Identities in the theory of monadic algebras are equivalent to simpler standard ones. This property is used to prove some well-known results as well as to determine the minimum number of variables needed in an identity characterizing an equational class. Identities are also shown to be preserved under certain types of extension. In this paper we prove that the equations of the theory of monadic algebras can be reduced to simpler standard ones. From this reduction, some properties already proved in [2] easily follow (see Theorem 3.3). We also obtain new results, mainly (1) The determination of the minimum number of variables needed in an equation characterizing a given equational class. (See Theorem 3.4; this question is raised in [2].) (2) A positive answer for the case a = 1 to the following two questions: Does every equation holding in a CA, also hold in its canonical embedding algebra? Does it also hold in its completion algebra? (See Corollary 3.6; these questions were raised by Monk at a seminar in algebraic logic at the University of Colorado.) I wish to thank Donald Monk for much valuable advice and Dan Demaree for useful conversations. I am also indebted to the referee whose suggestions have helped to improve a first version of this paper. 1. Preliminaries. A monadic algebra 91 is considered as a structure (A, +,,-, C) where IA, +,,-) is a boolean algebra and C is a quantifier on (A, +, -). CA1 is the class of all monadic algebras. The first-order language L1 of CA1 has vo, vl, * * * for individual variables but we use xo, . , yo' , etc. to represent arbitrary variables; is used for the formal equality; for the sake of simplicity we write +, ,-, and C for the formal operation symbols corresponding to the concrete operations; 0 and 1 are taken as abbreviations for vo0 vo and vo + -vO Received by the editors October 1, 1970. AMS 1969 subject class.fications. Primary 0248.