On Free Annotated Algebras

In [4] the authors proved that certain systems of annotated logics are algebraizable in the sense of [1]. Later in [5, 6] the study of the associated quasi{varieties of annotated algebras is initiated. In this paper we continue the study of the these classes of algebras, in particular, we report some recent results about the free annotated algebras. 1 Preliminaries For the sake of conciseness, we will assume that the reader is familiar with the general theory of algebraizability of deductive systems as developed in [1]. This paper is one in a series in which we have applied the theory to a family of deductive systems known as annotated logics We state here without proof the main results about them obtained in [4, 5, 6]. 1.1 Annotated Logics Annotated logics were introduced in [7] by V. S. Subrahmanian as logical foundations for computer programming. A complete study of these systems, from the model theoretical and proof theoretical points of view has been done in [3]. They show that most classical basic results in model theory can be adapted to these systems. However, these systems are not structural in the sense that their consequence relation is not closed under substitutions. Research supported by FONDECYT grants 1960931 and 1990433. 1 In order to work algebraically with them, in [4] we introduced a structural version SP of annotated logics and we proved that they are equivalent, in a very natural sense, to the original P systems. We studied the algebraizability of these systems using as main framework the theory of algebraization of deductive systems developed by Blok and Pigozzi in [1]. The main result proven in [4] is that these systems are algebraizable, moreover, they are nitely algebraizable if and only if the lattice of annotation constants is nite..