Modal-Like Operators in Boolean Lattices, Galois ... - IOS Press

Fundamenta Informaticae 76 (2007) 129–145

129

IOS Press

Modal-Like Operators in Boolean Lattices, Galois Connections and Fixed Points Jouni J¨arvinen∗ Turku Centre for Computer Science, University of Turku, 20014 Turku, Finland [email protected]

Michiro Kondo School of Information Environment, Tokyo Denki University, Inzai 270-1382, Japan

Jari Kortelainen Laboratory of Applied Mathematics, Lappeenranta University of Technology, P.O. Box 20, 53851 Lappeenranta, Finland

Abstract. In this work, four modal-like operators on Boolean lattices are introduced and their theory is presented from lattice-theoretical, topological and algebraic point of view. It is also shown how rough set approximation operators, modal operators in temporal logic, and linguistic modifiers determined by L-sets can be interpreted as modal-like operators.

Keywords: Galois connections, Closure operators, Fixed points, Boolean lattices with additional operators

1.

Background

Kripke interpreted in [18] the modal operators of possibility and necessity as relational operators similar to operators studied by Birkhoff in [3], and J´onsson and Tarski in [12]. It must be noted that already Ore [21] studied Galois connections of relational operators and their topological characteristics. Relational operators may be defined, for example, as follows (cf. [10]): Let U be a non-empty set, R ⊆ U × U , and for all x ∈ U , R(x) = {y ∈ U | x R y} . ∗

Address for correspondence: Turku Centre for Computer Science, University of Turku, 20014 Turku, Finland

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We define two operators on ℘(U ), where ℘(U ) is the power set of U , such that for all X ⊆ U , X N = {x ∈ U | R(x) ∩ X 6= ∅}

(1)

X H = {x ∈ U | R(x) ⊆ X} .

(2)

and

It is clear that also another pair of operators, say M and O, can be defined if we replace R by the inverse R−1 of R in formulae (1) and (2), respectively (see e.g. [11, 17]). Interestingly, the pairs (N,O) and (M,H) are Galois connections on ℘(U ) [6]. The operators N,M ,H and O may be called modal-like operators, because also other interpretations, different from that of Kripke, may be considered. Rough set theory introduced by Pawlak [22] has two key notions: indiscernibility relations and rough set approximations. Rough set approximations are defined in terms of an indiscernibility relation which is an equivalence such that two objects are related if we cannot distinguish them by using our information about their properties. Now X H, called the lower approximation of X, can be interpreted as the set of objects that surely are in X, and X N, X’s upper approximation, can be seen as the set of elements that possibly are in X. In the literature one can find several studies considering rough approximations determined by arbitrary relations. This motivates us to consider the approximations determined by any binary relation in connection with the approximations given by the inverse relation. Goguen generalized fuzzy sets to L-sets, where L is a ‘transitive partially ordered set’, in [8]. We have understood this slightly vague definition in such a way that the set L must have some kind of order structure and the ordering relation is at least transitive. We define an L-set ϕ on U as a mapping ϕ : U → L, where L is a preordered set. Now each L-set ϕ determines naturally a preorder . on U by x . y if and only if ϕ(x) ≤ ϕ(y). Typically, L may consist of attributes such as ‘good’, ‘excellent’, ‘poor’, and ‘adequate’, for example. Assume that ϕ : U → L is an L-set describing what is the ability of persons in U to speak Finnish. Then, for example, x ∈ X M if and only if x can speak Finnish at least as well as a person in X. Furthermore, x ∈ X H if and only if x . y implies y ∈ X, that is, there cannot be a person outside X speaking Finnish at least as well as x. So, also for L-sets, the modal-like operators have nice interpretations: N,M ,H, and O may be called compositional modifiers (see [11, 15, 20], for instance) and they can be interpreted to model linguistic hedges (see e.g. [19, 25]). If R is viewed as a relation ‘later than’, then the operators N, H, M, O can be interpreted as temporal operators ‘sometimes in the future’, ‘always in the future’, ‘sometimes in the past’ and ‘always in the past’, respectively. Note also that in [2, 9] both order-reversing and order-preserving Galois connections are studied in connection with standard modalities. However, in these two mentioned papers the role of Galois connections is essentially different – there Galois connections are interpreted, for example, as ‘intuitionistic’ complements (i.e., not possible) and they are used in connection with modalities, but in our considerations the modal-like operators themselves form Galois connection pairs. Already Birkhoff noticed in [3] that if R ⊆ U × U is a preorder, then N is a closure operator in a unique topology on U . Of course, H is an interior operator in that same topology, and these observations hold for M and O, too. It is well-known (see e.g. [3]) that the induced topologies satisfy the arbitrary inter-

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section property, thus, these topologies are called Alexandrov topologies (see [1]). In this case it is interesting to find out that we have for any X ⊆ U , X N = X NO,

X M = X MH,

X H = X HM,

X O = X ON,

as it can be deduced from [16], for example. In the current paper we study also the fixpoints of the modal-like operators. The fixpoints are important, since, for example, in rough set theory, the set X N is interpreted as a set of elements which possibly belong to the set X when objects are viewed through an indiscernibility relation. A fixpoint X = X N can now be seen as a ‘precise set’, since the set X and the set of elements which possibly are in X are equal. We show that if R ⊆ U × U is a reflexive relation, then {X ⊆ U | X = X N} is an Alexandrov topology on U ; this result appears also in [14]. It is interesting to notice that also {X ⊆ U | X = X N} = {X ⊆ U | X = X O}. Whenever R is also symmetric, the topology {X ⊆ U | X = X N} is closed with respect to complementation. Further, when R is reflexive and transitive, N is the smallest neighborhood operator in the topology {X ⊆ U | X = X N}. The paper is organized as follows: In Section 2 we recall Galois connections and present some properties of dual and conjugate maps. In Section 3 we give several fixpoint results for different types of Galois connections. Finally, in Section 4, we study modal-like operators defined on complete atomic Boolean lattices, and we give the main properties of these operators extending the work [10].

