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Introduction

Roger Maddux, retired from the Departments of Mathematics and Computer Science at Iowa State University, does research on relation algebras, part of Logic and Foundations of Mathematics and Theory of Computation. The recently completed project 'Relevance logic' is on the interconnection between relevance logic and relation algebras.

Additional affiliations

January 2008 - March 2008

January 2008 - March 2008

August 1977 - July 2012

## Publications

Publications (125)

It is well-known that general constraint satisfaction problems (CSPs) may be reduced to the binary case (BCSPs) [Pei92]. CSPs may be represented by binary constraint networks (BCNs), which can be represented by a graph with nodes for variables for which values are to be found in the domain of interest, and edges labelled with binary relations betwe...

Two theorems are presented that extend Arrow's Theorem to relations that are not necessarily complete. Let U be a set with three or more elements, let WW be the set of weak orderings of U , let LL be the set of linear orderings of U , and let f be an n -ary function mapping WnWn to WW. By Arrow's Impossibility Theorem, if f satisfies Arrow's Condit...

Algebras introduced by, or attributed to, Sugihara, Belnap, Meyer, and Church are representable as algebras of binary relations with set-theoretically defined operations. They are definitional reducts or subreducts of proper relation algebras. The representability of Sugihara matrices yields sound and complete set-theoretical semantics for R-mingle...

The Tarskian classical relevant logic TR arises from Tarski's work on the foundations of the calculus of relations and on first-order logic restricted to finitely many variables, presented by Tarski and Givant their book, A Formalization of Set Theory without Variables, and summarized in first nine sections. TR is closely related to the well-known...

The "unsharpness problem" is solved by the construction of a finite nonintegral relation algebra with atoms p, q, u, v, x, y such that certain conditions are satisfied. These conditions are expressed as equations. If the atoms were actually binary relation, the equations would assert that (1) p and q are functions, (2) p and q have the same domain,...

Part of this paper is a study of a few finite integral relation algebras that arose in a search for examples of small relation algebras that are weakly representable but not representable. We find several non-representable algebras by splitting atoms and varying the cycle structures of some integral relation algebras without 1-cycles. The absence o...

The Theorems of Pappus and Desargues (for the projective plane over a field) are generalized here by two identities involving determinants and cross products. These identities are proved to hold in the three-dimensional vector space over a field. They are closely related to the Arguesian identity in lattice theory and to Cayley-Grassmann identities...

The Tarskian classical relevant logic TR arises from Tarski’s work on the foundations of the calculus of relations and on first-order logic restricted to finitely many variables, presented by Tarski and Givant their book, A Formalization of Set Theory without Variables, and summarized in first nine sections. TR is closely related to the well known...

Canonical relativized cylindric set algebras are used to sharpen the relative representation theorem for weakly associative relation algebras, that every complete atomic weakly associative relation algebra is isomorphic with the relativization of a set relation algebra to a symmetric and reflexive binary relation, by insuring that the atoms of the...

The Theorems of Pappus and Desargues are generalized by two special formulas that hold in the three-dimensional vector space over a field.

Algebras introduced by, or attributed to, Sugihara, Belnap, Meyer,
and Church are representable as algebras of binary relations with
set-theoretically defined operations. They are definitional reducts
or subreducts of proper relation algebras. The representability of
Sugihara matrices yields sound and complete set-theoretical
semantics for R-mingle...

Tarski's classical relevant logic TR arises from his work on the foundations of the calculus of relations and on first-order logic restricted to finitely many variables. The theorems of TR are defined here as the formulas whose translations into first-order logic of binary relations can be proved using no more than four variables from the assumptio...

Report on two papers posted to arXiv in January and March 2019.

Canonical relativized cylindric set algebras and set relation algebras are used to prove that every complete atomic weakly associative relation algebra is isomorphic to the relativization of a set relation algebra to a symmetric and reflexive binary relation.

