Marek Nowak A Proof of Tarski's Fixed

Point Theorem by Application of Galois Connections

Abstract. Two examples of Galois connections and their dual forms are considered. One of them is applied to formulate a criterion when a given subset of a complete lattice forms a complete lattice. The second, closely related to the first, is used to prove in a short way the Knaster-Tarski's fixed point theorem.

Keywords: Closure and interior operation, Galois connection, Fixed point theorem.

1. Introduction

For given antimonotone Galois connection defined for the complete lattices, a dual form - an appropriate monotone Galois connection (a residuated pair of mappings) is considered. The pair of closure and interior operations induced on a complete lattice by such anti- and monotone Galois connections is of our interest. Two examples of Galois connections and their dual forms are introduced in the paper. First one, considered in Sect. 3, embraces a Galois connection responsible for the dual isomorphism between a complete lattice and a closure system of subsets of a meet-generating subset of the lattice. The induced closure and interior operations are of so general form that they enable to formulate a simple criterion saying when a subset B of given complete lattice (A, <) forms a complete lattice with respect to the ordering < (Lemma 1 and Proposition 2). This criterion is applied in Sect. 4 to prove in a simple short way the Knaster-Tarski's fixed point theorem [10] (Corollary 9). The proof is constructive in the sense that it shows the explicit form of supremum and infimum of a subset in the lattice of all fixed points of a monotone mapping (cf. [2, Theorem 5.1]). This form differs from that of [2], moreover from that of [6]. The proof is also based on some simple results (inter alia Proposition 8) concerning the second example of Galois connections introduced in the paper (Sect. 4). This example is responsible

Presented by Andrzej Indrzejczak; Received March 21, 2014

Studia Logica

DOI: 10.1007/s11225-014-9559 _y © The Author(s) 2014. This article is published with open access at Springerlink.com

Published online: 11 July 2014

for well-known isomorphisms between the lattice of all closure (interior) operations defined on a complete lattice (A, <) and the lattice of all closure (interior) systems of (A, <). The induced closure and interior operations are here defined on the complete lattice of all monotone mappings of a complete lattice (A, <) into itself. The closure operation C induced by the antimonotone Galois connection assigns to each monotone map a the least closure operation c defined on (A, <) such that a < c, where < is the pointwise order on mappings from A to A induced by lattice ordering of (A, <). In turn, the dual (monotone) Galois connection induces an interior operation Int assigning to each monotone mapping a the greatest interior operation I on (A, <) such that I < a. A crucial point of the proof of Knaster-Tarski's theorem presented here, is the fact that the set of all fixed points of a monotone map a turns out to be the intersection of the closure and interior systems of (A, <) corresponding to closure C(a) and interior I nt(a) operations, respectively.

2. Preliminaries

The paper deals mostly with the closure and interior operations defined on a complete lattice. Given a complete lattice (A, <) any mapping C : A —> A such that for each a E A, a < C(a), C(C(a)) < C(a) and C is monotone: a < b ^ C(a) < C(b), is called a closure operation defined on (A, <). Any subset B C A is said to be a closure system or Moore family of the lattice (A, <) if for each X C B, inf a X E B. Given a closure operation C on (A, <), the set of all its fixed points called closed elements: {a E A : a = C(a)}, is a closure system of (A, <). Conversely, given a closure system B of (A, <), the map C : A —> A defined by C(a) = infA{x E B : a < x}, is a closure operation on (A, <). The closure system B is just the set of all its closed elements. On the other hand, the closure system of all closed elements of a given closure operation C defines, in that way, just the operation C. Thus, there is a one to one correspondence between the class of all closure operations and of all closure systems of (A, <) (in fact it is a dual isomorphism between respective complete lattices of all closure operations and closure systems). Any closure system B of (A, <) forms a complete lattice with respect to the order < such that infB X = inf A X and supB X = C(supA X), for each X C B, where C is the closure operation corresponding to closure system B. Given a subset X of A, there exists the least closure system B of (A, <) such that X C B, called generated by X. It will be denoted here by [X]cl. It is simply the intersection of all the closure

systems of (A, <) containing X and is of the form: [X]ci = {inf a Y : Y C X}. The closure operation C corresponding to closure system [X]cl is expressed by C (a) = inf a {x E X : a < x}, any a E A.

