Abstract
Quilliot ( Discrete Math. 1982 .) showed that when the bowls of a connected graph satisfy the Helly property it is possible to deduce for this graph some fixed point and homomorphism extension theorems. For a partially ordered set E a special family of subsets is defined which, when it satisfies the Helly property, permits the deductions that every homomorphism from E into E has a fixed point, that every antitone function from E has “almost” a fixed point, and that there exists a simple criterion letting us know when a function f from a subset A of a partially ordered set G can be extended into a homomorphism from G to E.
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