Tarski's relevance logic is defined and shown to contain many formulas and derived rules of inference. The definition arises from Tarski's work on first-order logic restricted to finitely many variables. It is a relevance logic because it contains the Basic Logic of Routley-Plumwood-Meyer-Brady, has Belnap's variable-sharing property, and avoids th...

Tarski's relevance logic is defined and shown to contain many formulas and derived rules of inference. The definition arises from Tarski's work on first-order logic restricted to finitely many variables. It is a relevance logic because it contains the Basic Logic of Routley-Plumwood-Meyer-Brady, has Belnap's variable-sharing property, and avoids th...

Tarski's relevance logic is defined and shown to contain many formulas and derived rules of inference. The definition arises from Tarski's work on first-order logic restricted to finitely many variables. It is a relevance logic because it contains the Basic Logic of Routley-Plumwood-Meyer-Brady, has Belnap's variable-sharing property, and avoids th...

Sugihara's relation algebra is a complete atomic proper relation algebra that contains chains of relations isomorphic to Sugihara's original matrix. Belnap's relation algebra (better known as the Point Algebra) is a proper relation algebra containing chains of relations isomorphic to the 2-element and 4-element Sugihara matrices. In addition, the c...

In this note, we prove that two different finite relation algebras are representable over finite sets. We give an explicit group representation of 5265\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidema...

An error in a proof of a correct theorem in the classic paper, Boolean Algebras with Operators, Part I, by Jónsson and Tarski is discussed.

The variety of representable relation algebras is closed under canonical extensions but not closed under completions. What variety of relation algebras is generated by completions of representable relation algebras? Does it contain all relation algebras? It contains all representable finite relation algebras, and this paper shows that it contains m...

We prove that any equational basis that defines representable relation algebras (RRA) over weakly representable relation algebras (wRRA) must contain infinitely many variables. The proof uses a construction of arbitrarily large finite weakly representable but not representable relation algebras whose “small” subalgebras are representable.

The title theorem is proved by example: an algebra of binary relations, closed under intersection and composition, that is not isomorphic to any such algebra on a finite set.

This paper presents a proof of Gallai's Theorem, adapted from A. Soifer's
presentation in The Mathematical Coloring Book of E. Witt's 1952 proof of
Gallai's Theorem.

Many finite symmetric integral non-representable relation algebras, including
almost all Monk algebras, can be embedded in the completion of an atomic
symmetric integral representable relation algebra whose finitely-generated
subalgebras are finite.

We prove that any equational basis that defines \RRA\ over \wRRA\ must
contain infinitely many variables. The proof uses a construction of arbitrarily
large finite weakly representable but not representable algebras whose "small"
subalgebras are representable.

This is a one-page extended abstract for the conference, Lattices and Relations 2012.

Many finite symmetric integral non-representable relation alge-bras, including almost all Monk algebras, can be embedded in the completion of an atomic symmetric integral representable relation algebra whose finitely-generated subalgebras are finite.

It is known that for all finite n ≥ 5, there are relation algebras with n-dimensional
relational bases but no weak representations.
We prove that conversely, there are finite weakly
representable relation algebras with no n-dimensional relational
bases. In symbols: neither of
the classes RAn and wRRA contains the other.

The contemporary theory of relation algebras is a direct outgrowth of the nineteenth century calculus of relations. After
a few examples illustrating the calculus of relations (the most widely applied part of the subject), this chapter touches
upon some topics in the algebraic theory of relation algebras: basic definitions, examples, constructions,...

Sound and complete semantics for classical propositional logic can be obtained by interpreting sentences as sets. Replacing sets with commuting dense binary relations produces an interpretation that turns out to be sound but not complete for R. Adding transitivity yields sound and complete semantics for RM, because all normal Sugihara matrices are...

Relevance logics are known to be sound and complete for relational semantics with a ternary accessibility relation. This paper investigates the problem of adequacy with respect to special kinds of dynamic semantics (i.e., proper relation algebras and relevant families of relations). We prove several soundness results here. We also prove the complet...