2.

Conjugates, duals, and Galois connections

We perform our study in the setting of ordered sets, that is, sets equipped with a reflexive, antisymmetric, and transitive relation ≤. Most notions and results presented in this section can be found in [4, 5, 6, 12], for example. In order to make the paper self-contained, we also present some proofs of widely known propositions. Let P and Q be ordered sets. A map f : P → Q is called order-preserving, if for any x, y ∈ P , x ≤ y implies f (x) ≤ f (y) in Q. Further, a map f : P → Q is an order-embedding, if for any x, y ∈ P , x ≤ y is equivalent to f (x) ≤ f (x). Note that an order-embedding is always an injection. An order-embedding onto Q is called an order-isomorphism between P and Q. When there exists an order-isomorphism between P and Q, we say that P and Q are order-isomorphic and write P ∼ = Q. W f : P → Q is a complete join-morphism if whenever S ⊆ P and S exists in P , then W A map W W W f (S) = {f (x) | x ∈ S} exists in Q and f ( S) = f (S). Similarly, f : P → Q is a complete meet-morphism if it preserves every existing meet. If f is both a complete join-morphism and a complete meet-morphism, then it is a complete morphism. Notice that if P and Q are bounded, then f (0P ) = 0Q and g(1P ) = 1Q for any complete join-morphism f and a complete meet-morphism g. It is clear that every order-isomorphism is a complete morphism, and that every complete joinmorphism, as well as every complete meet-morphism, is order-preserving. Further, if P is a complete lattice and f : P → Q is a complete join-morphism or a complete meet-morphism, then f (P ) is a complete lattice. Let P be an ordered set and S ⊆ P . A map f : S → P is called extensive if x ≤ f (x) for all x ∈ S. Further, f is called a self-map on P if S = P . A self-map f on P is said to be idempotent if f (f (x)) = f (x) for all x ∈ P .

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A closure operator on an ordered set P is an idempotent, extensive and order-preserving self-map on P . A self-map f is called a topological closure operator (also called a Kuratowski closure operator) on a complete lattice L if it is idempotent, extensive, and satisfies f (0) = 0 and f (a ∨ b) = f (a) ∨ f (b) for all a, b ∈ L (cf. [4, 13]). Further, if f is a closure operator and a complete join-morphism on a complete lattice L, then, in this paper, f is called an Alexandrov closure operator on L (cf. [1, 3]). It is clear that for complete lattices, each Alexandrov closure operator is a Kuratowski closure operator and every Kuratowski closure operator is a closure operator. Moreover, the corresponding interior operators are defined as dual concepts canonically. Next we consider Galois connections. There are two theoretically equivalent ways to define Galois connections; the one adopted here, in which the maps are order-preserving, and the other in which they are order-reversing. The definition used here and the basic properties given in Propositions 2.1 and 2.2 can be found in [5, 6], for example. Definition 2.1. For two ordered sets P and Q, a pair (f, g) of maps f : P → Q and g : Q → P is called a Galois connection between P and Q if for all p ∈ P and q ∈ Q, f (p) ≤ q ⇐⇒ p ≤ g(q). The map g is called the adjoint and f is called the co-adjoint. Moreover, if (f, g) is a Galois connection, then we say that f has an adjoint and g has a co-adjoint. It is known that for f : P → Q and g : Q → P , a pair (f, g) is a Galois connection if and only if (a) the maps f and g are order-preserving; (b) p ≤ g(f (p)) for all p ∈ P and f (g(q)) ≤ q for all q ∈ Q. The following proposition presents some well-known basic properties of Galois connections. Proposition 2.1. Let (f, g) be a Galois connection between two ordered sets P and Q. (a) For all p ∈ P and q ∈ Q, f (g(f (p))) = f (p) and g(f (g(q))) = g(q). (b) The map f is a complete join-morphism and g is a complete meet-morphism. (c) The map g ◦ f is a closure operator on P and the map f ◦ g is an interior operator on Q. (d) The maps f (p) 7→ g(f (p)) and g(q) 7→ f (g(q)) are mutually inverse order-isomorphisms between f (P ) and g(Q). (e) The maps f and g uniquely determine each other by the equations ^ _ f (p) = {q ∈ Q | p ≤ g(q)} and g(q) = {p ∈ P | f (p) ≤ q} . The next well-known result states when a map on a complete lattice induces a Galois connection. Proposition 2.2. Let L and K be complete lattices. (a) A map f : L → K has an adjoint if and only if f is a complete join-morphism. (b) A map g : K → L has a co-adjoint if and only if g is a complete meet-morphism.

