Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userResults and speculations concerning Comer relation algebras and the flexible atom conjecture
280
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study some finite integral symmetric relation algebras whose forbidden cycles are all 2-cycles. These algebras arise from a finite field construction due to Comer. We consider conditions that allow other finite algebras to embed into these Comer algebras, and as an application give the first known finite representation of relation algebra $34_{65}$, one of whose atoms is flexible. We conclude with some speculation about how the ideas presented here might contribute to a proof of the flexible atom conjecture.
rate research
Read More
Robin Hirsch posed in 1996 the Really Big Complexity Problem: classify the computational complexity of the network satisfaction problem for all finite relation algebras $bf A$. We provide a complete classification for the case that $bf A$ is symmetric and has a flexible atom; the problem is in this case NP-complete or in P. If a finite integral relation algebra has a flexible atom, then it has a normal representation $mathfrak{B}$. We can then study the computational complexity of the network satisfaction problem of ${bf A}$ using the universal-algebraic approach, via an analysis of the polymorphisms of $mathfrak{B}$. We also use a Ramsey-type result of Nev{s}etv{r}il and Rodl and a complexity dichotomy result of Bulatov for conservative finite-domain constraint satisfaction problems.
We construct an explicit filtration of the ring of algebraic power series by finite dimensional constructible sets, measuring the complexity of these series. As an application, we give a bound on the dimension of the set of algebraic power series of bounded complexity lying on an algebraic variety defined over the field of power series.
Algebras introduced by, or attributed to, Sugihara, Belnap, Meyer, and Church are representable as algebras of binary relations with set-theoretically defined operations. They are definitional reducts or subreducts of proper relation algebras. The representability of Sugihara matrices yields sound and complete set-theoretical semantics for R-mingle.
For any pair of ordinals $alpha<beta$, $sf CA_alpha$ denotes the class of cylindric algebras of dimension $alpha$, $sf RCA_{alpha}$ denote the class of representable $sf CA_alpha$s and $sf Nr_alpha CA_beta$ ($sf Ra CA_beta)$ denotes the class of $alpha$-neat reducts (relation algebra reducts) of $sf CA_beta$. We show that any class $sf K$ such that $sf RaCA_omega subseteq sf Ksubseteq RaCA_5$, $sf K$ is not elementary, i.e not definable in first order logic. Let $2<n<omega$. It is also shown that any class $sf K$ such that $sf Nr_nCA_omega cap {sf CRCA}_nsubseteq {sf K}subseteq mathbf{S}_csf Nr_nCA_{n+3}$, where $sf CRCA_n$ is the class of completely representable $sf CA_n$s, and $mathbf{S}_c$ denotes the operation of forming complete subalgebras, is proved not to be elementary. Finally, we show that any class $sf K$ such that $mathbf{S}_dsf Ra CA_omega subseteq {sf K}subseteq mathbf{S}_csf RaCA_5$ is not elementary. It remains to be seen whether there exist elementary classes between $sf RaCA_omega$ and $mathbf{S}_dsf RCA_{omega}$. In particular, for $mgeq n+3$, the classes $sf Nr_nCA_m$, $sf CRCA_n$, $mathbf{S}_dsf Nr_nCA_m$, where $mathbf{S}_d$ is the operation of forming dense subalgebras are not first order definable.
We prove that if the set of unordered pairs of real numbers is colored by finitely many colors, there is a set of reals homeomorphic to the rationals whose pairs have at most two colors. Our proof uses large cardinals and it verifies a conjecture of Galvin from the 1970s. We extend this result to an essentially optimal class of topological spaces in place of the reals.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
