Abstract The transitive closure of a binary relation R can be thought of as the best possible approximation of R “from above” by a transitive relation. We consider the question of approximating a relation from below by transitive relations. Our main result is that every thick relation (a relation whose complement contains no infinite chain) on a countable set has a transitive thick subrelation. This allows for a solution to a problem arising from previous work by the author and Alan Taylor. We also exhibit a thick relation on an uncountable set with no transitive thick subrelation.

The universal Horn theory of relational Kleene algebra with tests (RKAT) is of practical interest, particularly for program semantics. We develop an (infinitary) proof system, based on well-founded trees of finite automata, which is sound and complete for this theory. A small modification of this system yields a proof system which is sound and complete for the universal Horn theory of *-continuous Kleene algebras with tests (KAT*). This sheds light on the relationship between RKAT and KAT*.

Given a universal Horn formula of Kleene algebra with hypotheses of the form r = 0, it is already known that we can efficiently construct an equation which is valid if and only if the Horn formula is valid. This is an example of elimination of hypotheses, which is useful because the equational theory of Kleene algebra is decidable while the universal Horn theory is not. We show that hypotheses of the form r = 0 can still be eliminated in the presence of other hypotheses. This lets us extend any technique for eliminating hypotheses to include hypotheses of the form r = 0.

This paper is a contribution to the project of determining which set existence axioms are needed to prove various theorems of analysis. For more on this project and its history we refer the reader to [1] and [2]. We work in a weak subsystem of second order arithmetic. The language of second order arithmetic includes the symbols 0, 1, =, <, +, ·, and ∈, together with number variables x, y, z, … (which are intended to stand for natural numbers), set variables X, Y, Z, … (which are intended to stand for sets of natural numbers), and the usual quantifiers (which can be applied to both kinds of variables) and logical connectives. We write ∀x < t φ and ∃x < t φ as abbreviations for ∀x(x < t → φ) and ∃x{x < t ∧ φ) respectively; these are called bounded quantifiers. A formula is said to be if it has no quantifiers applied to set variables, and all quantifiers applied to number variables are bounded. It is if it has the form ∃xθ and it is if it has the form ∀xθ, where in both cases θ is . The theory RCA0 has as axioms the usual Peano axioms, with the induction scheme restricted to formulas, and in addition the comprehension scheme, which consists of all formulas of the form where φ is , ψ is , and X does not occur free in φ(n). (“RCA” stands for “Recursive Comprehension Axiom.” The reason for the name is that the comprehension scheme is only strong enough to prove the existence of recursive sets.) It is known that this theory is strong enough to allow the development of many of the basic properties of the real numbers, but that certain theorems of elementary analysis are not provable in this theory. Most relevant for our purposes is the fact that it is impossible to prove in RCA0 that every continuous function on the closed interval [0, 1] attains maximum and minimum values (see [1]). Since the most common proof of the Mean Value Theorem makes use of this theorem, it might be thought that the Mean Value Theorem would also not be provable in RCA0. However, we show in this paper that the Mean Value Theorem can be proven in RCA0. All theorems stated in this paper are theorems of RCA0, and all of our reasoning will take place in RCA0.