Non-embeddable simple relation algebras

Abstract

This paper presents solutions or partial solutions for several problems in the theory of relation algebras. In a simple relation algebra $ \frak U $ an element x satisfying the condition (a) $ 0 \neq x; 1; \breve{x} + \breve{x}; 1; x \leq 1^{'} $ must be an atom of $ \frak U $ . It follows that x must also be an atom in every simple extension of $ \frak U $ . Andreka, Jonsson and Nemeti [1, Problem 4] (see [12, Problem P5]) asked whether the converse holds: if x is an atom in every simple extension of a simple relation algebra, must it satisfy (a)? We show that the answer is “no”.¶ The only known examples of simple relation algebras without simple proper extensions are the algebras of all binary relations on a finite set. Jonsson proposed finding all finite simple relation algebras without simple proper extensions [12, Problem P6]. We show how to construct many new examples of finite simple relation algebras that have no simple proper extensions, thus providing a partial answer for this second problem. These algebras are also integral and nonrepresentable.¶ Andreka, Jonsson, Nemeti [1, Problem 2] (see [12, Problem P7]) asked whether there is a countable simple relation algebra that cannot be embedded in a one-generated relation algebra. The answer is “yes”. Givant [3, Problem 9] asked whether there is some k such that every finitely generated simple relation algebra can be embedded in a k-generated simple relation algebra. The answer is “no”.