Relation algebra


In mathematics, a relation algebra is a residuated Boolean algebra supporting an involutary unary operation called converse. The motivating example of a relation algebra is the algebra 2^X² of all binary relations on a set X, with R•S interpreted as the usual composition of binary relations. Early forms of relation algebra emerged in the 19th century with the work of Augustus De Morgan, Charles Peirce, and Ernst Schröder. Its present-day purely equational form was developed by Alfred Tarski and his students starting in the 1940s.

Definition

A relation algebra (L, ∧, ∨, ¬, 0, 1, •, I, ▷, ◁,

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) is an algebraic structure such that Since x▷y can be defined in terms of composition and converse as x

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•y, and dually x◁y as x•y

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, it is not necessary to include ▷ or ◁ in the signature, which can therefore be simplified to (L, ∧, ∨, ¬, 0, 1, •, I,

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), the more usual form of the signature for relation algebras. On the other hand x

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is definable as either x▷I or I◁x, whence a relation algebra can have the same signature as a residuated Boolean algebra. With that definition the axioms become (x▷I)▷I = x = I◁(I◁x). But this simply asserts that ▷I and I◁ are involutions. Jónsson and Tsinakis have shown that if either one is an involution then so is the other and they are then the same operation, namely converse. This leads to a particularly straightforward definition: When x◁y is viewed as a form of quotient of x by y, with I as the corresponding multiplicative unit, x

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= I◁x can be understood as the reciprocal of x by syntactic analogy with 1/x, a term some authors use synonymously with converse.Since residuated Boolean algebras are axiomatized with finitely many equations, so are relation algebras, which therefore form a finitely axiomatized variety called RA, the variety of relation algebras.

Examples

1. Any Boolean algebra is made a relation algebra by taking composition (the monoid multiplication •) to be conjunction, i.e. x•y is defined as x∧y. This interpretation of composition uniquely determines converse to be identity (ў = y) and the residuals yx and x/y both to be implication y→x, that is, ¬y∨x.2. The motivating example of a relation algebra depends on the definition of a binary relation R on a set X as any subset R ⊆ X². The power set 2^X² consisting of all binary relations on X is a Boolean algebra. While 2^X² can be made a relation algebra by taking R•S = R∧S as for the preceding example, the standard interpretation of • is instead given by x(R•S)z = ∃y.xRySz. That is, the pair (x,z) belongs to the relation R•S just when there exists y ∈ X such that (x,y) ∈ R and (y,z) ∈ S. This interpretation uniquely determines R'S to consist of all pairs (y,z) such that for all x ∈ X, if xRy then xSz. Dually S/R consists of all pairs (x,y) such that for all z ∈ X, if yRz then xSz. The translation ў = ¬(y¬I') then establishes the converse R

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of R as consisting of all pairs (y,x) such that (x,y) ∈ R''.3. An important generalization of this example is the power set 2^E where E ⊆ X² is any equivalence relation on the set X. This is a generalization because X² is itself an equivalence relation, namely the complete relation consisting of all pairs. While 2^E is not a subalgebra of 2^X² when E ≠ X² (since in that case it does not contain the relation X², the top element 1 being E instead of X²), it is nevertheless made a relation algebra using the same definitions of the operations. Its importance resides in the definition of a representable relation algebra as any relation algebra isomorphic to a subalgebra of the relation algebra 2^E for some equivalence relation E on some set. It is straightforward to show that the class RRA of all representable relation algebras is the quasivariety generated by the relation algebras of the form 2^X² for some X. Less obvious is that RRA is in fact a variety, shown by Tarski in 1955. Earlier (1950) Lyndon had shown that RRA was a proper subclass of the variety RA (or in logical language, the RA axioms do not completely axiomatize concrete or representable relation algebras). Whereas RA is by definition finitely axiomatizable, RRA was shown by Monk in 1964 not to be finitely axiomatizable.

Expressing properties of binary relations in RA

Many properties of binary relations can be succinctly expressed as equations or inequalities using RA operations, as illustrated by the following.R is total iff I ≤ R•R

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.R is deterministic iff R

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•R ≤ I.A function is a binary relation that is total and deterministic. The next two properties generalize to all binary relations properties that are normally applied just to functions.R is surjective iff I ≤ R

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•R(equivalently if R

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is total).R is injective iff R•R

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≤ I(equivalently if R

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is deterministic).R is reflexive iff I ≤ R.R is transitive iff R•R ≤ R. A preorder is a reflexive transitive binary relation.R is antisymmetric iff R ∧ R

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≤ I. A partial order is an antisymmetric preorder.R is symmetric iff R ≤ R

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. An equivalence relation is a symmetric preorder.