An interior operation and an interior system are the dual concepts with respect to closure ones. That is, a monotone mapping I : A —> A such that for any a E A, I (a) < a, I (a) < I (I (a)) is said to be an interior operation defined on a complete lattice (A, <). Any subset B of A is called an interior system of the lattice (A, <) if for each X C B, supA X E B. Given an interior operation I on (A, <) the set of all its fixed poits called open elements: {a E A : a = I (a)}, is an interior system of (A, <). Conversely, given an interior system B of (A, <), the map I : A —> A defined by I (a) = supA{x E B : x < a}, is an interior operation on (A, <). The interior system B is just the set of all its open elements. On the other hand, the interior system of all open elements of a given interior operation I defines, in that way, just the operation I. So, as before, a similar correspondence between the class of all interior operations and interior systems, exists (which is an isomorphism of respective complete lattices of all interior operations and all interior systems of (A, <)). Any interior system B of (A, <) forms a complete lattice with respect to the order < such that supB X = supA X and inf b X = I (inf a X), for each X C B, where I is the interior operation corresponding to interior system B. Given a subset X of A, there exists the least interior system B of (A, <) such that X C B. Such an interior system is said to be generated by X and will be denoted as [X]in. It is the intersection of all the interior systems of (A, <) containing X and is of the form: [X]in = {supA Y : Y C X}. The interior operation I corresponding to interior system [X ]in is defined by I (a) = supA {x E X : x < a}, any a E A.

We shall consider the monotone and antimonotone Galois connections defined only for complete lattices. A general theory of Galois connections is to be found for example in [1,3-5,7].

Let us remind that while (A, <A), (B, <B) are the complete lattices, any pair of mappings f : A —> B, g : B —> A such that for each a E A,b E B : b <B f (a) iff a <a g(b), is called an antimonotone Galois connection for those lattices. Equivalently, such a Galois connection (f,g) fulfils the following conditions: a <a g(f(a)), b <B f (g(b)) for any a E A, b E B and f,g are antimonotone. When the pairs (f,gi), (f,g2) are Galois connections for the lattices (A, <A), (B, <B) then g1 = g2. The first element f of an antimonotone Galois connection (f, g) for the lattices (A, <a), (B, <b) is usually called a Galois function. A sufficient and necessary condition for a map f : A —> B to be a Galois function is of the form: f (supA X) = infB{f (a) : a E X}, for any X C A. Given a Galois function

f, the second unique element g of the Galois connection (f, g) is given by g(b) = supA{a E A : b <B f (a)}, for each b E B. This mapping g satisfies the condition: g(supB Y) = infA{g(b) : b E Y}, for any Y C B. Given a Galois connection (f,g) for the lattices (A, <A), (B, <B), the ranges f [A], g[B] of the mappings f and g are the sets of all closed elements with respect to the closure operations Cl2, Cl1 respectively that are induced on B and A in the following way: for each a E A, b E B, Cl2(b) = f (g(b)), Cl1(a) = g(f (a)). Since for each a E g[B], b E f [A] : g(f (a)) = a, f (g(b)) = b and moreover for any a1,a2 E g[B] : ai <a a2 iff f (a2) <B f (ai), so the complete lattices (g[B], <A), (f [A], <B) are dually isomorphic (with f being a dual isomorphism).

