Yasuo Kawahara

An algebraic formalization of fuzzy relations

Abstract This paper provides an algebraic formalization of mathematical structures formed by fuzzy relations with sup-min composition. A simple proof of a representation theorem for Boolean relation algebras satisfying Tarski rule and point axiom has been given by Schmidt and Strohlein. Unlike Boolean relation algebras, fuzzy relation algebras are not Boolean but equipped with semi-scalar multiplication. First, we present a set of axioms for fuzzy relation algebras and improve the definition of point relations. Then by using relational calculus, a representation theorem for such relation algebras is deduced without Tarski rule.

Pushout-complements and basic concepts of grammars in toposes

Abstract An existence theorem of pushout-complements is given in an elementary topos by using category theory of binary relations, called relational calculus, and it is also shown more explicitly in the category of directed graphs, which is a typical example of toposes, as an application. Moreover an embedding theorem and Church-Rosser theorem on grammars (derivations) in a topos are proved.

On the cardinality of relations

This paper will discuss and characterise the cardinality of boolean (crisp) and fuzzy relations. The main result is a Dedekind inequality for the cardinality, which enables us to manipulate the cardinality of the composites of relations. As applications a few relational proofs for the basic theorems on graph matchings, and fundamentals about network flows will be given.

A small final coalgebra theorem

This paper presents an elementary and self-contained proof of an existence theorem of final coalgebras for endofunctors on the category of sets and functions.

Categorical representation theorems of fuzzy relations

Abstract This paper provides a notion of Zadeh categories as a categorical structure formed by fuzzy relations with sup-min composition, and proves two representation theorems for Dedekind categories (relation categories) with a unit object analogous to one-point set, and for Zadeh categories without unit objects.

On reversible cellular automata with finite cell array

Discrete quantum cellular automata are cellular automata with reversible transition. This paper deals with 1d cellular automata with finite cell array and triplet local transition rules. We present the necessary condition of local transition rules for cellular automata to be reversible, and prove the reversibility of some cellular automata.

Relational graph rewritings

This note presents a new formalization of graph rewritings which generalizes traditional graph rewritings. Relational notions of graphs and their rewritings are introduced and several properties about graph rewritings are discussed using relational calculus (theory of binary relations). Single pushout approaches to graph rewritings proposed by Raoult and Kennaway are compared with our rewritings of relational (labeled) graph. Moreover, a more general sufficient condition for two rewritings to commute and a theorem concerning critical pairs useful to demonstrate the confluency of graph rewriting systems are also given.

Dedekind categories with cutoff operators

Dedekind categories provide a suitable categorical framework for lattice-valued binary relations. It is known that the notion of crispness cannot be described by the basic tools of this theory only. In this paper we will study Dedekind categories with a cutoff operator in order to circumvent this shortage. We will introduce and investigate the properties of the operator and its relationship with other tools previously used for the same purpose. The main result of this paper is a representation theorem for Dedekind categories with a cutoff operator satisfying the point axiom.

Period lengths of cellular automata on square lattices with rule 90

This paper studies two‐dimensional cellular automata ca−90(m,n) having states 0 and 1 and working on a square lattice of size (m−1)×(n−1). All their dynamics, driven by the local transition rule 90, can be simply formulated by representing their configurations with Laurent polynomials over a finite field F2={0,1}. The initial configuration takes the next configuration to a particular configuration whose cells all have the state 1. This paper answers the question of whether the initial configuration lies on a limit cycle or not, and, if that is the case, some properties on period lengths of such limit cycles are studied.

Lattices in dedekind categories

Lattice structures are fundamental and useful in mathematics and theoretical computer science. It is well-known that lattice structures with meet and join operations satisfying associative, commutative and absorption laws are equivalent to lattice structures defined by ordering relations having joins and meets. This paper defines a notion of lattices in Dedekind categories and studies some basic properties of lattice structures. Following relational calculus, an element-free representation of these properties is discussed.