Expressive power

The metamathematics of RA are discussed at length in Tarski and Givant (1987). RA consists entirely of equations manipulated using nothing more than uniform replacement and the substitution of equals for equals. Both rules are wholly familiar from school mathematics and from abstract algebra generally. Hence RA proofs are carried out in a manner familiar to all mathematicians, unlike the case in mathematical logic generally.RA can express any (and up to logical equivalence, exactly the) first-order logic (FOL) formulas containing no more than three variables (a given variable can be quantified multiple times as long as the quantifiers do not nest). Surprisingly, this fragment of FOL suffices to express Peano arithmetic and almost all axiomatic set theories ever proposed. Hence RA is, in effect, a way of algebraizing nearly all mathematics, while dispensing with FOL and its connectives, quantifiers, turnstiles, and modus ponens. Because RA can express Peano arithmetic and set theory, the classic theorems of Gödel apply to it; RA is incomplete, incompletable, and undecidable. (N.B. None of these properties describe the Boolean algebra fragment of RA.)The representable relation algebras, forming the class RRA, are those relation algebras isomorphic to some relation algebra comprised of binary relations on some set and closed under the standard interpretations of the RA operations. It is easily shown, e.g. using the method of pseudoelementary classes, that RRA is a quasivariety, that is, axiomatizable by a universal Horn theory. In 1950, Roger Lyndon proved the existence of equations holding in RRA that did not hold in RA, that is, the variety generated by RRA is a proper subvariety of the variety RA. In 1955 Alfred Tarski showed that RRA is itself a variety, which however, as shown by Donald Monk in 1964, has no finite axiomatization, unlike RA which is finitely axiomatized by definition. That not every relation algebra is representable is one of the fundamental differences from Boolean algebras, all of which are representable as sets of subsets of some set closed under union, intersection, and complement.

Software

• RelMICS / Relational Methods in Computer Science maintained by Wolfram Kahl
• Carsten Sinz: ARA / An Automatic Theorem Prover for Relation Algebras

Historical remarks

DeMorgan founded RA in 1860, but Charles Peirce took it much further and became fascinated with its philosophical power. The work of DeMorgan and Peirce came to be known mainly in the extended and definitive form Ernst Schröder gave it in his Vorlesungen (1890-1905). Principia Mathematica drew strongly on Schröder's RA, but acknowledged him only as the inventor of the notation. The foundational paper for RA as presented here is Tarski (1941); he devised the above axioms, and he and his students have continued to write on RA down to the present day. Much of the content of Tarski and Givant (1987) was worked out by Tarski alone in the 1940s, who returned to the subject in the 1970s with the help of Steven Givant. For more on the history of RA, see Maddux (2006).

See also

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• Allegory (category theory)
• Binary relation
• Cartesian product
• Cartesian square
• Charles Peirce
• Converse of a relation
• Extension in logic

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• Logic of relatives
• Relation
• Relation construction
• Relative product of relations
• Relation reduction
• Relational calculus
• Relational algebra

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• Relative product of relations
• Spatial-temporal reasoning
• Theory of relations
• Triadic relation
• Residuated lattices
• Cylindric algebras

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References

• Carnap, Rudolf, 1958. Introduction to Symbolic Logic with Applications, Dover.
• Halmos, P. R., 1960. Naive Set Theory. Van Nostrand.
• Bjarni Jónsson and Constantine Tsinakis, Relation algebras as residuated Boolean algebras, Algebra Universalis, 30 (1993) 469-478.
• Lucas, John Randolph, 1999. Conceptual Roots of Mathematics. Routledge.
• Roger Maddux, 2006. Relation Algebras, vol. 150 in Studies in Logic and the Foundations of Mathematics. Elsevier Science.
• Leon Henkin, Alfred Tarski, and Monk, J. D., Cylindric Algebras Part 1 (1971) and Part 2 (1985). North Holland.
• Hirsch R., and Hodkinson, I., 2002 "Relation Algebra by Games," v. 147 in Studies in Logic and the Foundations of Mathematics. Elsevier Science.
• Alfred Tarski,1941, "On the calculus of relations," Journal of Symbolic Logic 6: 73-89.
• ------, and Givant, Steven, 1987. A Formalization of Set Theory without Variables. Providence RI: American Mathematical Society.

External links

• Yohji AKAMA, Yasuo Kawahara, and Hitoshi Furusawa, "Constructing Allegory from Relation Algebra and Representation Theorems."
• Richard Bird, Oege de Moor, Paul Hoogendijk, "Generic Programming with Relations and Functors."
• R.P. de Freitas and Viana, "A Completeness Result for Relation Algebra with Binders."
• Peter Jipsen:
  • Relation algebras. In Mathematical structures. If there are problems with LaTeX, see an old HTML versionhere.
  • "Foundations of Relations and Kleene Algebra."
  • "Computer Aided Investigations of Relation Algebras."
  • "A Gentzen System And Decidability For Residuated Lattices."
• Vaughan Pratt:
  • "Origins of the Calculus of Binary Relations." A historical treatment.
  • "The Second Calculus of Binary Relations."
• Priss, Uta, "An FCA interpretation of Relation Algebra."
• Kahl, Wolfram, and Schmidt, Gunther, "Exploring (Finite) Relation Algebras Using Tools Written in Haskell." See homepage of the whole project.

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