In turn, a pair f : A —> B, g : B —> A such that for each a E A, b E B : b <B f (a) iff g(b) <a a, is called a monotone Galois connection or a residuated pair of mappings for the lattices (A, <a), (B, <B). Equiva-lently, a monotone Galois connection (f,g) fulfils the following conditions: g(f(a)) <a a, b <B f(g(b)) for any a E A, b E B and f,g are monotone functions. When (f,g1), (f,g2) are residuated pairs for the lattices (A, <A ), (B, <B) then g1 = g2. The first element f of a monotone Galois connection (f,g) for the lattices (A, <a), (B, <B) is usually called a residuated function while the unique second one g-a residual of f. A sufficient and necessary condition for a map f : A —> B to be a residuated function is of the form: f (inf a X) = infB{f (a) : a E X}, for any X C A. Given a residuated function f, its residual g is expressed by g(b) = inf A{a E A : b <B f (a)}, for each b E B. This mapping g satisfies the condition: g(supB Y) = supA{g(b) : b E Y}. Given a residuated pair (f,g) for the lattices (A, <A), (B, <B), the ranges f [A], g[B] are, respectively, the sets of all closed and open elements with respect to the following closure and interior operations Cl, Int: for each a E A, b E B, Cl(b) = f (g(b)), Int (a) = g(f (a)). Since for each a E g[B], b E f[A] : g(f(a)) = a, f(g(b)) = b and moreover for any a1,a2 E g[B] : a1 <a a2 iff f (a1) <B f(a2), so the complete lattices (g[B], <a), (f [A], <B) are isomorphic (with f being an isomorphism).

From the very definition of Galois connections it follows that any antimonotone Galois connection (f,g) for the lattices (A, <a), (B, <B) is simultaneously a residuated pair for the lattices (A, <A), (B, <B), where is the converse ordering to <A. Taking this into account, having defined a Galois function f<A : A —> B for the complete lattices (A, <a), (B, <B) (we write down the parameter: <a, on which the function may depend as an essential one, however in general there are the other parameters which may occur in a definition of Galois function) let us consider a mapping

f<2 : A —> B which is defined exactly in the same way as the function f<A except that instead of the parameter <a the converse relation is applied. Notice that when f<A being a Galois function fulfils the condition: f<A (sup<A X) = inf<B{f<A (a) : a E X}, the mapping f<j has to satisfy the following one: f<~ (inf<A X) = inf<B {f<j (a) : a E X}, any X C A, that is, f<2 is a residuated function for the lattices (A, <A), (B, <b). Let us call such a residuated function the dual residuated function with respect to f<A. Moreover, when (f,g) is an antimonotone Galois connection let us call the residuated pair (fd,gd), where fd is the dual residuation function with respect to f, the dual residuated pair (or the dual Galois connection) with respect to (f, g). Obviously, one can start not from a Galois but a resid-uated function (residuated pair) and define the dual Galois function (the dual antimonotone Galois connection).

Having at our disposal the Galois connections: (f, g), (fd, gd) for the complete lattices (A, <a), (B, <B) we are especially interested in the interior-closure pair (Int,C) of operations on (A, <a), where Int = fd o gd and C = f o g (the closure operation C was denoted by Cl\ above).

In the sequel we consider two important examples of antimonotone Galois connections and their dual forms. First one enables to formulate a simple criterion saying when a given subset of a complete lattice forms a complete lattice. The second example, closely related to the first, has rather unexpected applications. It enables a very simple proving of the Knaster-Tarski's fixed point theorem.

3. A Criterion of Being a Complete Lattice

Let (A, <) be any complete lattice and B C A. The following pair of mappings: f : A —> p(B), g : p(B) —> A defined by f (a) = {x E B : a < x}, any a E A and g(X) = inf a X, any X C B, forms an antimonotone Galois connection for the lattices (A, <), (p(B), C). The dual residuated function with respect to f is then of the form: fd(a) = {x E B : x < a} and its residual is defined by gd(X) = infA{a E A : X C fd(a)} = infA{a E A : X C{x E B : x < a}} = supA X, as one could expect.

