It is well known that clopen subgroups of finitely generated free profinite groups are again finitely generated free profinite groups. Clopen submonoids of free profinite monoids need not be finitely generated nor free. Margolis, Sapir and Weil proved that the closed submonoid generated by a finite code (which is in fact clopen) is a free profinite monoid generated by that code. In this note we...

In this paper, we prove the following form of Stone’s representation theorem: Let ∑ be a σ -algebra of subsets of a set X . Then there exists a totally disconnected compact Hausdorff space K for which ( ∑ ,∪,∩) and (C(K),∪,∩) , where C(K) denotes the set of all clopen subsets of K , are isomorphic as Boolean algebras. Furthermore, by defining appropriate joins and meets of countable families in...

We say that E ⊆ X × X is a clopen graph on X iff E is symmetric and irreflexive and clopen relative to X\∆ where ∆ = {(x, x) : x ∈ X} is the diagonal. Equivalently E ⊆ [X] and for all x 6= y ∈ X there are open neighborhoods x ∈ U and y ∈ V such that either U × V ⊆ E or U × V ⊆ X\E. For clopen graphs E1, E2 on spaces X1, X2, we say that E1 continuously reduces to E2 iff there is a continuous map...

A graph G on a topological space X as its set of vertices is clopen if the edge relation of G is a clopen subset of X2 without the diagonal. We study clopen graphs on Polish spaces in terms of their finite induced subgraphs and obtain information about their cochromatic numbers. In this context we investigate modular profinite graphs, a class of graphs obtained from finite graphs by taking inve...

While there is, up to homeomorphism, only one Cantor space, i.e. one zero-dimensional, perfect, compact, nonempty metric space, there are many measures on Cantor space which are not topologically equivalent. The clopen values set for a full, nonatomic measure μ is the countable dense subset {μ(U) : U is clopen} of the unit interval. It is a topological invariant for the measure. For the class o...

The existing topological representation of an orthocomplemented lattice via the clopen orthoregular subsets a Stone space depends upon Alexander’s Subbase Theorem, which asserts that X is compact if every subbasic open cover admits finite subcover. This easy consequence Ultrafilter Theorem—whose proof Zorn’s Lemma, well known to be equivalent Axiom Choice. Within this work, we give choice-free ...

We propose Clopen Knowledge Bases (CKBs) as a new formalism combining Answer Set Programming (ASP) with ontology languages based on first-order logic. CKBs generalize the prominent r-hybrid and DL+LOG languages of Rosati, and are more flexible for specification of problems that combine open-world and closed-world reasoning. We argue that the guarded negation fragment of first-order logic (GNFO)...