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The effective Borel hierarchy

M. Vanden Boom (2007)

Fundamenta Mathematicae

Let K be a subclass of Mod() which is closed under isomorphism. Vaught showed that K is Σ α (respectively, Π α ) in the Borel hierarchy iff K is axiomatized by an infinitary Σ α (respectively, Π α ) sentence. We prove a generalization of Vaught’s theorem for the effective Borel hierarchy, i.e. the Borel sets formed by union and complementation over c.e. sets. This result says that we can axiomatize an effective Σ α or effective Π α Borel set with a computable infinitary sentence of the same complexity. This result...

The elementary-equivalence classes of clopen algebras of P-spaces

Brian Wynne (2008)

Fundamenta Mathematicae

Two Boolean algebras are elementarily equivalent if and only if they satisfy the same first-order statements in the language of Boolean algebras. We prove that every Boolean algebra is elementarily equivalent to the algebra of clopen subsets of a normal P-space.

The induced paths in a connected graph and a ternary relation determined by them

Ladislav Nebeský (2002)

Mathematica Bohemica

By a ternary structure we mean an ordered pair ( X 0 , T 0 ) , where X 0 is a finite nonempty set and T 0 is a ternary relation on X 0 . By the underlying graph of a ternary structure ( X 0 , T 0 ) we mean the (undirected) graph G with the properties that X 0 is its vertex set and distinct vertices u and v of G are adjacent if and only if { x ∈ X 0 T 0 ( u , x , v ) } ∪ { x ∈ X 0 T 0 ( v , x , u ) } = { u , v } . A ternary structure ( X 0 , T 0 ) is said to be the B-structure of a connected graph G if X 0 is the vertex set of G and the following statement holds for all u , x , y ∈ X 0 : T 0 ( x , u , y ) if and only if u belongs to an induced x - y ...

The isomorphism relation between tree-automatic Structures

Olivier Finkel, Stevo Todorčević (2010)

Open Mathematics

An ω-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for ω-tree-automatic structures. We prove first that the isomorphism relation for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is not determined by the axiomatic system ZFC. Then we prove that...

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