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Proof: We prove (a); the proof for (b) is analogous. If f has an adjoint f a , that is, (f, f a ) is a Galois connection, then by Proposition 2.1(b), f is a complete join-morphism. On the other hand, let f be a complete join-morphism. Let us define for all y ∈ K, f a (y) =

_

{z ∈ L | f (z) ≤ y} .

(3)

W Let x ∈ L and y ∈ K. IfWf (x) ≤ y, then trivially x ≤ {z ∈ L | f (z) ≤ y} = f a (y). Conversely, if x ≤ f a (y), then f (x) ≤ {f (z) | f (z) ≤ y} ≤ y. Thus, (f, f a ) is a Galois connection. u t For complete lattices L and K, the previous proposition shows that for each complete join-morphism f : L → K, the pair (f, f a ), where f a is defined in (3), is a Galois connection. Similarly, for each complete meet-morphism g : K → L, the pair (g a , g) is a Galois connection, where for any x ∈ L, g a (x) =

^

{z ∈ K | x ≤ g(z)} .

In the following, we study conjugate maps on a Boolean lattice. We show, for example, that there is a correspondence between Galois connections and conjugate pairs. We recall the following definition from [12]. Definition 2.2. Let f and g be self-maps on a Boolean lattice B. We say that g is a conjugate of f , if for any x, y ∈ B, we have x ∧ f (y) = 0 ⇐⇒ y ∧ g(x) = 0. It is obvious that if g is a conjugate of f , then f is a conjugate of g. Therefore, in the following we shall say ‘f and g are conjugate’ instead of ‘g is a conjugate of f ’. Furthermore, each map has at most one conjugate. In particular, if a map f is the conjugate of itself, then we call f self-conjugate. Properties of self-conjugate maps are studied in [23]. The following proposition appears in [12], but in a slightly different form. It connects for a map on a complete Boolean lattice, the existence of the conjugate to the property of being a complete joinmorphism. Note that this results holds only for complete Boolean lattices, not generally for complete lattices, as Proposition 2.2 does. Proposition 2.3. Let f be a self-map on a complete Boolean lattice. Then f has a conjugate if and only if f is a complete join-morphism. It is clear that if f : B → B is a complete join-morphism on a complete Boolean lattice B, then f has a conjugate that also is a complete join-morphism on B. Further, the conjugate g of f is defined by g(x) =

^

y 0 | f (y) ∧ x = 0

for all x ∈ B, where y 0 denotes the complement of y ∈ B. Next we introduce the notion of the dual of the map needed for connecting Galois connections to conjugates.

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Definition 2.3. Let B be a Boolean lattice and let f and g be self-maps on B. We say that g is the dual of f , if for any x ∈ B, f (x0 ) = g(x)0 . For any map f , we denote by f ∂ the dual of f . It is obvious that if g = f ∂ , then f = g ∂ . Therefore, we usually say ‘f and g are dual’ instead of ‘g is the dual of f ’. It is also obvious that each map f on B has exactly one dual. The following obvious lemma connects complete join-morphisms and meet-morphisms together by applying the notion of duality. Lemma 2.1. Let B be a complete Boolean lattice. A self-map f on B is a complete join-morphism if and only if f ∂ is a complete meet-morphism. In Proposition 2.2 we showed for complete lattices that each complete join-morphism induces a Galois connection, and a similar result holds also for complete meet-morphisms. By Proposition 2.3 we also know for complete Boolean lattices that each complete join-morphism has a conjugate. We end this section by presenting a result connecting conjugate maps and Galois connections. Proposition 2.4. Let B be a complete Boolean lattice. (a) For any complete join-morphism f on B, the adjoint of f is the dual of the conjugate of f . (b) For any complete meet-morphism g on B, the co-adjoint of g is the conjugate of the dual of g. Proof: We prove (a). Let f W : B → B be a complete join-morphism. Then it has the adjoint f a : B → B a defined On the other hand, conjugate g of f W is defined as g(x) = V 0 by f (x)0 = {y | f (y) ≤ x}. V the 0 ∂ 0 0 0 00 dual of g is g (x) = g(x ) = ( {y | f (y) ≤ x }) = {y 00 | f (y) ≤ x} = W {y | f (y) ≤ x }. The a {y | f (y) ≤ x} = f (x). The proof for (b) is analogous. u t Our following remark summarizes the previous discussion.

Remark 2.1. Let B be a complete Boolean lattice and f : B → B be a complete join-morphism. Then, f has a unique adjoint, and we may define the conjugate of f as dual of the adjoint.

3.

Fixpoints of Galois connections

In this section we study fixpoints of Galois connections formed by self-maps on an ordered set. Recall that if (f, g) is a Galois connection on an ordered set P , then f is a complete join-morphism, g is a complete meet-morphism, and thus, both f and g are order-preserving. Further, if P is bounded, then f (0) = 0 and g(1) = 1. Also recall that each complete join- or meet-morphism on a complete lattice determines a Galois connection. We first give some simple lemmas showing that properties of the maps forming Galois connections are closely connected. Then, we present some results concerning the structure of the ordered set of fixpoints of different types of complete join-morphisms and Galois connections. Lemma 3.1. Let (f, g) be a Galois connection on an ordered set P . The following are equivalent: (a) x ≤ f (x) for all x ∈ P ;