These Galois connections are responsible for well-known isomorphisms of a complete lattice and a lattice of subsets of a given meet- or join-generating subset of the lattice. A subset B of a complete lattice (A, <) is said to be join-generating (meet-generating, cf. for example [5]) or join-dense (meet-dense, e.g. [8]) iff for each a E A, there is an X C B such that a = supA X (a =

infA X). For example, the set of all compact elements of an algebraic lattice is just its join-generating subset.

It is clear that the restriction of the map f to the set {infA X : X C B} (which is the closure system generated by B) is a dual isomorphism of the lattice ({infA X : X C B}, <) of all closed elements with respect to the closure operation CB = f o g to the lattice ({B n [a) : a E A}, C) (which is the closure system of (p(B), C) corresponding to closure operation g o f; here [a) = {x E A : a < x}). Similarly, the restriction of the map fd to the set {supA X : X C B} (which is the interior system generated by B) is an isomorphism of the lattice ({supA X : X C B}, <) of all open elements with respect to the interior operation IB = fd o gd to the lattice ({B n (a] : a E A}, C) (being the closure system of (p(B), C) corresponding to closure operation gd o fd; here (a] = {x E A : x < a}).

One can easily see from their definitions that the operations IB ,CB are of the following general form, for any a E A:

(1) IB(a) = supA{x E B : x < a},

(2) Cb(a) = infa{x E B : a < x}.

They simply corrrespond to the interior and to closure systems of (A, <) generated by B, respectively. The pair (IB,CB) is a generalization of the notion of so-called pair of interior-closure operations associated on a given subset of a complete lattice, introduced in [9] and widely applied there. In case a subset B forms a complete sublattice of the lattice (A, <), the pair (IB ,Cb) becomes just an interior-closure pair of operations associated on B. The existence of an interior-closure pair of operations associated on B is a necessary and sufficient condition for (B, <) to be a complete sublattice of (A, <) (cf. [9]). This criterion will be now generalized in order to provide the sufficient and necessary conditions for the poset (B, <) to be a complete lattice. Let us start from the crucial lemma.

Lemma 1. Let D,O C A be any closure and interior systems of a complete lattice (A, <), respectively. Then the following conditions are equivalent:

(i) for each a E O, CD (a) E O,

(ii) for each a E A, CD (IO (a)) E O,

(iii) for each a E A, IO (CD (a)) E D,

(iv) for each a E D, IO (a) E D,

where the operations IO ,CD are defined by (1) and (2), respectively, for the sets O,D instead of B. Moreover, any of these conditions implies that the

poset (D n O, <) is a complete lattice in which for any X C D n O, sup X = CD (supA X) and inf X = IO (inf A X). The inverse implication in general does not hold.

Proof. Suppose that the subsets D and O of A are closure and interior systems of a complete lattice (A, <), respectively. The equivalences (i) &

(ii), (iii) & (iv) are obvious. In order to show the implication (ii) ^ (iii) assume that for each a E A, CD(IO(a)) = IO(CD(IO(a))). Then given a E A we have CD(Io(CD(a))) = Io(CD(Io(CD(a)))). Since Io(CD(a)) < CD(a) so CD(IO(CD(a))) < CD(a) (CD is monotone and idempotent). Therefore, IO(Cd(IO(Cd(a)))) < IO(Cd(a)) (by monotonicity of IO) which together with the last identity implies that CD(IO(CD(a))) < IO(CD(a)) so we obtain

(iii). The proof from (iii) to (ii) goes analogously (by dual argument).

In order to prove the second part of lemma suppose (i) and consider an X C D n O. Then since O is an interior system we have supA X E O. So from (i) it follows that CD (supA X) E D n O. Now, given any a E X we have a < supA X < CD (supA X), so CD (supA X) is an upper bound of X in the poset (D n O, <). When z E D n O is such an upper bound we obtain: supA X < z, therefore CD(supA X) < CD(z) = z. In this way, CD(supA X) is the least upper bound of X in (D n O, <). The form of inf X in this poset follows from the condition (iv) in a similar way.

