Archive for Mathematical Logic最新文献

We prove two results intended to streamline proofs about cellularity that pass through mutual algebraicity. First, we show that a countable structure M is cellular if and only if M is (omega )-categorical and mutually algebraic. Second, if a countable structure M in a finite relational language is mutually algebraic non-cellular, we show it admits an elementary extension adding infinitely many infinite MA-connected components. Towards these results, we introduce MA-presentations of a mutually algebraic structure, in which every atomic formula is mutually algebraic. This allows for an improved quantifier elimination and a decomposition of the structure into independent pieces. We also show this decomposition is largely independent of the MA-presentation chosen.

Given a countable transitive model of set theory and a partial order contained in it, there is a natural countable Borel equivalence relation on generic filters over the model; two are equivalent if they yield the same generic extension. We examine the complexity of this equivalence relation for various partial orders, focusing on Cohen and random forcing. We prove, among other results, that the former is an increasing union of countably many hyperfinite Borel equivalence relations, and hence is amenable, while the latter is neither amenable nor treeable.

We formulate a restriction of Hindman’s Finite Sums Theorem in which monochromaticity is required only for sums corresponding to rooted finite paths in the full binary tree. We show that the resulting principle is equivalent to (Sigma ^0_2)-induction over (mathsf {RCA}_0). The proof uses the equivalence of this Hindman-type theorem with the Pigeonhole Principle for trees ({mathsf {T},}{mathsf {T}}^1) with an extra condition on the solution tree.

We establish that the isomorphism problem for the classes of computable nilpotent rings, distributive lattices, nilpotent groups, and nilpotent semigroups is (Sigma _{1}^{1})-complete, which is as complicated as possible. The method we use is based on uniform effective interpretations of computable binary relations into computable structures from the corresponding algebraic classes.

This paper studies various equivalent characterizations of left semisimple rings from the standpoint of reverse mathematics. We first show that (mathrm ACA_{0}) is equivalent to the statement that any left module over a left semisimple ring is semisimple over (mathrm RCA_{0}). We then study characterizations of left semisimple rings in terms of projective modules as well as injective modules, and obtain the following results: (1) (mathrm ACA_{0}) is equivalent to the statement that any left module over a left semisimple ring is projective over (mathrm RCA_{0}); (2) (mathrm ACA_{0}) is equivalent to the statement that any left module over a left semisimple ring is injective over (mathrm RCA_{0}); (3) (mathrm RCA_{0}) proves the statement that if every cyclic left R-module is projective, then R is a left semisimple ring; (4) (mathrm ACA_{0}) proves the statement that if every cyclic left R-module is injective, then R is a left semisimple ring.

We show that ( VTC ^0), the basic theory of bounded arithmetic corresponding to the complexity class (mathrm {TC}^0), proves the ( IMUL ) axiom expressing the totality of iterated multiplication satisfying its recursive definition, by formalizing a suitable version of the (mathrm {TC}^0) iterated multiplication algorithm by Hesse, Allender, and Barrington. As a consequence, ( VTC ^0) can also prove the integer division axiom, and (by our previous results) the ( RSUV )-translation of induction and minimization for sharply bounded formulas. Similar consequences hold for the related theories (Delta ^b_1text{- } CR ) and (C^0_2). As a side result, we also prove that there is a well-behaved (Delta _0) definition of modular powering in (IDelta _0+ WPHP (Delta _0)).