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(b) g(x) ≤ x for all x ∈ P . Proof: Assume that (a) holds. Then g(x) ≤ f (g(x)) for every x ∈ P . Since (f, g) is a Galois connection, we have f (g(x)) ≤ x, and hence g(x) ≤ x. Conversely, suppose that (b) holds. For every x ∈ P , g(f (x)) ≤ f (x) and x ≤ g(f (x)), which imply x ≤ f (x). u t An element x ∈ P is a fixpoint of a mapping f : P → P if f (x) = x. Next we show that an extensive complete join-morphism has the same fixpoints as its adjoint. Lemma 3.2. Let (f, g) be a Galois connection on an ordered set P . If f is extensive, then f and g have exactly the same fixpoints. Proof: If x is a fixpoint of f , then f (x) ≤ x implies x ≤ g(x) ≤ x. Conversely, if y is a fixpoint of g, then y ≤ g(y) and f (y) ≤ y ≤ f (y). u t Example 3.1. That f is extensive is necessary for Lemma 3. Let us consider the interval [0, 1] with its usual order. If f (x) = min{1/2, x}, then f is clearly a complete join-morphism which is not extensive. The map _ g(x) = {y ∈ [0, 1] | min{1/2, y} ≤ x} is the adjoint of f , and we have f (1/2) = 1/2 and g(1/2) = 1. Hence, 1/2 is a fixpoint of f , but not of g. Corollary 3.1. Let (f, g) be a Galois connection on an ordered set P . If f is extensive, then the following are equivalent: (a) x is a fixpoint of f ; (b) x is a fixpoint of g; (c) f (x) = g(x). Proof: By Lemma 3.2, (a) and (b) are equivalent and they imply (c). If f (x) = g(x) for some x ∈ P , then x ≤ f (x) = g(x) ≤ x, which means that (c) implies both (a) and (b). u t Lemma 3.3. Let (f, g) be a Galois connection on an ordered set P . The following are equivalent: (a) f (f (x)) ≤ f (x) for all x ∈ P ; (b) g(x) ≤ g(g(x)) for all x ∈ P . Proof: Assume that (a) holds. Now f (g(x)) ≤ x implies f (f (g(x))) ≤ x, from which we get f (g(x)) ≤ g(x) and g(x) ≤ g(g(x)). The other direction can be proved analogously. u t Lemmas 3.1 and 3.3 have the following obvious corollary.

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Corollary 3.2. Let (f, g) be a Galois connection on an ordered set P . The following are equivalent: (a) f is a closure operator; (b) f is an Alexandrov closure operator; (c) g is an interior operator; (d) g is an Alexandrov interior operator. Note that Lemma 3.2 and Corollary 3.2 imply that if f is a closure operator, then for all x ∈ P , g(f (x)) = f (x) and f (g(x)) = g(x).

(4)

For any self map f on a set P , let us denote F = {x ∈ P | f (x) = x}, that is, F is the set of all fixpoints of f . Next we will study the order structure of F more carefully. We begin by recalling the well-known Knaster–Tarski Fixpoint Theorem [24], which states that if f is an order-preserving self-map on a complete lattice L, then F is a complete lattice. If f is order-preserving and also extensive, the following stronger result concerning the structure of F can be presented (see [7], for example). Lemma 3.4. If f is an extensive and order-preserving self-map on a complete lattice L, then F is a complete meet-sublattice of L. Proof: V V V V Let S ⊆ F. Because f is extensive, S ≤ f ( S). For all x ∈ S, we have S ≤ x and f ( S) ≤ V V V f (x) = x. Thus, f ( S) ≤ S, and so S ∈ F. u t Let f be an extensive and order-preserving self-map on a complete lattice L. Let us define V n(x) = L {a ∈ F | x ≤ a}. By Lemma 3.4, n(x) is the smallest fixpoint of f above x. Recall that if (f, g) is a Galois connection and f is extensive, then F is also the set of fixpoints of g. We may also present the following simple observation. Lemma 3.5. Let (f, g) be a Galois connection on a complete lattice L. If f is a closure operator, then n(x) = f (x) for all x ∈ L. Proof: Since n(x) is a fixpoint of f , we have f (x) ≤ f (n(x)) = n(x). Also f (x) is a fixpoint of f above x, which gives n(x) ≤ f (x). u t For complete join-morphisms, F is a complete join-sublattice of L (cf. Remark 2.4 in [7]). Lemma 3.6. If f is a complete join-morphism on a complete lattice L, then F is a complete joinsublattice of L. Proof: W W W W SinceWf is a complete join-morphism, f ( S) = {f (x) | x ∈ S} = {x | x ∈ S} = S. Hence, also S ∈ F. u t

