Algebra Universalis, 34 (1995) 404-423 0002-5240/95/030404-20501.50 + 0.20/0 9 1995 Birkh/iuser Verlag, Basel Total tense algebras and symmetric semiassociative relation algebras P. JIPSEN, R. L. KRAMER AND R. D. MADDUX Abstract. It is well known that the lattice ARA of varieties of relation algebras has exactly three atoms. An unsolved problem, posed by B. J6nsson, is to determine the varieties of height two in ARA. This paper solves the corresponding question for varieties generated by total tense algebras. More specifically,we show that there are exactly four finitelygenerated varieties and infinitelymany nonfinitely generated varieties of height two. In the second half of the paper we show that total tense algebras are term equivalent to certain generalized relation algebras and extend our results to varieties of these algebras. 1. Introduction and definitions Tense algebras are the algebraic counterpart of tense logic, which has been studied and used by modal logicians to reason about tenses in a language and about temporal processes. We first describe a typical tense algebra. Let U be a set and R a binary relation on U. The relation R gives rise to two unary operations f and g on the set of all subsets of U, namely f(X) = the image of X under R = {u 9 U :xRu for some x 9 X} g(J() = the preimage of X under R = {u 9 U: uRx for some x 9 X}. If U is interpreted as a set of events, and R is interpreted as a temporal relation between events in U, then f(J() is the set of events in the future of some event in X, and g(X) is the set of events in the past of some event in X. A tense algebra is obtained if the operations f and g are added to the Boolean algebra of all subsets Presented by B. J6nsson. Received April 8, 1994; accepted in final form December 1, 1994. 404