Finite Monadic Algebras

Introduction. The elementary structure theory for finite Boolean algebras is, for our purposes, summarized in the following theorem, (see [1, p. 163], or [2, p. 344]). (*) If B is a finite Boolean algebra, then it is completely characterized by the number of its elements, a number which must be of the form 2", where n is the number of atoms of B. In this case B is faithfully represented as the Boolean algebra of all subsets of a set with n elements. Moreover, B is a free Boolean algebra if and only if n is itself of the form 2r, in which case B is free on r generators. In particular, a finitely generated Boolean algebra is finite. Recent work of Halmos, Tarski, and others in algebraic logic has engendered interest in the study of more general systems in which the Boolean structure is augmented by introducing certain additional operators. For instance, a closure algebra A is a Boolean algebra with a unary operator 3: .4—>A satisfying the axioms,