On representable relation algebras.

Monk, Donald

doi:10.1307/mmj/1028999131

Cited by 117 publications

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“…It follows that the axioms (i)-(xv) are incomplete, in the sense that there are equations which hold in every algebra of the form Re(U) but cannot be derived from (i)-(xv). J. Donald Monk [27] proved that the equations that hold in every algebra of the form Re(U) does not have a finite axiomatization.…”

Section: Relation Algebrasmentioning

confidence: 99%

Relation Algebras for Reasoning about Time and Space

Maddux

1994

Algebraic Methodology and Software Technology (AMAST’93)

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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, called a 'constraint satisfaction problem', can be defined for arbitrary relation algebras. It will be shown here that the constraint satisfiability problem is NP-complete for almost all compass and interval algebras. Relation AlgebrasComposition of binary relations was introduced to logic by Augustus De Morgan [28] [29] (see [30, pp. 55-57, 208, 221, etc.]). De Morgan observed that the syllogism "every A is a B, every B is a C, so every A is a C" remains valid if the copula "is" is replaced by any transitive relation L. De Morgan went further, noting that if LM is the composition of the relation L with the relation M , that is, A is an LM of B just in case A is an L of an M of B, then the following syllogism is valid: "if every A is an L of a B, and every B is an M of a C, then every A is an LM of a C." De Morgan [29] (see [30, p. 222]) denoted the converse of the relation L by L −1 and its contrary by not-L, and observed that these operations commute: the converse of the contrary of L is the contrary of the converse of L. Around the same time, George Boole [7] [6] created algebra from the logic of classes. Starting with [31] in 1870, Charles Sanders Peirce applied Boole's ideas to create algebra from De Morgan's logic of relations, "and after many attempts produced a good general algebra of logic, together with another algebra specially adapted to dyadic relations (Studies in Logic, by members of the Johns Hopkins University, 1883, Note B,[187][188][189][190][191][192][193][194][195][196][197][198][199][200][201][202][203]

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Section: Relation Algebrasmentioning

confidence: 99%

Relation Algebras for Reasoning about Time and Space

Maddux

1994

Algebraic Methodology and Software Technology (AMAST’93)

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“…Monk proved that no finite set of first-order sentences can define the class RRA of representable relation algebras [21]. Various strengthenings of this result have been obtained [13,1,25].…”

Section: Introductionmentioning

confidence: 99%

Strongly representable atom structures of relation algebras

Hirsch

Hodkinson

2001

Proc. Amer. Math. Soc.

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Abstract. A relation algebra atom structure α is said to be strongly representable if all atomic relation algebras with that atom structure are representable. This is equivalent to saying that the complex algebra Cm α is a representable relation algebra. We show that the class of all strongly representable relation algebra atom structures is not closed under ultraproducts and is therefore not elementary. This answers a question of Maddux (1982).Our proof is based on the following construction. From an arbitrary undirected, loop-free graph Γ, we construct a relation algebra atom structure α(Γ) and prove, for infinite Γ, that α(Γ) is strongly representable if and only if the chromatic number of Γ is infinite. A construction of Erdös shows that there are graphs Γr (r < ω) with infinite chromatic number, with a non-principal ultraproduct Q D Γr whose chromatic number is just two. It follows that α(Γr) is strongly representable (each r < ω) but Q D (α(Γr)) is not.

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“…Monk proved that RRA is indeed a canonical variety (reported in [15,Theorem 2.12]). On the other hand, relation algebras are badly behaved in a number of ways: RRA cannot be defined by finitely many axioms [17]; the equational theories of RA and RRA are not decidable [18,20,3]; RRA cannot be defined by any set of canonical equations [13] nor by any set of equations using only finitely many variables [14]; the problem of determining whether a finite relation algebra is representable is an undecidable problem [10]. An important line of research is to restrict the signature of relation algebras and study the behaviour of the corresponding representation class.…”

Section: Introductionmentioning

confidence: 99%