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By combining Lemmas 3.4 and 3.6, we may write the following corollary. Corollary 3.3. Let f be a complete join-morphism on a complete lattice L. If f is extensive, then F is a complete sublattice of L. Next we present a more concrete description of the smallest fixpoint n(x) of an extensive complete join-morphism f above x. For that we need the Kleene’s Fixpoint Theorem, which can be found in [5], for example. Let f be a self-map on an ordered set P . Let us define for any integer i ≥ 0, the i-fold composite f i (x) by f 0 (x) = x and f i+1 (x) = f (f i (x)) for all x ∈ P . Theorem 3.1. (Kleene’s Fixpoint Theorem) Let f : L → L be a complete join-morphism on a complete lattice L. Then _ {f i (0L ) | i ≥ 0} is the least fixpoint of f . By Kleene’s fixpoint result, our following result is obvious. Proposition 3.1. Let (f, g) be a Galois connection on a complete lattice L such that f is extensive. Then _ n(x) = {f i (x) | i ≥ 0}. Proof: Let x ∈ L and [x) = {y ∈ L | x ≤ y}. Then clearly for all y ∈ [x), x ≤ Wf (x) ≤ f (y), that is, f (y) ∈ [x). Let fx be the restriction of f to [x). By the Kleene’s fixpoint result, {f i (x) | i ≥ 0} is the smallest fixpoint of fx , which gives that it is also the smallest fixpoint of f above x. u t The proof of our next proposition is adapted from a more general result showing that the family of continuous maps between CPOs (i.e., complete partially ordered sets) is a CPO (cf. Theorem 8.9 in [5], for example). Proposition 3.2. Let (f, g) be a Galois connection on a complete lattice L. If f is extensive, then the map x 7→ n(x) is an Alexandrov closure operator. Proof: The map n is obviously extensive. Since n(x) ∈ F, n(n(x)) = n(x) is trivial, and hence n is also idempotent. Further, because f is a complete join-morphism it must be for any S ⊆ P ,  _
_ _
_ _ n S = fi S = f i (S) . i≥0

i≥0

W W i W Clearly, f i (x) i ≥ 0 and so W W ≤ n(x) ≤ n(S) for all i ≥ 0 and x ∈ S. Hence, f (S) ≤ Wn(S) for all W n( S) ≤ n(S). Finally, we show that n is order-preserving, which implies n(S) ≤ n( S). If x ≤ V V y, then {a ∈ F | x ≤ a} ⊇ {a ∈ F | y ≤ a} and n(x) = {a ∈ F | x ≤ a} ≤ {a ∈ F | y ≤ a} = n(y). u t

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Let (f, g) be a Galois connection on a complete lattice L. We have already noted that f (0) = 0 and g(1) = 1. For an extensive f , it must also be f (1) = 1 and g(0) = 0. Therefore, 0 and 1 are fixpoints of f and g, and so by Corollary 3.3, F is a complete sublattice of L with the bottom and top elements 0L and 1L , respectively. In general, as the following example shows, the set F may not be closed under complementation even if L is a complete Boolean lattice. Example 3.2. Let L be the 4-element complete Boolean lattice {0, a, b, 1} ∼ = 2 × 2. Let f and g be defined as follows: f (0) = 0, f (a) = 1, f (b) = b, f (1) = 1, g(0) = 0, g(a) = 0, g(b) = b, g(1) = 1. Now f is extensive and (f, g) is a Galois connection. In this case, since F = {0, b, 1}, we have b ∈ F but b0 = a ∈ / F. Proposition 3.3. Let (f, g) be a Galois connection on a complete Boolean lattice B. If f is extensive and self-conjugate, then F is a complete Boolean sublattice of B. Proof: By Corollary 3, F is a complete sublattice of B. Since f is self-conjugate, its adjoint g is equal to the dual f ∂ of f by Proposition 2.4. If x ∈ F, then x0 is a fixpoint of f ∂ because f ∂ (x0 ) = f (x)0 = x0 . Now, x0 ≥ f (f ∂ (x0 )) = f (x0 ). Since f is extensive, also x0 ≤ f (x0 ) holds. So, x0 ∈ F. u t Let (f, g) be a Galois connection on a complete Boolean lattice B such that f is extensive. We complete this section by giving F more topological flavour. Now for each x ∈ B, we can consider n(x) as the smallest ‘open set’ above x. Let us assign Γ = {x0 | x ∈ F}. Then Γ can be seen as the set of ‘closed elements’ of F. Proposition 3.4. Let (f, g) be a Galois connection on a complete Boolean lattice B. If f is extensive, then Γ = {x ∈ B | x = g ∂ (x)}, that is, Γ consists of the fixpoints of the conjugate of f . Proof: Γ = {x0 | x ∈ F} = {x0 | x = g(x)} = {x | x0 = g(x0 )} = {x | x0 = g ∂ (x)0 } = {x | x = g ∂ (x)}.

4.

u t

Modal-like operators on Boolean lattices

Let B be a complete Boolean lattice and A(B) the set of atoms of B. AW Boolean lattice B is atomic if every element x of B is the supremum of the atoms below it, that is, x = {a ∈ A(B) | a ≤ x}. In this section we assume that the considered Boolean lattices are always atomic. In the rest of the paper, B is a complete atomic Boolean lattice, ϕ and ψ are two given maps A(B) → B such that for all a, b ∈ A(B). a ≤ ϕ(b) ⇐⇒ b ≤ ψ(a). (5) The idea behind this requirement is that if R is any binary relation on a universe U , and ϕ and ψ are the mappings x 7→ R(x) and x 7→ R−1 (x), respectively, then ϕ and ψ can be considered as mappings A(℘(U )) → ℘(U ) satisfying (5).

