Algebra Universalis, 26 (1989) 234-246 0002-5240/89/020234-13501.50 + 0.20/0 (~ 1989 Birkh/iuser Verlag, Basel Varieties with decidable finite algebras I: Linearity PAWEt. M. IDZIAK Abstract. The aim of this paper is to prove that every congruence distributive variety containing a finite subdirectly irreducible algebra whose congruences are not linearly ordered has an undecidable first order theory of its finite members. This fills a gap which kept us from the full characterization of the finitely generated, arithmetical varieties (of finite type) having a decidable first order theory of their finite members. Progress on finding this characterization was made in the papers [14] and [15]. 1. Introduction The study of the decision problem for equationally definable classes of algebras has a vast literature. After the results concerning Boolean algebras (A. Tarski [21], 1949), lattices (A. Grzegorczyk [11], 1951), unary algebras (B. A. Trakhenbrot [24], 1953; A. Ehrenfeucht [7], 1959), semigroups (A. Tarski [23], 1953), groups (A. Tarski [23], 1953; W. Szmielew [19], 1955; Ju. L. Ershov [10], 1972; A. P. Zamjatin [27], 1978), commutative semigroups (A. Tarski [22], and M. A. Taitslin [20], 1962), relatively complemented distributive lattices (Ju. L. Ershov [8], 1964), varieties generated by a primal algebra (Ju. L. Ershov [9], 1967), vector spaces over a finite field (P. Eklof and E. Fisher [6], 1972), rings (S. D. Comer [5], 1974; A. P. Zamjatin [26], 1976), monadic algebras (S. D. Comer [4], 1975; M. Rubin [18], 1976), the sheaf-technique of S. D. Comer [5] was extended by H. Werner [25] to prove that every finitely generated discriminator variety has a decidable first order theory. Excellent results of S. D. Comer, H. Werner and A. P. Zamjatin were used by S. Burris and R. McKenzie [1] and R. McKenzie and M. Valeriote [16] to effectively reduce the problem of classifying the finitely generated varieties of finite type with decidable first order theories to the (still open) problem of classifying the finite rings 9] such that the class of unitary left 9t-modules has a decidable first order theory. Moreover, from [1] it follows that among congruence distributive finitely generated varieties of finite type the only ones with decidable theories are exactly discriminator varieties. Presented by Stanley Burris. Received February 22, 1988 and in final form November 10, 1988. 234