arXiv subjects

Mirai Ikebuchi

Publications and source records attributed to Mirai Ikebuchi.

6 recordsLinked to original sources

arXiv2026

Anick Resolution for Lawvere Theories from Algebraic Discrete Morse Theory

Inspired by Brown's collapsing method (or discrete Morse theory) to obtain a free resolution of $\bbZ$ over the monoid ring $\bbZ M$, we apply algebraic discrete Morse theory to compute the homology groups of Lawvere theories, which is defined as Tor of a certain module. We reinterpret known partial free resolutions arising from complete term rewriting systems in terms of collapsing of the normalized bar resolution. This perspective yields homological inequalities that bound the number of equational axioms in presentations and recovers classical results, such as lower bounds for group axiomatizations. Our main contribution is to extend these resolutions to higher dimensions.

math.KT↗

arXiv2025

Cohomology of Small Cartesian Closed Categories

We show the isomorphism between the Quillen cohomology and the Baues-Wirsching cohomology of a cartesian closed category (CCC). This is an extension of the results of Dwyer-Kan for small categories and Jibladze-Pirashvili for small categories with finite products. These results implies that The Quillen cohomology of a CCC C coincides with that of C as a category with finite products, and also that of C as a small category

math.CT↗

arXiv2025

Homological Invariants of Higher-Order Equational Theories

Many first-order equational theories, such as the theory of groups or boolean algebras, can be presented by a smaller set of axioms than the original one. Recent studies showed that a homological approach to equational theories gives us inequalities to obtain lower bounds on the number of axioms. In this paper, we extend this result to higher-order equational theories. More precisely, we consider simply typed lambda calculus with product and unit types and study sets of equations between lambda terms. Then, we define homology groups of the given equational theory and show that a lower bound on the number of equations can be computed from the homology groups.

cs.LO↗

arXiv2022

A Lower Bound of the Number of Rewrite Rules Obtained by Homological Methods

It is well-known that some equational theories such as groups or boolean algebras can be defined by fewer equational axioms than the original axioms. However, it is not easy to determine if a given set of axioms is the smallest or not. Malbos and Mimram investigated a general method to find a lower bound of the cardinality of the set of equational axioms (or rewrite rules) that is equivalent to a given equational theory (or term rewriting systems), using homological algebra. Their method is an analog of Squier's homology theory on string rewriting systems. In this paper, we develop the homology theory for term rewriting systems more and provide a better lower bound under a stronger notion of equivalence than their equivalence. The author also implemented a program to compute the lower bounds, and experimented with 64 complete TRSs.

cs.LO↗

arXiv2020

On properties of $B$-terms

$B$-terms are built from the $B$ combinator alone defined by $B\equivλfgx. f(g~x)$, which is well known as a function composition operator. This paper investigates an interesting property of $B$-terms, that is, whether repetitive right applications of a $B$-term cycles or not. We discuss conditions for $B$-terms to have and not to have the property through a sound and complete equational axiomatization. Specifically, we give examples of $B$-terms which have the cyclic property and show that there are infinitely many $B$-terms which do not have the property. Also, we introduce another interesting property about a canonical representation of $B$-terms that is useful to detect cycles, or equivalently, to prove the cyclic property, with an efficient algorithm.

cs.LO↗

arXiv2017

On repetitive right application of B-terms

B-terms are built from the B combinator alone defined by B f g x = f (g x), which is well-known as a function composition operator. This paper investigates an interesting property of B-terms, that is, whether repetitive right applications of a B-term circulates or not. We discuss conditions for B-terms to and not to have the property through a sound and complete equational axiomatization. Specifically, we give examples of B-terms which have the property and show that there are infinitely many B-terms which does not have the property. Also, we introduce a canonical representation of B-terms that is useful to detect cycles, or equivalently, to prove the property, with an efficient algorithm.

cs.LO↗