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Abstmct. 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 infinite relations on which no iterative local path-consistency algorithm terminates. We give a class of examples over a fixed finite algebra on which all iterative local algorithms, whether parallel or sequential, must take quadratic time. Specific relation algebras arising from interval constraint problems are also studied: the Interval Algebra, the Point Algebra, and the Containment Algebra.

ABSTRACT. Three varieties of algebras are introduced which extend the variety RA of relation algebras. They are obtained from RA by weakening the associative law for relative product, and are consequently called nonassociative, weakly-associative and semiassociative relation algebras, or NA, WA, and SA, respectively. Each of these varieties arises naturally in solving various problems concerning relation algebras. We show, for example, that WA is the only one of these varieties which is closed under the formation of complex algebras of atom structures of algebras, and that WA is the closure of the variety of representable RA's under relativization. The paper also contains a study of the elementary theories of these varieties, various representation theorems, and numerous examples.O. Introduction. Relation algebras (RA 's) have a binary operation; which serves as an abstract algebraic analogue of the relative product of binary relations. (The relative product of R, S C U X U, is R I S = {(x, z):(x, y) E Rand (y, z) E S for some y E U}.) The relative product is associative, and one of the postulates for RA's is that ; is associative. The associativity of relative product can be expressed by a sentence in a first-order language with binary relation symbols, namelyAlthough this sentence has three variables, it cannot be proved from the ordinary axioms of first-order logic without using four variables. In contrast, all the other postulates for relation algebras can not only be expressed but proved using only three variables. (These facts were first proved by Tarski. For a proof that (1) requires four variables to prove, see [3].) Tarski asked whether there are any other equations whose translations into first-order sentences can be expressed and proved using only three variables, but which are not derivable from the postulates for

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 relation algebras, a method for obtaining cylindric algebras from relation algebras, and the use of relation algebras in defining algebraic semantics for first-order logic.

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 landscape gives rise to a discrete power law for species-area relationships. However, this result requires a specific scheme for bisecting the landscape to generate the rectangles. Some additional, more general consequences of the probability rule are presented here. These include the result that the number of species in a well-shaped rectangle depends on its location, not just on its area. In addition, a self-similar landscape contains well-shaped rectangles that are, in fact, not self-similar. The probability rule in general produces testable predictions about how and where species are distributed that are independent of the power law.