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As usual, the symbol chosen for the bound variable does not matter, so that x. The extremal fixed points of the identity mapping Ax. Let E be a complete lattice and let f, g : E -4 E. Now we prove t h a t the least resp.

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The proof is similar for the greatest fixed point by the principle of symmetry. W1 16 1. Let g' - g. The proof for v is similar by the principle of symmetry.

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KI Note that, with a slight abuse of notation, the equality of the previous proposition can also be presented as x. As a consequence of the previous propositions, we get the following property. Let E be a complete lattice and D be an ordered set. It is natural to denote the extremal fixed points x. Another important case is when D is a product of complete lattices. In a general setting, let E l ,. By Proposition 1. If, moreover, for some k not equal to i or j, Ek -- Ei - Ej, then the mappings Oxj. Hence, the mappings denoted by xk. We shall consider such mappings in Section 1.

The extremal fixed points x. Observe first that if a is a fixed point of f, i. Assume t h a t the lattice E of the Example 1. By Theorem 1. S gf,. In all this section, E is a given arbitrary complete lattice, all the mappings we consider are monotonic with respect to all their arguments and the number of their arguments should be clear from the context. We also use the same symbol We shall also make large use of Corollary 1. From the very definition of O x. The following property is so useful that it deserves to be named the "golden lemma" of the p-calculus. We give the proof for 0 -.

The proof for 0 - y is similar, by the principle of symmetry. Thus b - x. Let h' x - O'y.

We have to prove a - b. By monotonicity of h' we deduce h' b 1. Let [0, 1] be the segment of the real line, ordered by the natural ordering, which is a complete lattice. It is obvious that x resp. It follows t h a t x. D The two monotonic binary operators V and A of the lattice E can also be considered as functions that are written infixed.

Then, when we write Ox. Let D be any set. Then Ox. Since f y 1. Another consequence is a generalization of Proposition 1. Thus the result follows by the remark that O y. This is because if g xl, Then ,Zn -- X. First equality Let h x -- 0 1 Z l. Obviously, - 01z1.

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By the previous proposition, X. Ol zl.

Ol Zl. Ol and. We give the proof for 0 - , the case Let a - x. However, we are usually interested in solving systems of equations rather t h a n single equations. This amounts to computing fixed points in product lattices. We have remarked already in Section 1.

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By the above, we can present f by f x , x , where x ranges over E1 x - - - x En and x over E, but we can also, equivalently, present it by f l X l ,. We will therefore denote an extremal fixed point O x. Let g x be equal to x! In other words, we substitute gp X for some occurrences of variables indexed by p. The following proposition shows that this transformation does not alter the result.

We give the proof for 0 - p, the case 0 - y is similar by the principle of symmetry. Hence, x.

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Complete lattices and fixed-point theorems On the other hand, let b" - y. It follows that a', b" I and a", b' are fixed points of f , thus both greater than the least fixed point a, bI. Then 01[ Xl ] Yl " 01[ 1]. By the same technique as above, we get al -- OlXl. Complete lattices and fixed-point theorems Conversely, we know t h a t a l ,. Indeed, it is enough to show t h a t if b l ,. T h e proof for the case 0 - v is symmetrical 9. Then b - Oy. V1 This proposition is called the Gauss elimination principle because it allows us to compute a l ,. Let us compute it x, y, z.

Then we compute p y , z. To do that, we compute g2 z - y. One can find this alternative viewpoint used in [1], [59] and [89], for instance.

A system of equations is a sequence 01 XI -- fl Xl,.. T h e solution of this system is the element h i x ,. We substitute this solution for Xl in all the remaining equations, and we get the system 02 X2 -- fl gl X2, 9 9 9 , Xn, X , X2, More formally, the solution of a system of equations can be defined by induction on n. On ,Xn, X ,. For n - 1, the solution of Xl -- f l X l is 01zl. Xn -- fn g X2, Complete lattices and fixed-point theorems By Proposition 1.

Z no 9 '"''. The solution of the system 0 Xl -- f l X l ,.

Rudiments of Calculus, Volume 146

Note that this corollary gives an alternative proof of the Gauss elimination principle. As a consequence of Proposition 1.

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S2 Xl,.