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Note that for all atoms a and b of B, the conditions a ≤ ϕ(b) and a ∧ ϕ(b) 6= 0 are equivalent, because a ≤ ϕ(b) implies a ∧ ϕ(b) = a 6= 0, and 0 6= a ∧ ϕ(b) ≤ a gives a = a ∧ ϕ(b), that is, a ≤ ϕ(b). Of course, also for all atoms a and b, a ≤ ψ(b) ⇐⇒ a ∧ ψ(b) 6= 0. Thus, it is clear that (5) is equivalent to the condition a ∧ ϕ(b) = 0 ⇐⇒ b ∧ ψ(a) = 0, which resembles the condition defining conjugates. Note that ϕ and ψ have only the atoms as their domains. First, we present a simple existence result. Lemma 4.1. Let B be a complete atomic Boolean lattice. For any map ϕ : A(B) → B, there exists a unique map ψ : A(B) → B satisfying (5). Proof: Let us define ψ(a) =

_

{c ∈ A(B) | a ≤ ϕ(c)} .

If a ≤ ϕ(b), then trivially b ≤ ψ(a). Conversely, if b ≤ ψ(a), then _ b ∧ ψ(a) = b ∧ {c ∈ A(B) | a ≤ ϕ(c)} _ = {b ∧ c | c ∈ A(B) and a ≤ ϕ(c)} = b > 0. Because x ∧ y = 0 for all atoms x 6= y, we must have a ≤ ϕ(b). The uniqueness of ψ is trivial.

u t

As we already noted, the maps ϕ and ψ can be viewed as a generalization of a binary relation and its inverse. Therefore, we may also define the four modal-like operators introduced in Section 1 in the more general setting of complete atomic Boolean lattices. Definition 4.1. Let B be a complete atomic Boolean lattice. For any x ∈ B, we define _ xH = {a ∈ A(B) | ϕ(a) ≤ x}; _ xN = {a ∈ A(B) | ϕ(a) ∧ x 6= 0}; _ xO = {a ∈ A(B) | ψ(a) ≤ x}; _ xM = {a ∈ A(B) | ψ(a) ∧ x 6= 0}. The operators N,M ,H and O are called modal-like Boolean operators. In the following we list some basic properties of modal-like Boolean operators.

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Lemma 4.2. Let B be a complete atomic Boolean lattice. For all a ∈ A(B) and x ∈ B, (a) a ≤ xH ⇐⇒ ϕ(a) ≤ x

and

a ≤ xN ⇐⇒ ϕ(a) ∧ x 6= 0;

(b) a ≤ xO ⇐⇒ ψ(a) ≤ x

and

a ≤ xM ⇐⇒ ψ(a) ∧ x 6= 0;

(c) The maps x 7→ xN and x 7→ xH are mutually dual; (d) The maps x 7→ xM and x 7→ xO are mutually dual; (e) aM = ϕ(a); (f) aN = ψ(a); _ (g) xM = {ϕ(a) | a ∈ A(B) and a ≤ x}; _ (h) xN = {ψ(a) | a ∈ A(B) and a ≤ x}. Proof: The proofs for (a) and (c) can be found Win [10], and (b) and (d) can be W proved as (a) and (c), respectively. W (e) Let a ∈ A(B). Then ϕ(a) = {b ∈ A(B) | b ≤ ϕ(a)} = {b ∈ A(B) | a ≤ ψ(b)} = {b ∈ A(B) | a ∧ ψ(b) 6= 0} = aM. Claim (f) can be proved in a similar manner. W W M M M M (g) Let x ∈ B. Because W is a complete join-morphism, x = ( {a ∈ A(B) | a ≤ x}) = {a | a ∈ A(B) and a ≤ x} = {ϕ(a) | a ∈ A(B) and a ≤ x}. The proof for (h) is analogous. u t Now we W can show the following two important Galois W connections determined by ϕ and ψ. Recall that xM = {ϕ(a) | a ∈ A(B) and a ≤ x} and xN = {ψ(a) | a ∈ A(B) and a ≤ x}. Proposition 4.1. The pairs (M,H) and (N,O) are Galois connections. Proof: Let xM ≤ y, that is, xM = W B be a complete atomic Boolean lattice and x, y ∈ B. Assume that {ϕ(a) | a ≤ x} ≤ y. If a ≤ x, then ϕ(a) ≤ y, which gives x ≤ y H. Conversely, assume that W x ≤ y H. Then for all b ∈ A(B), b ≤ x implies ϕ(b) ≤ y. Suppose that a ≤ xM = {ϕ(b) | b ≤ x}. Then there exists b ∈ A(B) such that a ≤ ϕ(b) and b ≤ x. This gives a ≤ ϕ(b) ≤ y, that is, xM ≤ y. The proof for the other pair is similar. u t By the previous proposition, (M,H) is a Galois connection, that is, H is the adjoint, but H is also equal to the dual of the conjugate of M. Because N is the dual of H, we obtain the following result. Corollary 4.1.

M

and N are conjugate.

Next we list the obvious properties of modal-like Boolean operators that are clear by Proposition 4.1. Lemma 4.3. Let B be a complete atomic Boolean lattice. (a) xN ≤ y ⇐⇒ x ≤ y O

and

xM ≤ y ⇐⇒ x ≤ y H.

(b) xN = xNON

and

xM = xMHM.

(c) xH = xHMH

and

xO = xONO.

(d) The maps x 7→ xN and x 7→ xM are complete join-morphisms. (e) The maps x 7→ xH and x 7→ xO are complete meet-morphisms.