Let K N denote the complete graph on N vertices with vertex set V=V(K N ) and edge set E=E(K N ). For x,y∈V, let xy denote the edge between the two vertices x and y. Let L be any finite set and ℳ⊆L 3 . Let c:E→L. Let [n] denote the integer set {1,2,⋯,n}. For x,y,z∈V, let c(xyz) denote the ordered triple (c(xy),c(yz),c(xz)). We say that c is good wi...

The class of finite symmetric integral relation algebras with no 3-cycles is a particularly interesting and easily analyzable
class of finite relation algebras. For example, it contains algebras that are not representable, algebras that are representable
only on finite sets, algebras that are representable only on infinite sets, algebras that are r...

Review begins on p.308.
Eric Schechter. Classical and nonclassical logic: an introduction to the mathematics of propositions.
Princeton University Press, Princeton and Oxford, 2005, x + 507 pp.

Review of Tarski's biography begins on page 535.

Review for the Bulletin of Symbolic Logic of the book:
Alfred Tarski, Life and Logic Cambridge University Press, Cambridge, 2004, vi + 425 pp., by Anita Burdman Feferman and Solomon
Feferman

There are nonrepresentable relation algebras generated by their functional elements. This solves a problem posed many years ago. The number of generating functional elements can be as low as 2. This leaves open the problem whether there is a nonrepresentable relation algebra generated by a single functional element.

There are nonrepresentable relation algebras generated by their functional elements. This solves a problem posed many years ago. The number of generating functional elements can be as low as 2. This leaves open the problem whether there is a nonrepresentable relation algebra generated by a single functional element. 1.

Self-similar distributions of species across a landscape have been proposed as one potential cause of the well-known species-area relationship. The best known of these proposals is in the form of a probability rule for species occurrence. The application of this rule to the number of species occurring in primary well-shaped rectangles within the la...

Volume 9, Number 4, Dec. 2003 REVIEWS The Association for Symbolic Logic.
Review of Hirsch-Hodkinson begins on page 515.

Review for the Bulletin of Symbolic Logic of the book: Relation Algebras by Games, by Robin Hirsch and Ian Hodkinson

The Bulletin of Symbolic Logic, Volume 9, Number 1, March 2003.
Review begins on first page (p.37)

Review for the Bulletin of Symbolic Logic of the monograph:
Decision problems for Equational Theories of Relation Algebras. Memoirs of the American Mathematical Society, vol. 126, no. 604. American Mathematical Society, Providence, March 1997, xiv + 126 pp., by Hajnal Andréka , Steven Givant, and Istvan Németi.

For every finite n 4 there is a logically valid sentence 'n with the following properties: 'n contains only 3 variables (each of which occurs many times); 'n contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol); 'n has a proof in first-order logic with equality that contains exactly n va...

We confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal α ≥ 3 there is a logically valid sentence X, in a...

Relation were invented by Tarski and his collaborators in the middle of the twentieth century. The concept of integrality arose naturally early in the history of the subject, and so did various constructions of nite integral relation algebras. Later the concept of nite-dimensionality was introduced for classifying nonrepresentable relation algebras...

Review of four papers. Three by Jean Larson, one by Paul Erdos, Vance Faber, and Jean Larson. All four involve cylindric set algebras.

The relational calculus MU was presented in Willem-Paul de Roever's dissertation as a framework for describing and proving properties of programs. MU is axiomatized by de Roever in stages. The next-to-last stage is the calculus MU2, namely MU without the recursive μ-operator. Its axioms include typed versions of Tarski's axioms for the calculus of...

Harte et al . ([1][1]) assumed the probability rule: if a species occurs in an area A , then the probability that it occurs in half of that area is a constant, a , independent of area A , satisfying 0.5 ≤ a ≤ 1. From this rule, Harte et al . ([1][1]) give a mathematical proof of the power law

Constraint networks over relation algebras are defined. Compass algebras are introduced for reasoning about space. They are related to the interval algebras, which are used for reasoning about time. The problem of determining whether a network has a closed zeroless reduction is shown to NP-complete for almost all compass and interval algebras. This...

