IntroductionA subresidwted lattice (abbreviated s.r. lattice) is a pair (A, Q), where A is a bounded distribut.ive lattice (with largest element 1 and smallest element 0), and Q is a sublattice of A containing 0, 1 such that for each x, y E A there is an element x E Q with the property that for all q E Q, x A q 5 y if and only if q z. This z is denoted byx --+ y, or simply x 4 y. We shall sometimes refer to the 8.r. lattice by A. When Q = A, A is usually called a HEYTDIG azgebra: (or ~&-BOOLEW &Igebra). "he set of complemented elements of A is called the center of A , and is denoted by B(A). If x E B(A), its complement is denoted by --x.Suppose we have a propositional calculus with symbols &, v, 3 andfor conjunction, disjunction, implication and negation, and possibly with a symbol 0 for affirmation or necessitation. If we assign to the propositional variables values in a s.r. lattice, then we obtain a valuation w(a) for eaoh formula 01 by the. oc is said to be wlid in A if w(a) = 1 for every assignment in A. A logic is said t o be chrmkrized by a class K of s.r. lattices if it consists of those formulas which are valid in every member of K.This framework provides a unified method of classifying several known calculi and leads naturally to new calculi which are of some algebraic and philosophic interest. For example, it is well known that the intuitionist propositional calculus is characterized by the class of all HEYTING algebras.I n the LEWIS systems 54 and S 5 of modal logic, there are two kinds of implication:classical or material implication, denoted by oc 3 B, and strict implication, denoted by u (a r, 8). There are also classical negation, denoted by NLX, and strict negation, denoted by 0oc. It will be convenient for us to change the notation as follows:we shall use a =I /? and -a for strict implication and strict negation. The notation for classical negation will be -a and classical implication, previously denoted by u 2 b will be denoted by -a vb. With this notation, the set of all theorems of 54 (or 55) which involve the strict connectives 3 and N together with & and v is called the LEWY caZcu2us1) for S4 (or S5) by HACKING [4]. We shall use R4 and R6 to denote these LEWY calculi. HACJUNG gave a set of axioms for R4 and R5 using GENTZEN methods. We shall see that R4 is characterized by the class of all 8.r. lattices, and R5 is characterized by the class of all sx. lattices (A, Q) such that Q is a BooLEan subalgebra of the center of A.The LEWIS S4 and 55 systems are characterized by the class of all s.r. lattices (A, Q) such that A is a Booman algebra (and for the case of S5, Q is a BooLEan subalgebra v 1) After C s s m LEWY, who should not be confused with C. I. LEWIS. X -+ y h< (Z V 2) + (y V 2). Also (i) holds because (1 + q) A (q + 1 ) = q for all q E Q.Corollary 3. A 8 . 9 ' . lattice (A, Q ) is subdh'ectly irreducible if and only if (x E Q : x < 1) has a largest element.Proof. By Theorem 2, (A, &) is subdirectly irreducible if and only if Q has a smallest filter properly containing (1). This is eas...

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Rosenbloom's axioms are based on a minimum of undefined operations and are therefore quite complicated. This complexity also hinders his development of the theory. In this paper, a set of axioms for Post algebras is presented which makes use of a greater number of operations, as well as certain constants. These operations Co, ■ ■ ■ , Cn-i are generalized complementation operators, where 77 is the order of the Post algebra. The axioms, given in terms of these operators, are very simple. In addition, the simplicity of the operations makes a large part of the theory much more transparent. Another striking feature of the development is the role played by the underlying Boolean algebra of the Post algebra. The existence of this Boolean algebra has been known for a long time, but this fact has not been as fully exploited as in this approach. It will be shown, for example, that the representation theory for Post algebras follows immediately from the corresponding theory for Boolean algebras. No further use of the Axiom of Choice is needed. In addition many properties of a Post algebra, such as completeness, infinite distributivity, and the atomistic property, are fully mirrored by the corresponding properties for the underlying Boolean algebra. The notation is explained in §2. §3 presents the axioms, and various theorems and remarks concerning the arithmetic and structure of the algebra. §4 discusses Post functions and their reduction to a given form. Examples are given in §5. The representation theory is described in §6, and §7 discusses completeness properties of infinite Post algebras. 2. Notation. The usual lattice notation is employed. The supremum of x and y is denoted by xVy, and the infimum of x and y is denoted by xAy, or more briefly, by xy. The symbols Vx< and Ax,-denote, respectively, the supremum and infimum of the x,-over a specified index set. The symbols Vl; Xi and Ai; Xi emphasize that the supremum or infimum is taken in the lattice P. If x has a complement, it is denoted by {x}~ or, if convenient, by x. 3. Formulation. Let 77 be a fixed integer satisfying 77 2:2. Let L he a distributive lattice with zero 0 and unit u, and satisfying the following conditions : Axiom 1. For every element x£P there exist 77 elements C0(x), Ci(x), • • • , C_i(x) which are pairwise disjoint and whose supremum is u; that is, C,-(x)Cy(x) =0 for i?*j and VJTo1 C,(x) =u.

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