Finally, in order to show that none of the conditions (i) — (iv) needs to be true when a poset (D n O, <) is a complete lattice, take for example a 4-element chain: 0 < a < b < 1 and consider D = {0, b, 1}, O = {0, a, 1}. ■

Now let us formulate our criterion saying when a subset of given complete lattice (A, <) forms a complete lattice with respect to the order <.

Proposition 2. Let (A, <) be a complete lattice and B C A. Consider the operations IB,CB defined by (1), (2). The following conditions are equivalent:

(a) for each a E A, CB(IB(a)) E B,

(b) for each a E A, IB(CB(a)) E B,

(c) (B, <) is a complete lattice such that for any X C B, sup X = CB (supA X) and inf X = IB (inf A X).

Proof. Let B C A. Put D = [B]cl, O = [B]in. Then we have immediately B C D n O and Cd = Cb , Io = Ib .

(a) ^ (b) & (c): Assume that (a) holds. Then the condition (ii) of Lemma 1 is satisfied. Moreover, taking any a E D n O we have

CB(IB(a)) = a so from (a) it follows that a E B, consequently, B = DnO. Thus, on one hand, from (ii) and Lemma 1 it follows that (iii) of Lemma 1 holds which leads to (b). On the other hand, simultaneously from (ii) and Lemma 1 it follows that (c) holds true.

(b) ^ (a): By the dual argument with respect to the proof of implication (a) ^ (b).

(c) ^ (a): Suppose that (c) holds. Let a E A. Since IB(a) E [B]in so IB(a) = supA X for some X C B. Therefore, CB(IB(a)) = CB(supAX) = supX by (c). Thus, CB(IB(a)) E B. ■

4. The Galois Connections Involving Monotone Mappings on Complete Lattices

Let (A, <) be a complete lattice and Mon-the class of all monotone mappings from A to A. Obviously, the poset (Mon, <) is a complete sublattice of the complete lattice (AA, <) of all the mappings from A to A, where for any a,/3 E AA, a < /3 iff for all x E A, a(x) < /3(x). For any F C Mon, (supF)(a) = supA{a(a) : a E F} and (inf F)(a) = infA{a(a) : a E F}, for each a E A.

The main goal of this section is to prove the Knaster-Tarski's fixed point theorem using a special Galois connection. This Galois connection turns out to be significant also from the other point of view. It is responsible for well-known dual isomorphism between the complete lattice of all closure operations defined on the complete lattice (A, <) and the complete lattice of all closure systems of (A, <). The connection is of the form: f : (Mon, <) —> (p(A), C) is a mapping defined by f (a) = {x E A : a(x) < x} and g : (p(A), C) —> (Mon, <) is such that for any B C A, g(B) : A —> A is defined by g(B)(a) = infA{x E B : a < x} = infA(Bn[a)). It is obvious that g(B) for each B C A is monotone. Notice simply that given B C A, g(B) is just the closure operation CB from the previous section.

Lemma 3. (f,g) is a Galois connection, i.e., f,g are antimonotone, for each a E Mon, a < g(f (a)) and for any B C A, B C f (g(B)).

Proof. The proof that both f,g are antimonotone is straightforward. In order to show that given a E Mon, a < g(f(a)), notice that given a E A, g(f (a))(a) = infA{x E A : a(x) < x & a < x}. Consider any x E A such that a(x) < x and a < x. Then since the map a is monotone we have: a(a) < a(x) which implies that a(a) < x. This means that a(a) is a lower

bound of the set {x E A : a(x) < x & a < x} in the lattice (A, <). Therefore, a(a) < infA{x E A : a(x) < x & a < x}, that is a(a) < g(f (a))(a). To the end, in order to prove that for all B C A, B C f (g(B)) take any a E B. Our goal is to show that g(B)(a) < a. However, in case a E B we have: infA{x E B : a < x} = a, so g(B)(a) = a. ■

