Download ON THE NUMBER OF QUASI

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