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Bulletin of the Section of Logic Volume 14/3 (1985), pp. 99–101 reedition 2007 [original edition, pp. 99–102] Jacek Malinowski ON THE NUMBER OF QUASI-MODAL ALGEBRAS The number of quasi-modal algebras defined on a finite Boolean algebra is determined. Let A be a Boolean algebra with additional unary operation 2. A is called a quasi-modal algebra iff 2(a ∧ b) = 2a ∧ 2b, for all a, b ∈ A. A quasi-modal algebra in which 21 = 1 is called a modal algebra. Our considerations are based on the following fairly simple fact. Lemma 1. Every finite Boolean algebras A is isomorphic with the algebra of all subsets 2n of an n-element set, for some n. Let A be a finite Boolean algebra. Thus A has cardinality 2n for some n ∈ ω. We inductively define the notion of rank of an element a ∈ A. An element a ∈ A has rank 0 iff it is the unit of A, a = 1. An element a has rank k + 1 if it is covered (in the order of A) by an element of rank k. Thus a has rank 1 if it is a co-atom in A (i.e. the complement of an atom). If we identity A with the algebra 2n , then it is easy to verify that the elements of 2n of rank k are precisely the (n − k)-element subsets of 2n . Consequently, (1) if A has cardinality 2n , then the number of elements of A of rank k n! equals (nk ) = k!(n − k)! and (2) given an element a of rank k, the cardinality of the principle ideal (a] = {x ∈ A : x ≤ a} equals 2k . 100 Jacek Malinowski Lemma 2. Let A be a finite Boolean algebra of cardinality 2n and let CAt = {c1 , . . . , cn } be the set of all co-atoms of A. Assume that 2∗ is an unary function defined on CAt ∪ {1} with values in A such that 2∗ ci ≤ 2∗ 1, for i = 1, . . . , n. Then there is an unique extension of 2∗ to an operation 2 : A → A satisfying the equation (3) 2(x ∧ y) = 2x ∧ 2y (i.e. the resulting algebra is a quasi-modal algebra). Proof. If a < 1 in A then there are elements ck1 , . . . , ckm in CAt (uniquely determined by a) such that a = ck1 ∧ . . . ∧ ckm . We set 2a =df 2∗ ck1 ∧ . . . ∧ 2∗ ckm . A straighforward calculation shows that Box is indeed an extension of 2∗ and that 2 obeys the law (3). Corollary 3. The number of quasi-modal algebras defined on a finite Boolean algebra A is equal to the number of all functions 2∗ : CAt ∪ {1} → A which satisfy the condition: (4) 2∗ c ≤ 2∗ 1, for all c ∈ CAt. Theorem 4. Let A be an 2n -element Boolean algebra. The number of all n P quasi-modal algebras on A equals (1 + 2n )n = (nk ) · 2k·n . k=0 Proof. Let F (k) be the family of all functions 2∗ : CAt ∪ {1} → A such that rank(2∗ 1) = k and 2∗ c ≤ 2∗ 1, for all c ∈ CAt. We claim that (5) |F (k)| = (nk ) · 2k·n . Indeed, assume that an element 2∗ 1 of rank k has been assigned to the unit element 1. Then, by (2), the principle ideal (2∗ 1] in A has 2n−k elements. Consequently, the element 2∗ 1 can be precisely assigned 2 (2n−k )n = 2n −nk functions 2∗ 1 : CAt → (2∗ 1]. Since by (1), there are (nk ) elements of A of rank k, it follows that the cardinality of F (k) equals (nk ) · 2k·n . Form (5) our theorem readily follows. On the Number of Quasi-Modal Algebras 101 Corollary 5. The number of modal algebras defined on a Boolean algebra 2 A of cardinality 2n is equal to 2n . Remark. Corollary 5 also follows from the well-known fact that there is a bijective correspondence between modal operators 2 on A and binary relations on A. A modal algebra is called a Feys algebra if it satisfies the inequality: 2a ≤ a, for all a. Corollary 6. The number of Feys algebras defined on a Boolean algebra of cardinality 2n is equal to 2(n−1)n . (Note that 2(n−1)n is the number of reflexive relations on a set of power n). The Section of Logic Institute of Philosophy and Sociology Polish Academy of Sciences Lódź, Poland