Now, consider the closure operations induced by the Galois connection (f,g), Cli : Mon —> Mon and Cl2 : p(A) —> p(A) defined by Cli(a) = g(f(a)), for any a E Mon and Cl2(B) = f(g(B)), for each B C A. Obviously, {a E Mon : Cli (a) = a} = g[p(A)] and {B C A : Cl2(B) = B} = f [Mon]. Moreover, the mapping f restricted to the set {a E Mon : Cli(a) = a} is a dual isomorphism between the posets ({a E Mon : Cli(a) = a}, <), ({B C A : Cl2(B) = B}, C).

One may characterize the sets of all closed elements with respect to the first and to the second closure operations in the following way.

Proposition 4. (1) For any a E Mon, Cli(a) = a iff a is a closure operation on (A, <).

(2) For any B C A, Ch(B) = B iff for any X C B, inf a X e B, that is B is a closure system of the lattice (A, <).

Proof. For (1) (^): Assume that Cli(a) = a. Then a = g(B) for some B C A. So a is the closure operation CB on (A, <) from the previous section.

(^): Assume that a is a closure operation on (A, <). Our goal is to show that g(f (a)) < a. For each a E A we have g(f (a))(a) = infA{x E A : a(x) < x & a < x}. From the assumption it follows that given a E A, a(a(a)) < a(a) and a < a(a), so a(a) E {x E A : a(x) < x & a < x}, thus infA{x E A : a(x) < x & a < x}< a(a), that is g(f (a))(a) < a(a).

For (2) (Assume that Cl2(B) = B and X C B. Then obviously, B = f (a) for some a E Mon, that is, B = {x E A : a(x) < x} for some a E Mon. So we have furthermore X C{x E A : a(x) < x}. Hence, taking any a E X into account we have a(a) < a while from the monotonicity of a it follows that a(infA X) < a(a) (for inf A X < a). Thus a(inf A X) < a, so a(inf A X) is a lower bound of X, therefore, a(inf A X) < inf A X. This means that inf a X E B.

(^): Assume that for all X C B, inf A X E B. It is sufficient to show that f (g(B)) C B. We have f (g(B)) = {a E A : g(B)(a) < a} = {a E A : infA{x E B : a < x} < a} = {a E A : infA{x E B : a < x} = a} = {a E A : inf a(B n [a)) = a}. So let a E f (g(B)). Then inf a(B n [a)) = a. Since B n [a) C B so from the assumption it follows that infA(B n [a)) E B, that is, a E B. ■

As one may see, Proposition 4 yields the above-mentioned correspondence between the closure operations and closure systems of given complete lattice.

Corollary 5. (1) For any monotone mapping a : A —> A, Cl1(a) is the least closure operation c : A —> A such that a < c. Explicitly, for any a E A : Cl1(a)(a) = Cf(a)(a) = infA{x E A : a(x) < x & a < x}.

(2) For any B C A, Cl2(B) is the least closure system Z C A such that B C Z (i.e. Cl2(B) = [B]cl). Explicitly, Ch(B) = {infaX : X C B}.

Proof. It is obvious that given any poset (Y, <) and a closure operation Cl : Y —> Y, for any y E Y, Cl(y) is the least element y' E {x E Y : x = Cl(x)} such that y < y'. So we obtain the first statements of (1) and (2) due to Proposition 4 since {a E Mon : Cl1(a) = a} is the class of all the closure operations mapping A into A, and {B C A : Cl2(B) = B} is the family of all the closure systems contained in A. The explicit form of the operation Cl1 immediately follows from its definition (comp. the proof for (1) of Proposition 4). In order to show the explicit form of Cl2 we have to show, according to the proof for (2) of Proposition 4, that {a E A : inf a (B n [a)) = a} = {inf a X : X C B}. The inclusion (C) is obvious. In order to prove the inverse inclusion take any X C B. Then X C {x E B : inf a X < x} = B n [inf a X). Hence inf a(B n [inf a X)) < inf a X. However, on the other hand, the element inf a X is a lower bound of the set B n [inf a X). So inf a X < inf a(B n [inf a X)) and finally inf a X = inf a (B n [inf a X)). Thus, inf a X E{a E A : infa(B n [a)) = a}. ■