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(f) The maps x 7→ xMH and x 7→ xNO are closure operators. (g) The maps x 7→ xHM and x 7→ xON are interior operators. (h) The ordered sets (B N, ≤), (B H, ≤), (B M, ≤), and (B O, ≤) are complete lattices. (i) (B N, ≤) ∼ = (B O, ≤). = (B M, ≥) ∼ = (B H, ≥) ∼ Since the maps ϕ and ψ mimic binary relations and their inverse relations, we study different types of these maps. Recall that a map ϕ : A(B) → B is extensive if a ≤ ϕ(a) for all a ∈ A(B). Now, we give the following definition. Definition 4.2. Let B be a complete atomic Boolean lattice. A map ϕ : A(B) → B is (a) symmetric, if a ≤ ϕ(b) implies b ≤ ϕ(a) for all a, b ∈ A(B); (b) closed, if a ≤ ϕ(b) implies ϕ(a) ≤ ϕ(b) for all a, b ∈ A(B). It is clear that a binary relation is reflexive if and only if its inverse is reflexive, and similar conditions holds with respect to symmetry and transitivity, too. Our next lemma shows that analogous conditions hold between ϕ and ψ. Lemma 4.4. Let B be a complete atomic Boolean lattice. (a) ϕ is extensive ⇐⇒ ψ is extensive. (b) ϕ is symmetric ⇐⇒ ψ is symmetric ⇐⇒ ϕ = ψ. (c) ϕ is closed ⇐⇒ ψ is closed. Proof: Let a, b, c be atoms of B. (a) Obviously, a ≤ ϕ(a) if and only if a ≤ ψ(a). (b) Suppose ϕ is symmetric and that a ≤ ψ(b). Then b ≤ ϕ(a) and a ≤ ϕ(b). Thus, also b ≤ ψ(a) holds, that is, ψ is symmetric. If ψ is symmetric, then for all a, b ∈ A(B), b ≤ ψ(a) ⇐⇒ a ≤ ψ(b) ⇐⇒ b ≤ ϕ(a). Hence, ϕ = ψ. Finally, if ϕ = ψ, then b ≤ ϕ(a) implies a ≤ ψ(b) = ϕ(b), that is, ϕ is symmetric. So, the three conditions are equivalent. (c) Let the map ϕ be closed. Assume that b ≤ ψ(a). If c ≤ ψ(b), then b ≤ ϕ(c) and a ≤ ϕ(b). Since ϕ is closed, a ≤ ϕ(b) ≤ ϕ(c). Hence, also c ≤ ψ(a) holds and ψ(b) ≤ ψ(a), that is, ψ is closed. The proof for the other direction is analogous. u t Next we present some ‘correspondence’ results. Proposition 4.2. If B is a complete atomic Boolean lattice, the following conditions are equivalent: (a) ϕ is extensive; (b) xH ≤ x for all x ∈ B; (c) x ≤ xN for all x ∈ B.

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Proof: Assume that ϕ is extensive. If a ∈ A(B), then a ≤ xH means that a ≤ ϕ(a) ≤ x. Thus, (a) implies (b). If (b) holds, then x0 ≥ x0 H = xN0 , that is, x ≤ xN. This means that (b) implies (c). Suppose that ϕ is not extensive. Then a 6≤ ϕ(a) and a ∧ ϕ(a) = 0 for some a ∈ A(B). This means a 6≤ aN. So, also (c) implies (a). u t

BM

Symmetry is closely related to being a self-conjugate. Note also that if ϕ is symmetric, then B N = and B H = B O.

Proposition 4.3. If B is a complete atomic Boolean lattice, the following conditions are equivalent: (a) ϕ is symmetric; (b)

N

(c)

(N,H)

is self-conjugate; is a Galois connection.

Proof: Suppose ϕ is symmetric, then by Lemma 4.4, ϕ = ψ and xN = xM for all x ∈ B. So, (a) implies (b) It is known that for a complete join-morphism, the dual of its conjugate equals its adjoint. Hence, (b) implies (c). Assume that ϕ is not symmetric. Then there exist a, b ∈ A(B) such that a ≤ ϕ(b), but b 6≤ ϕ(a). Now, ϕ(a) ∧ b = 0 and a 6≤ bN. Thus, also ϕ(b) 6≤ bN, that is, b 6≤ bNH. This gives that (c) implies (a). u t Proposition 4.4. If B is a complete atomic Boolean lattice, the following conditions are equivalent: (a) ϕ is closed; (b) xH ≤ xHH for all x ∈ B; (c) xNN ≤ xN for all x ∈ B. Proof: Suppose that ϕ is closed. Let a ∈ A(B). If a ≤ xH, then ϕ(a) ≤ x. Assume that a 6≤ xHH, that is, ϕ(a) 6≤ xH. Thus, there exists b ∈ A(B) such that b ≤ ϕ(a) and b 6≤ xH. Now, b 6≤ xH means ϕ(b) 6≤ x. On the other hand, b ≤ ϕ(a) gives ϕ(b) ≤ ϕ(a) ≤ x, a contradiction! Thus, (a) implies (b). Since xN0 = x0 H ≤ x0 HH = xNN0 , which is equivalent to xNN ≤ xN, we have that (b) implies (c). Suppose that ϕ is not closed. Then there exist a, b ∈ A(B) such that a ≤ ϕ(b), but ϕ(a) 6≤ ϕ(b). This means that there exists c ∈ A(B) such that c ≤ ϕ(a) and c 6≤ ϕ(b). Thus, ϕ(b) ∧ c = 0 and b 6≤ cN. On the other hand, c ≤ ϕ(a) implies ϕ(a) ∧ c 6= 0 and a ≤ cN. Hence, ϕ(b) ∧ cN ≥ a ∧ cN = a, that is, b ≤ cNN. Thus, cNN 6≤ cN and (c) implies (a). u t Recall for the next corollary that every complete join-morphism which is a closure operator must be an Alexandrov closure operator. Corollary 4.2. For a B complete atomic Boolean lattice B, the following conditions are equivalent: (a) ϕ is extensive and closed; (b) The map N : B → B is an Alexandrov closure operator;

