Miroslav Olšák — Page 2

arXiv2018

Loop conditions with strongly connected graphs

We prove that the existence of a term $s$ satisfying $s(r,a,r,e) = s(a,r,e,a)$ in a general algebraic structure is equivalent to an existence of a term $t$ satisfying $t(x,x,y,y,z,z)=t(y,z,z,x,x,y)$. As a consequence of a general version of this theorem and previous results we get that each strongly connected digraph of algebraic length one, which is compatible with an operation $t$ satisfying an identity of the from $t(\ldots)=t(\ldots)$, has a loop.

math.LO↗

arXiv2018

Equations in oligomorphic clones and the Constraint Satisfaction Problem for $ω$-categorical structures

There exist two conjectures for constraint satisfaction problems (CSPs) of reducts of finitely bounded homogeneous structures: the first one states that tractability of the CSP of such a structure is, when the structure is a model-complete core, equivalent to its polymorphism clone satisfying a certain non-trivial linear identity modulo outer embeddings. The second conjecture, challenging the approach via model-complete cores by reflections, states that tractability is equivalent to the linear identities (without outer embeddings) satisfied by its polymorphisms clone, together with the natural uniformity on it, being non-trivial. We prove that the identities satisfied in the polymorphism clone of a structure allow for conclusions about the orbit growth of its automorphism group, and apply this to show that the two conjectures are equivalent. We contrast this with a counterexample showing that $ω$-categoricity alone is insufficient to imply the equivalence of the two conditions above in a model-complete core. Taking a different approach, we then show how the Ramsey property of a homogeneous structure can be utilized for obtaining a similar equivalence under different conditions. We then prove that any polymorphism of sufficiently large arity which is totally symmetric modulo outer embeddings of a finitely bounded structure can be turned into a non-trivial system of linear identities, and obtain non-trivial linear identities for all tractable cases of reducts of the rational order, the random graph, and the random poset. Finally, we provide a new and short proof, in the language of monoids, of the theorem stating that every $ω$-categorical structure is homomorphically equivalent to a model-complete core.

cs.LO↗

arXiv2018

Loop conditions

We discuss such Maltsev conditions that consist of just one linear equation, we call them loop conditions. To every such condition can be assigned a graph. We provide a classification of conditions with undirected graphs. It follows that the Siggers term is the weakest non-trivial loop condition.

math.LO↗

arXiv2016

The weakest nontrivial idempotent equations

An equational condition is a set of equations in an algebraic language, and an algebraic structure satisfies such a condition if it possesses terms that meet the required equations. We find a single nontrivial equational condition which is implied by any nontrivial idempotent equational condition.

math.LO↗