Now let us consider the dual residuated pair of mappings with respect to Galois connection (f,g). The dual residuated function fd should be defined by changing in the definition of f the order < defined on Mon into its inverse order. But the order in the complete lattice of all monotone mappings from A to A is in turn defined by the order of the lattice (A, <). So taking the inverse order on mappings means to take into consideration the inverse order of < on A. Therefore we put fd(a) = {x E A : x < a(x)}. One can check that so defined map fulfils the condition for being a residuated function for the complete lattices (Mon, <), (p(A), C): given F C Mon, fd(inf F) = {x E A : x < (inf F)(x)} = {x E A : x < infA{a(x) : a E F}} = f|{{x E A : x < a(x)} : a E F} = f){fd(a) : a E F}.

According to the general definition of a residual, we have for any B C A : gd(B) = inf {a E Mon : B C fd(a)}. So for each a E A, gd(B)(a) = infA{a(a) : a E Mon & B C fd(a)} = infA{a(a) : a E Mon & B C{x E A : x < a(x)}}. It is easily seen that given a E A, gd(B)(a) is an upper bound of

the set {x E B : x < a} in the lattice (A, <). On the other hand, consider any upper bound £ of the set {x E B : x < a}, that is, Vx E B (x < a ^ x < z). Then a monotone mapping az defined on A by az (x) = z whenever x < a otherwise az(x) = 1A (the unit of the complete lattice (A, <)), is such that B C {x E A : x < az(x)}. From this and the fact: az(a) = z, it follows that z E {a(a) : a E Mon & B C {x E A : x < a(x)}} and consequently, gd(B)(a) < z. Finally, gd(B)(a) = supA{x E B : x < a}. So, given B C A, the mapping gd(B) is just the interior operation IB from the previous section so it is monotone.

Now, one can consider the interior operation Int : Mon —> Mon and the closure operation Cl : p(A) —> p(A) induced by the residuated pair (fd,gd) that is defined by Int (a) = gd(fd(a)), for any a E Mon and Cl(B) = fd(gd(B)), for each B C A. Furthermore, firstly, {a E Mon : Int (a) = a} = gd[p(A)] and {B C A : Cl(B) = B} = fd[Mon]. Secondly, the mapping fd restricted to the set {a E Mon : Int (a) = a} is an isomorphism between the posets ({a e Mon : Int (a) = a}, <), ({B C A : Cl(B) = B}, C). Thus, the following proposition is responsible for an isomorphism between the complete lattices of all interior operations and interior systems defined on given complete lattice.

Proposition 6. (1) For any a E Mon, Int (a) = a iff a is an interior operation on (A, <).

(2) For any B C A, Cl(B) = B iff for each Y C B, supA Y E B, that is B is an interior system in the lattice (A, <).

Proof. Analogous to the proof of Proposition 4, by dual argument. ■

Corollary 7. (1) For any monotone mapping a : A —> A, Int (a) is the greatest interior operation I : A —> A such that I < a. Explicitly, for any a E A : Int(a)(a) = Ifd(a)(a) = supA{x E A : x < a(x) & x < a}.

(2) For any B C A, Cl(B) is the least interior system Z C A such that B C Z (that is Cl(B) = [B]in). Explicitly, Cl(B) = {supA Y : Y C B}.

Proof. Analogous to the proof of Corollary 5, by dual argument. ■

Example. Consider the lattice (Mon, <) of all monotone mappings defined on 4-element lattice ({0, a, b, 1}, <) with a,b - incomparable elements (Fig. 1). There are 7 closure and 7 interior operations in Mon (Fig. 2). The set Mon may be divided into 7 equivalent classes modulo the equivalence relation Of induced on Mon by f (that is adf / iff f (a) = f (/)) as well as by fd.

