Quasi-identities of Finite Semigroups and Symbolic Dynamics Stuart W. Margolis Mark V. Sapir ∗ September 17, 1993 1 Introduction Let us first recall some basic facts and definitions from universal algebra (see [Malcev], [Cohn]). A variety is a class of universal algebras given by identities, i.e. formulas of the type (∀x 1 ,...,x n ) u = v where u = u(x 1 ,...,x n ) and v = v(x 1 ,...,x n ) are terms. For example, the class of all abelian groups is a variety of groups given by the identity (∀x, y ) xy = yx. A quasi-variety is a class of universal algebras given by quasi-identities, i.e. formulas of the type (∀x 1 ,...,x n ) u 1 = v 1 & ... &u m = v m → u = v where u i ,v i , u, v are terms of variables x 1 ,...,x n (see [Malcev] for details). For example, the class of all torsion free groups is a quasi-variety of groups given by the following infinite set of quasi-identities: {x p =1 → x =1|p is a prime}. * Research of both authors were supported in part by NSF and the Center for Commu- nication and Information Sciences of the University of Nebraska at Lincoln. 1