Excerpts from the book "Relation Algebras", Elsevier, 2006.

The first half is a tutorial on orderings, lattices, Boolean algebras, operators on Boolean algebras, Tarski's fixed point theorem, and relation algebras. In the second half, elements of a complete relation algebra are used as ''meanings'' for program statements. The use of relation algebras for this purpose was pioneered by de Bakker and de Roever...

The central result of this paper is that every pair-dense relation algebra is completely representable. A relation algebra is said to be pair-dense if every nonzero element below the identity contains a "pair". A pair is the relation algebraic analogue of a relation of the form fha; ai ; hb; big (with a = b allowed). In a simple pair-dense relation...

The calculus of relations was created and developed in the second half of the nineteenth century by Augustus De Morgan, Charles Sanders Peirce, and Ernst Schroder. In 1940 Alfred Tarski proposed an axiomatization for a large part of the calculus of relations. In the next decade Tarski's axiomatization led to the creation of the theory of relation a...

Review for the Journal of Symbolic Logic of the book with ten papers: Arrow logic and multi-modal logic, edited by Maarten Marx, László Pólos, and Michael Masuch, Studies in logic, language and information, CSLI Publications, Stanford, and FoLLI, 1996, also distributed by Cambridge University Press, New York, xiv + 247 pp.
Ten papers by Venema, M...

The relational calculus was presented in Willem-Paul de Roever's dissertation as a framework for describing and proving properties of programs. is axiomatized by de Roever in stages. The next-to-last stage is the calculus , namely without the recursive -operator. Its axioms include typed versions of Tarski's axioms for the calculus of relations, to...

The sequential calculus of von Karger and Hoare (18) is designed for reasoning about sequential phenomena, dynamic or temporal logic, and concurrent or reactive systems. Unlike the classical calculusof relations,it has no operationfor forming the converse of a relation. Sequentialalgebras(15) are algebras that satisfy certain equations in the seque...

This paper presents solutions or partial solutions for several problems in the theory of relation algebras. In a simple relation
algebra an element x satisfying the condition (a) must be an atom of . It follows that x must also be an atom in every simple extension of . Andréka, Jónsson and Németi [1, Problem 4] (see [12, Problem P5]) asked whether...

Let k be any cardinal. The free semiassociative relation algebra, free
relation algebra, and free representable relation algebra on k
generators are characterized as algebras of equivalence classes of formulas
that arise naturally from the first-order language with equality and k
binary relation symbols by varying the number of variables in the l...

this paper are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of Kestrel Institute, or any agency of the United States Government

Copyright c flPeter Ladkin and Roger Maddux 1986 1

The concepts of binary constraint satisfaction problems can be naturally generalized to the relation algebras of Tarski. The concept of path-consistency plays a central role. Algorithms for path-consistency can be implemented on matrices of relations and on matrices of elements from a relation algebra. We give an example of a 4-by-4 matrix of infin...

Applying combinatorial methods, we prove that the symmetric relation algebra ɛn+1(1, 2, 3) ofn+1 atoms is finitely representable for alln ≳ 1, on at most (2+o(1))n2 elements asn → ∞. We explicitly construct a representation of size 4.5n2, for every n >1.

It is well known that the lattice
RA
of varieties of relation algebras has exactly three atoms. An unsolved problem, posed by B. Jnsson, is to determine the varieties of height two in
RA
.This paper solves the corresponding question for varieties generated by total tense algebras. More specifically, we show that there are exactly four finitely ge...

Abstract There are eighteen isomorphism types of finite relation algebras with eight or fewer elements, and all of them are representable. We determine all the cardinalities of sets on which these algebras have representations. 1Introduction We say that a relation algebra is small if it has no more than eight elements. A relation algebra is a Boole...

If K is a class of semiassociative relation algebras and K contains the relation algebra of all binary relations on a denumerable set, then the word problem for the free algebra over K on one generator is unsolvable. This result implies that the set of sentences which are provable in the formalism ℒw
× is an undecidable theory. A stronger algebraic...

