Algebraic Geometry over Heyting Algebras

Universal algebraic geometry is a new area of modern algebra, whose subject is basically the study of equations over an arbitrary algebraic structure A (see [11]). In the classical algebraic geometry A of type L is a field. Many articles already published about algebraic geometry over groups, see [1, 8, 16], and [10]. O. Kharlampovich and A.Miyasnikov developed algebraic geometry over free groups to give affirmative answer for an old problem of Alfred Tarski concerning elementary theory of free groups (see [7] and also [15] for the independent solution of Z. Sela). Also in [9], a problem of Tarski about decidablity of the elementary theory of free groups is solved. Algebraic geometry over algebraic structures (universal algebraic geometry) is also developed for algebras other than groups. A systematic study of universal algebraic geometry is done in a series of articles by V.Remeslennikov, A. Myasnikov and E. Daniyarova in [2–4], and [5]. The notations of the present paper are standard and can be find in [2] or [11]. Our main aim in this article is to deal with the equational conditions in the universal algebraic geometry over Heyting algebras, i.e. different conditions relating systems of equations especially conditions about systems and sub-systems of equations over algebras. The main examples of such conditions are equational noetherian property and qω-compactness. We begin with a review of basic concepts of universal algebraic geometry and we describe the properties of being being equational noetherian, qω-compact. We will show that only finite Heyting algebras have these properties.