We reprove that the class $\mathsf{CRCA}_n$ of completely representable $\mathsf{CA}_n$s, and the class $\sf CRRA$ of completely representable $\mathsf{RA}$s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety $\sf V$ between polyadic algebras of dimension $n$ and diagonal free $\mathsf{CA}_n$s. We show that that the class of completely and strongly representable algebras in $\sf V$ is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class $\sf CRRA$ is not closed under $\equiv_{\infty,\omega}$. In contrast, we show that given $\alpha\geq \omega$, and an atomic $\mathfrak{A}\in \mathsf{PEA}_{\alpha}$, then for any \(n/p

Strongly representable atom structures of relation algebras

Accepted versio

Blow up and Blur constructions in Algebraic Logic

The idea in the title is to blow up a finite structure, replacing each 'colour or atom' by infinitely many, using blurs to represent the resulting term algebra, but the blurs are not enough to blur the structure of the finite structure in the complex algebra. Then, the latter cannot be representable due to a {finite- infinite} contradiction. This structure can be a finite clique in a graph or a finite relation algebra or a finite cylindric algebra. This theme gives examples of weakly representable atom structures that are not strongly representable. Many constructions existing in the literature are placed in a rigorous way in such a framework, properly defined. This is the essence too of construction of Monk like-algebras, one constructs graphs with finite colouring (finitely many blurs), converging to one with infinitely many, so that the original algebra is also blurred at the complex algebra level, and the term algebra is completey representable, yielding a representation of its completion the complex algebra. A reverse of this process exists in the literature, it builds algebras with infinite blurs converging to one with finite blurs. This idea due to Hirsch and Hodkinson, uses probabilistic methods of Erdos to construct a sequence of graphs with infinite chromatic number one that is 2 colourable. This construction, which works for both relation and cylindric algebras, further shows that the class of strongly representable atom structures is not elementary.Comment: arXiv admin note: text overlap with arXiv:1304.114