Annals of Pure and Applied Logic最新文献

Semiconic idempotent logic is a common generalization of intuitionistic logic, relevance logic with mingle, and semilinear idempotent logic. It is an algebraizable logic and it admits a cut-free hypersequent calculus. We give a structural decomposition of its characteristic algebraic semantics, conic idempotent residuated lattices, showing that each of these is an ordinal sum of simpler partially ordered structures. This ordinal sum is indexed by a totally ordered residuated lattice, which serves as its skeleton and is both a subalgebra and nuclear image. We equationally characterize the totally ordered residuated lattices appearing as such skeletons. Further, we describe both congruence and subalgebra generation in conic idempotent residuated lattices, proving that every variety generated by these enjoys the congruence extension property. In particular, this holds for semilinear idempotent residuated lattices. Based on this analysis, we obtain a local deduction theorem for semiconic idempotent logic, which also specializes to semilinear idempotent logic.

We study expansions of a vector space V over a field $F$, possibly with extra structure, with a generic submodule over a subring of $F$. We construct a natural expansion by existentially defined functions so that the expansion in the extended language satisfies quantifier elimination. We show that this expansion preserves tame model theoretic properties such as stability, NIP, NTP₁, NTP₂ and NSOP₁. We also study induced independence relations in the expansion.

We prove that if A is a computable Hopfian finitely presented structure, then A has a computable d-${\Sigma }_2$ Scott sentence if and only if the weak Whitehead problem for A is decidable. We use this to infer that every hyperbolic group as well as any polycyclic-by-finite group has a computable d-${\Sigma }_2$ Scott sentence, thus covering two main classes of finitely presented groups. Our proof also implies that every weakly Hopfian finitely presented group is strongly defined by its ${\exists }^{+}$-types, a question which arose in a different context.

We explore different generalizations of the classical concept of independent families on ω following the study initiated by Kunen, Fischer, Eskew and Montoya. We show that under ${\left(D\ell \right)}_{\kappa }^{⁎}$ we can get strongly κ-independent families of size $2^{\kappa }$ and present an equivalence of $\mathrm{GCH}$ in terms of strongly independent families. We merge the two natural ways of generalizing independent families through a filter or an ideal and we focus on the $C$-independent families, where $C$ is the club filter. Also we show a relationship between the existence of $J$-independent families and the saturation of the ideal $J$.

We develop a transfer principle of structural Ramsey theory from finite structures to ultraproducts. We show that under certain mild conditions, when a class of finite structures has finite small Ramsey degrees, under the (Generalized) Continuum Hypothesis the ultraproduct has finite big Ramsey degrees for internal colorings. The necessity of restricting to internal colorings is demonstrated by the example of the ultraproduct of finite linear orders. Under CH, this ultraproduct $L^{⁎}$ has, as a spine, ${\eta }_1$, an uncountable analogue of the order type of rationals η. Finite big Ramsey degrees for η were exactly calculated by Devlin in [5]. It is immediate from [39] that ${\eta }_1$ fails to have finite big Ramsey degrees. Moreover, we extend Devlin's coloring to ${\eta }_1$ to show that it witnesses big Ramsey degrees of finite tuples in η on every copy of η in ${\eta }_1$, and consequently in $L^{⁎}$. This work gives additional confirmation that ultraproducts are a suitable environment for studying Ramsey properties of finite and infinite structures.

In this paper, we establish an explicit higher reciprocity law for the polynomial ring over a nonprincipal ultraproduct of finite fields. Such an ultraproduct can be taken over the same finite field, which allows to recover the classical higher reciprocity law for the polynomial ring $F_q\left[t\right]$ over a finite field $F_q$ that is due to Dedekind, Kühne, Artin, and Schmidt. On the other hand, when the ultraproduct is taken over finite fields of unbounded cardinalities, we obtain an explicit higher reciprocity law for the polynomial ring over an infinite field in both characteristics 0 and $p>0$ for some prime p. We then use the higher reciprocity law to prove an analogue of the Grunwald–Wang theorem for such a polynomial ring in both characteristics 0 and $p>0$ for some prime p.

Shelah showed that the existence of free subsets over internally approachable subalgebras follows from the failure of the PCF conjecture on intervals of regular cardinals. We show that a stronger property called the Approachable Bounded Subset Property can be forced from the assumption of a cardinal λ for which the set of Mitchell orders $\left\{o\right(\mu \left)\right|\mu <\lambda \}$ is unbounded in λ. Furthermore, we study the related notion of continuous tree-like scales, and show that such scales must exist on all products in canonical inner models. We use this result, together with a covering-type argument, to show that the large cardinal hypothesis from the forcing part is optimal.

In this paper, we unify the study of classical and non-classical algebra-valued models of set theory, by studying variations of the interpretation functions for = and ∈. Although, these variations coincide with the standard interpretation in Boolean-valued constructions, nonetheless they extend the scope of validity of $\mathrm{ZF}$ to new algebra-valued models. This paper presents, for the first time, non-trivial paraconsistent models of full $\mathrm{ZF}$. Moreover, due to the validity of Leibniz's law in these structures, we will show how to construct proper models of set theory by quotienting these algebra-valued models with respect to equality, modulo the filter of the designated truth-values.

We make use of a finite support product of the Jensen-type forcing notions to define a model of the set theory $\text{ZFC}$ in which, for a given $n\ge 3$, there exists a good lightface ${\Delta }_n^1$ well-ordering of the reals but there are no any (not necessarily good) well-orderings in the boldface class ${\Delta }_{n-1}^1$.

We determine sufficient structure for an elementary topos to emulate Nelson's Internal Set Theory in its internal language, and show that any topos satisfying the internal axiom of choice occurs as a universe of standard objects and maps. This development allows one to employ the proof methods of nonstandard analysis (transfer, standardisation, and idealisation) in new environments such as toposes of G-sets and Boolean étendues.