This paper presents a brief introduction to relation algebras, including some examples motivated by work in computer science, namely, the `interval algebras', relation algebras that arose from James Allen's work on temporal reasoning, and by `compass algebras', which are designed for similar reasoning about space. One kind of reasoning problem, cal...

This review of the complete works of Tarski includes detailed content information not in the original and corrects an error by which a non-Tarski problem was substituted for one of Tarski's problems.

This is a supplement to the paper “Finitary Algebraic Logic” [1]. It includes corrections for several errors and some additional results. MSC: 03G15, 03G25.

This paper presents a brief introduction to relation algebras, in- cluding some examples motivated by work in computer science, namely, the 'interval algebras', relation algebras that arose from James Allen's work on temporal reasoning, and by 'compass alge- bras', which are designed for similar reasoning about space. One kind of reasoning problem,...

Venema proved the true relational equations can be proved from the axioms of relation algebras together with a special rule. The axioms can be weakened and Venema's rule can be weakened, but not both.

Conjecture (1) of [Ma83] is confirmed here by the following result: if $3 \leq \alpha < \omega$, then there is a finite relation algebra of dimension $\alpha$, which is not a relation algebra of dimension $\alpha + 1$. A logical consequence of this theorem is that for every finite $\alpha \geq 3$ there is a formula of the form $S \subseteq T$ (asse...

The central result of this paper is that every pair-dense relation algebra is completely representable. A relation algebra is said to be pair-dense if every nonzero element below the identity contains a “pair”. A pair is the relation algebraic analogue of a relation of the form (⟨a, a⟩, ⟨b, b⟩) (with a = b allowed). In a simple pair-dense relation...

By constructing special relation algebras we show that if 3<α<ω, then SNr 3 CA α ≠SNr 3 CA 3α-7 and there is a logically valid first-order sentence containing at most three variables with a proof in which every sentence has at most 3α-7 variables, but no proof in which every sentence has at most α variables.

By constructing special relation algebras we show that if 3 < α <ω, then SNr3CAα ≠ SNr3CA3α − 7 and there is a logically valid first-order sentence containing at most three variables with a proof in which every sentence has at most 3α − 7 variables, but no proof in which every sentence has at most a variables.

We define a way, which we call splitting, of getting new relation algebras from old ones. We characterize those algebras to which splitting can be applied. We show how to split representable relation algebras in order to obtain nonrepresentable ones, and we give many examples.

These are notes for a short course on relation algebras, finite-dimensional cylindric algebras, and their interconnections, delivered at the Conference on Algebraic Logic, Budapest, Hungary, August 8--14, 1988, sponsored by the the Janos Bolyai Mathematical Society.
Published in Algebraic Logic (Proc. Conf. Budapest 1988) ed. by H. Andreka, J. D....

There are six finite nonintegral representable relation algebras such that every nonintegral simple semiassociative relation algebra has a nontrivial subalgebra isomorphic to one of those six.

There is an incomplete atomic relation algebra which is not the reduct of any 4-dimensional cylindric algebra. This completes the answer to a problem in [Mo61].

The set of equations which use only one variable and hold in all representable relation algebras cannot be derived from any finite set of equations true in all representable relation algebras. Similar results hold for cylindric algebras and for logic with finitely many variables. The main tools are a construction of nonrepresentable one-generated r...

For every suitable relational structure there is a canonical relativized cylindric set algebra. This construction is used to obtain a generalization of Resek's relative representation theorem, and a stronger version of the "Stone type representation theorem" by Andreka and Thompson.

For every suitable relational structure there is a canonical relativized cylindric set algebra. This construction is used to obtain a generalization of Resek's relative representation theorem, and a stronger version of the «Stone type representation theorem» by Andreka and Thompson

## Projects

Projects (3)

Characterize formulas of relevance logic provable with 1, 2, 3,or 4 variables. Show algebras (matrices) appearing in relevance logic research are representable with binary relations. Establish connections between relevance logic, relation algebras, and the Peirce-Schroder-Tarski calculus of binary relations.