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**Fixed point theorems for ordered sets.**
*(English)*
Zbl 0554.06005

An ordered set P is said to have the fixed point property (f.p.p.) if every isotone transformation has a fixed point. The aim of this paper is to investigate conditions under which the f.p.p. holds. The paper includes the following results, most of which were published by author without proofs in (K): [Spisy Přirod Fak. Univ. Brno 457, 469-472 (1964)]:

1. Let P satisfy the condition (*) For each chain C in P, \(C^*=\{x\in P| a\in C\to a\leq x\}\) is down directed. Then P has the f.p.p. iff every chain (including \(\emptyset)\) in P has a l.u.b. ((K), Th. 3). A transformation f of P is called directable if it is isotone and \(\{\) \(x\in P| x\leq f(x)\}\) is an up-directed subset of P.

2. ((K), Th. 2). Let each directable transformation of P have a fixed point. Then for every chain C in P (including \(\emptyset)\), the set \(C^*\) is not empty and for each element \(x\in C^*\) a minimal element m of \(C^*\) exists with \(m\leq x\). In particular, the conclusion holds if P has the f.p.p.

3.(Corollary). Let P have the f.p.p. then for every \(x\in P\) there is a minimal element u and a maximal element v of P with \(u\leq x\leq v.\)

A subset S of P is said to satisfy the finite minimal condition if for each \(x\in S\) a minimal element m of S exists such that \(m\leq x\), and the set of all minimal elements of S is finite.

4((K), Corollary of Th. 4). Let for every chain C in P the set \(C^*\) be not empty and satisfy the finite minimal condition. Then every entire isotone transformation f of P has a fixed point (f is said to be entire if \(x\leq f(x)\) whenever \(x\leq f^ n(x)\) for some n).

5((K), Th. 1). Let P have 0 and satisfy the condition: If l.u.b. of a chain C in P does not exist then \(C^*\) is not empty and every chain (including \(\emptyset)\) in \(C^*\) has a g.l.b. then P has the f.p.p.

6. Let P be bounded above and the set max(P) of maximal elements of P be finite. If for each non-empty subset S of max(P) the set \(\{\) \(x\in P| a\in S\to x\leq a\}\) has the f.p.p. then P has, too.

7. Let P be down-directed and compact in its interval topology. Then P has the f.p.p.

8. Let a \(\vee\)-semilattice P satisfy the condition (*). Then P has the f.p.p. iff it is compact in its interval topology.

The other results of the paper are concerned to properties of the sets of fixed points of isotone transformations for P containing 0 and such that l.u.b. of every non-empty chain in P exists.

Reviewer’s remarks. The result 1(2) is highly proximated to theorem 6(1) from the reviewer’s paper [Usp. Mat. Nauk 19, No.2(116), 147-150 (1964; Zbl 0134.257)]. The proof of this theorem is also the proof of the result 1(2). The result 3 is an immediate consequence of theorem 1 from this paper.The results 7, 8 are immediate consequences of theorem 2 from the reviewer’s paper [ibid., 143-145 (1964; Zbl 0141.009)] and theorem 2 from the above mentioned paper. (Both papers were received November 21, 1961.)

1. Let P satisfy the condition (*) For each chain C in P, \(C^*=\{x\in P| a\in C\to a\leq x\}\) is down directed. Then P has the f.p.p. iff every chain (including \(\emptyset)\) in P has a l.u.b. ((K), Th. 3). A transformation f of P is called directable if it is isotone and \(\{\) \(x\in P| x\leq f(x)\}\) is an up-directed subset of P.

2. ((K), Th. 2). Let each directable transformation of P have a fixed point. Then for every chain C in P (including \(\emptyset)\), the set \(C^*\) is not empty and for each element \(x\in C^*\) a minimal element m of \(C^*\) exists with \(m\leq x\). In particular, the conclusion holds if P has the f.p.p.

3.(Corollary). Let P have the f.p.p. then for every \(x\in P\) there is a minimal element u and a maximal element v of P with \(u\leq x\leq v.\)

A subset S of P is said to satisfy the finite minimal condition if for each \(x\in S\) a minimal element m of S exists such that \(m\leq x\), and the set of all minimal elements of S is finite.

4((K), Corollary of Th. 4). Let for every chain C in P the set \(C^*\) be not empty and satisfy the finite minimal condition. Then every entire isotone transformation f of P has a fixed point (f is said to be entire if \(x\leq f(x)\) whenever \(x\leq f^ n(x)\) for some n).

5((K), Th. 1). Let P have 0 and satisfy the condition: If l.u.b. of a chain C in P does not exist then \(C^*\) is not empty and every chain (including \(\emptyset)\) in \(C^*\) has a g.l.b. then P has the f.p.p.

6. Let P be bounded above and the set max(P) of maximal elements of P be finite. If for each non-empty subset S of max(P) the set \(\{\) \(x\in P| a\in S\to x\leq a\}\) has the f.p.p. then P has, too.

7. Let P be down-directed and compact in its interval topology. Then P has the f.p.p.

8. Let a \(\vee\)-semilattice P satisfy the condition (*). Then P has the f.p.p. iff it is compact in its interval topology.

The other results of the paper are concerned to properties of the sets of fixed points of isotone transformations for P containing 0 and such that l.u.b. of every non-empty chain in P exists.

Reviewer’s remarks. The result 1(2) is highly proximated to theorem 6(1) from the reviewer’s paper [Usp. Mat. Nauk 19, No.2(116), 147-150 (1964; Zbl 0134.257)]. The proof of this theorem is also the proof of the result 1(2). The result 3 is an immediate consequence of theorem 1 from this paper.The results 7, 8 are immediate consequences of theorem 2 from the reviewer’s paper [ibid., 143-145 (1964; Zbl 0141.009)] and theorem 2 from the above mentioned paper. (Both papers were received November 21, 1961.)

Reviewer: S.R.Kogalovskij

### MSC:

06A06 | Partial orders, general |

54H25 | Fixed-point and coincidence theorems (topological aspects) |

06A12 | Semilattices |