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(c) The map H : B → B is an Alexandrov interior operator. Notice that we could also present correspondences similar to Propositions 4.2, 4.3, 4.4, and Corollary 4.2 between ψ and the maps M and O. Note also that by Lemma 4.4, ϕ and ψ have the same properties. Proposition 4.5. Let B be a complete atomic Boolean lattice. If ϕ is extensive and closed, then the following equations hold for all x ∈ B: (a) xNO = xN and xON = xO; (b) xMH = xM and xHM = xH. Proof: By Corollary 4.2, N and M are closure operators. Claims are obvious by (4).

u t

It is now clear that if ϕ (and thus ψ) is extensive and closed, then for a complete atomic Boolean lattice B we have B N = B O and B M = B H. Corollary 4.3. Let B be a complete atomic Boolean lattice. If ϕ is extensive, symmetric and closed, then the following assertions hold: (a) xNH = xN and xHN = xH for all x ∈ B; (b) xMO = xM and xOM = xO for all x ∈ B; (c) B N = B H = B M = B O. Let FN be the set of fixpoints of the map N : B → B. Let ϕ be extensive. Then by Corollary 3.1, FN consists elements x ∈ B such that xO = xN. Let us denote ΓN = {x0 | x ∈ F}. Then ΓN = {x ∈ B | xM = x} by Proposition 8, and if ϕ is symmetric, FN = ΓN. Further, if ϕ is also closed, we have FN = B N. Now we can present the following stocktaking. Proposition 4.6. Let B be a complete atomic Boolean lattice. (a) FN is a complete lattice. (b) If ϕ is extensive, then FN is a complete sublattice of B. (c) If ϕ is extensive and symmetric, then FN is a complete Boolean sublattice of B. (d) If ϕ is extensive and closed, then xN is the smallest fixpoint of N above x ∈ B. We end our discussion with the following example connecting the study of this section to the modallike operators on sets. Example 4.1. Let U be a non-empty set. The power set ℘(U ) is a complete atomic Boolean lattice with respect to the set-inclusion relation. Since the atoms {x} of ℘(U ) can be identified with the elements of U , each map ϕ : U → ℘(U ) may be considered to be of the form ϕ : A(B) → B, where B equals ℘(U ) (cf. [10]). By the previous discussion it is clear that the self-maps N and M on ℘(U ) are complete join-morphisms, and the pairs (N,O) and (M,H) are Galois connections on ℘(U ) satisfying all properties of Sections 3 and 4.

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In particular, for any binary relation R on U we may define ϕ : U → ℘(U ), x 7→ R(x). Then ψ(x) = R−1 (x) for all x ∈ U . The following observations are obvious: (a) R is reflexive ⇐⇒ ϕ is extensive ⇐⇒ ψ is extensive; (b) R is symmetric ⇐⇒ ϕ is symmetric ⇐⇒ ψ is symmetric; (c) R is transitive ⇐⇒ ϕ is closed ⇐⇒ ψ is closed. Let us consider sets X ⊆ U such that X N = X, that is, the fixpoints of the map X 7→ X N. These are important, for example, in rough set theory, because the set X N is interpreted as a set of elements possibly belonging to X when objects are observed by the accuracy given by an indiscernibility relation. A fixpoint X = X N is now precise in the sense that the set X and the set X N of elements possibly in X are equal. Let us denote the sets of fixpoints by FN, that is, FN = {X ⊆ U | X = X N}. If R is reflexive, the family FN consists of fixpoints of both N and O. In fact, we know that FN is the family of sets such that X O = X N. Further, FN is a complete sublattice of ℘(U ), that is, an Alexandrov topology on U . For an Alexandrov topology T on U , also the family T ∂ = {U − X | X ∈ T } is an Alexandrov topology, called the dual topology of T (see [1]). It is now clear that the dual topology of FN is FM = {X ⊆ U | X = X M} = {X ⊆ U | X = X H}. It is important to notice that if ϕ is extensive and closed, thus R is a preorder on U , then the Alexandrov topology FN is such that N is its smallest neighborhood operator. Furthermore, O is the interior operator and M is the closure operator (cf. [11]). Additionally, N is the closure operator, H is the interior operator and M is the smallest neighborhood operator in FM. Recall that each L-set induces a preorder . and we can easily show that the Alexandrov topology FN consists of down-sets of ., because for any open set X ∈ FN we have X = X O. Therefore, x ∈ X and x & y implies y ∈ X. We end this work by noting, that if R is reflexive and symmetric, then FN is closed under setcomplementation, and hence it is the dual topology of itself. It is important to notice that if R is an  alsoN equivalence, FN is a complete atomic Boolean lattice such that {x} x∈U is the set of its atoms.

Acknowledgements We would like to take the opportunity to thank the anonymous referees for their insightful comments and suggestions.

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