arXiv subjects

Colin Bloomfield

Publications and source records attributed to Colin Bloomfield.

3 recordsLinked to original sources

arXiv2026

Dialectica Categories over Heyting Algebras

Categorification---the process of constructing a categorical model of a piece of mathematics---often identifies a common abstraction that connects formerly unrelated but known structures. In the case of de Paiva's categorification of G\"odel's Dialectica interpretation, we find that its specialization to partial orders produces (functorial) embeddings of Heyting algebras into residuated lattices that appear to have been overlooked. For the non-categorical audience, we present this specialization and take care to reproduce the original proofs in the algebraic setting. Along the way we obtain results particular to this algebraic setting: an embedding lacking an evident adjoint in de Paiva's general construction acquires a definable one here; a single Dialectica tensor validates contraction in the intuitionistic construction D yet refutes it in the classical variant G; and, over ZF, the poset reflection PD(Set) collapses onto the four-element algebra PD(2) exactly when the Axiom of Choice holds.

math.LO↗

arXiv2022

Fibered Universal Algebra for First-Order Logics

We extend Lawvere-Pitts prop-categories (aka. hyperdoctrines) to develop a general framework for providing "algebraic" semantics for nonclassical first-order logics. This framework includes a natural notion of substitution, which allows first-order logics to be considered as structural closure operators just as propositional logics are in abstract algebraic logic. We then establish an extension of the homomorphism theorem from universal algebra for generalized prop-categories and characterize two natural closure operators on the prop-categorical semantics. The first closes a class of structures (which are interpreted as morphisms of prop-categories) under the satisfaction of their common first-order theory and the second closes a class of prop-categories under their associated first-order consequence. It turns out these closure operators have characterizations that closely mirror Birkhoff's characterization of the closure of a class of algebras under the satisfaction of their common equational theory and Blok and J\'onsson's characterization of closure under equational consequence, respectively. These "algebraic" characterizations of the first-order closure operators are unique to the prop-categorical semantics and do not have analogs, for example, in the Tarskian semantics for classical first-order logic. The prop-categories we consider are much more general than traditional intuitionistic prop-categories or triposes (i.e., topos representing indexed partially ordered sets). Nonetheless, to our knowledge, our results are still new even when restricted to these special classes of prop-categories.

math.LO↗

arXiv2021

Radio-k-Labeling of Cycles for Large k

Let $G$ be a simple connected graph. For any two vertices $u$ and $v$, let $d(u,v)$ denote the distance between $u$ and $v$ in $G$. A radio-$k$-labeling of $G$ for a fixed positive integer $k$ is a function $f$ which assigns to each vertex a non-negative integer label such that for every two vertices $u$ and $ v $ in $G$, $|f(u)-f(v)| \geq k - d(u,v) +1$. The span of $f$ is the difference between the largest and smallest labels of $f(V)$. The radio-$k$-number of a graph $G$, denoted by $rn_k(G)$, is the smallest span among all radio-$k$-labelings admitted by $G$. A cycle $C_n$ has diameter $d=\lfloor n/2 \rfloor$. In this paper, we combine a lower bound approach with cyclic group structure to determine the value of $ rn_k(C_n)$ for $k \geq n-3$. For $d \leq k < n-3$, we obtain the values of $rn_k(C_n)$ when $n$ and $k$ have the same parity, and prove partial results when $n$ and $k$ have different parities. Our results extend the known values of $rn_d (C_n)$ and $rn_{d+1} (C_n)$ shown by Liu and Zhu, and by Karst, Langowitz, Oehrlein and Troxell, respectively.

math.CO↗