The homomorphism equation on semilattices
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Aequationes Mathematicae
The homomorphism equation on semilattices Lucio R. Berrone
Abstract. Several results concerning the homomorphism functional equation f (x ∨ y) = f (x) ∨ f (y) , in the class of semilattices, as well as other similar equations, are presented in the paper. Mathematics Subject Classification. 39B52, 06A12. Keywords. Functional equation, Semilattice, Homomorphism.
1. Introduction The standard concepts from order and semilattice theories, which this paper deals with are covered by the initial chapters of virtually every text on these matters: [1], Chap. 1; [2], Chaps. 1 and 2; [3], Chaps. 1 and 2; [4], Chap. 1; [6], Chaps. 1-3; and [7], Chaps. 1 and 2 is a fairly incomplete list of them. However, a few conventions are in order on the terminology and notation as used below. Two elements x, y belonging to a partially ordered set (poset) (P, ≤) are said to be comparable when x ≤ y or y ≤ x. Otherwise we write x y and the elements x, y are said to be parallel. We write x < y to specify that x ≤ y but x = y. A poset C is called a chain when every pair of elements of C is comparable. Let us denote by Cn the finite chain of n + 1 elements: Cn = {a0 , a1 , . . . , an } with the linear order a0 < a1 < · · · < an . If P is a poset and a ∈ P , then we write ↓ a := {x ∈ P : x ≤ a}, ↑ a := {x ∈ P : a ≤ x} and a = {x ∈ P : x a}. A function f : P1 → P2 between the posets (P1 , ≤) and (P2 , ) is said to be isotone when f (x) f (y) provided that x ≤ y, while it is called antitone in the case that f (y) f (x). Thus, the order is preserved by isotone functions and it is reversed by antitone ones. A function which is isotone or antitone is
L. R. Berrone
AEM
said to be monotone. For example, given two semilattices (S1 , ◦) and (S2 , ∗), a solution f to the homomorphism functional equation f (x ◦ y) = f (x) ∗ f (y) , x, y ∈ S1 ,
(1)
is always a monotone function. Indeed, f will be isotone or antitone depending on how orders on the semilattices S1 and S2 are considered. When S1 and S2 are both considered as join-semilattices; i.e., when the order on S1 is defined by x ≤ y if and only if x ◦ y = y and analogously for the order of S2 , then f (y) = f (x ◦ y) = f (x) ∗ f (y) provided that x ≤ y, whence f (x) f (y) and f turns out to be isotone. The same holds when S1 and S2 are both considered as meet-semilattices (which means that the order is defined by x ≤ y if and only if x ◦ y = x), while a similar reasoning shows that f must be antitone when S1 is a join-semilattice but S2 is a meet-semilattice (or vice versa). In what follows, the operation of a general semilattice S will be denoted by “∨” or “∧” depending on whether S is a join or meet-lattice, and the use of the same symbol “∨” to denote the operation in two different join-semilattices S1 and S2 should not cause confusions. Further simplifications of the notation will be introduced when needed. Thus, Eq. (1) written in the form f (x ∨ y) = f (x) ∨ f (y) , x, y ∈ S1 ,
(2)
means that joins are preserved by the function f : (S1 , ∨) → (S2 , ∨). When S
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