O _ closure operations □ - interior operations

Figure 1. The lattice (Mon, <) for the lattice ({0,a,6,1}, <)

Here we write down the explicit form of each equivalence class modulo Of : {1111, a111, a1a1,b111,bb11}, {aa11, aaa1, aaaa}, {0111,01a1,0b11, 0ba1}, {b1b1, bbb1, bbbb}, {0a11,0aa1, 0011,0aaa, 00a1, 00aa}, {01b1,0101,0bb1, 0b01,0bbb, 0b0b}, {0ab1, 0a01, 00b1,0a0a,0001,00bb, 000a, 000b, 0000}. In each class at the first place a closure operation occurs. This is the unique

Figure 2. The lattice of all closure and interior operations on the lattice ({0, a,b,1}, <)

closure operation in a given class, being the greatest element of it, denoted so far as Cl1(a) or Cf (a) for any map a from the equivalence class.

Now let us proceed to a proof of Knaster-Tarski's theorem. To this aim first let us remind that given a monotone mapping a : A —> A we have: Cl1(a) = g(f (a)) = Cf(a). Explicitly, for each a E A, Cf(a)(a) = infA{x E f (a) : a < x} = infA{x E A : a(x) < x & a < x}. The set f (a) = {x E A : a(x) < x}, is the closure system corresponding (by dual isomorphism g) to closure operation Cf(a), so f(Cf(a)) = {x E A : Cf(a)(x) < x} = {x E A : Cf (a)(x) = x} = f (a). Moreover, Int (a) = gd (fd(a)) = Ifd(a), that is

Ifd(a)(a) = supA{x E fd(a) : x < a} = supA{x E A : x < a(x) & x < a}. The set fd(a) = {x E A : x < a(x)}, is the interior system corresponding (by isomorphism gd) to interior operation Ifd(a), so fd(Ifd(a)) = {x E A : x < Ifd(a)(x)} = {x E A : Ifd(a)(x) = x} = fd(a). Since a is monotone, both systems: f (a), fd(a) are closed on a conceived as an unary operation on A.

Proposition 8. For all a E Mon :

(1) the interior system fd(a) is closed on the operation Cf (a): for any a E fd(a), Cf (a) (a) E fd(a),

(2) the closure system f (a) is closed on the operation Ifd(a): for any a E

f (a), Ifd(a) (a) E f (a).

Proof. Assume that a : A —> A is any monotone mapping. In order to show (1) suppose that a E fd(a). Hence and from the assumption it follows that a < a(a) < a(Cf (a)(a)). Moreover, a(Cf (a)(a)) E f (a) for Cf (a)(a) is a closed element and the set f (a) of all closed elements with respect to Cf (a) is closed on a. In this way, a(Cf (a)(a)) E {x E f (a) : a < x}. Therefore, infA{x E f (a) : a < x}< a(Cf(a)(a)), that is, Cf(a)(a) < a(Cf(a)(a)). This means that Cf (a)(a) E fd(a). Analogously for (2). Obviously, the conditions (1), (2) are equivalent due to Lemma 1 (i) & (iv). ■

Corollary 9. (The Knaster-Tarski's fixed point theorem [10]) Given a complete lattice (A, <) and a monotone function a : A —> A, the poset (B, <), where B = {x E A : x = a(x)}, is a complete lattice in which for any X C B, sup X = Cf (a)(supA X) and inf X = Ifd(a) (inf A X).

Proof. By simple application of Lemma 1 for D = f (a), O = fd(a). Any of conditions (i) — (iv) of Lemma 1 is satisfied due to Proposition 8. ■

Open Access. This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

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m. Nowak Department of Logic University of Lodz Kopcinskiego 16/18 Lodz, Poland

marnowak@filozof.uni.lodz.pl