Free algebras with an arbitrary number of free generators in varieties of BL-algebras generated by one BL-chain that is an ordinal sum of a finite MV-chain Ln and a generalized BL-chain B are described in terms of weak Boolean products of BL-algebras that are ordinal sums of subalgebras of Ln and free algebras in the variety of basic hoops generated by B. The Boolean products are taken over the...

A classical theorem of Kuratowski says that every Baire one function on a Gδ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this heirarchy depending on its oscillation index β(f). We p...

We prove the following theorem: Given a ⊆ ω and 1 ≤ α < ωCK 1 , if for some η < א1 and all u ∈ WO of length η, a is Σα(u), then a is Σα. We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: Σ1-Turing-determinacy implies the existence of 0#. A major step in delineating the precise connections between large cardinals and game-determinacy hypotheses is the well-known ...

We construct an infinite dimensional quasi-polyadic equality algebra A such that its cylindric reduct is representable, while A itself is not representable. 1 The most well known generic examples of algebraizations of first order logic are Tarski’s cylindric algebras (CA) and Halmos’ polyadic equality algebras (PEA). The theory of cylindric algebras is well developed in the treatise [10], [11],...

We construct, for each countable ordinal ξ, a closed graph with Borel chromatic number two and Baire class ξ chromatic number א0. 2010 Mathematics Subject Classification. Primary: 03E15, Secondary: 54H05