Journal of Mathematical Sciences, Vol. 246, No. 5, May, 2020 Isotone extensions and complete lattices Oleksiy Dovgoshey Abstract. The set of necessary and sufficient conditions under which an isotone mapping from a subset of a poset X to a poset Y has an isotone extension to an isotone mapping from X to Y is found. Keywords. Isotone mapping, complete lattice, linearly ordered set, generalized lattice, universal poset. 1. Introduction Let (X,  X ) and (Y,  Y ) be partially ordered sets (posets). A mapping f : X → Y is isotone, if the statement (x  X y) ⇒ (f (x)  Y f (y)) holds for all x,y ∈ X . In particular, if f is an isotone bijection and f −1 is isotone, then f is an isomorphism between (X,  X ) and (Y,  Y ). Let A ⊆ X, and let f : A → Y be isotone as a mapping of the poset (A,  A ) with the order  A := X ∩(A × A). A mapping g : X → Y is said to be an isotone extension of f, if g is isotone and f (x)= g(x) holds for every x ∈ A. The problem of extension of a mapping f : A → Y to an isotone mapping g : X → Y is usually considered under additional restrictions on f , g, X , A, and Y . For example, an isotone extension was constructed in [3] in the case where X and Y are closed cones in topological vector spaces, A is the interior of the cone X , and g is continuous or semicontinuous. Another typical situation is an extension of an isotone continuous mapping defined on the ordered or preordered topological space to an isotone continuous mapping on the compactification of this space. As is noted in [17], such a kind of investigations is motivated, in particular, by attempts to transfer the casual relations on the ideal bondaries of Lorentz manifolds. It was proved by P. Pongsriiam and I. Termwuttipong [19] that a function f : [0, ∞) → [0, ∞) is ultrametric-preserving if and only if f −1 (0) = {0} and f is increasing (see also [6]). A generalization of this result to the case of functions, which preserve a given class of ultrametrics, is naturally based on the isotone extension of the corresponding mappings. The problem of the monotone interpolation of monotone data, which arises in numerical analysis (see, e.g., [25]), and the theorem on extension of a measure from a Boolean algebra to the corresponding σ-algebra (see, e.g., [18, §1.5]) are also should be pointed out. Moreover, it is necessary to note that an isotone extension of origin isotone data is carried out by the interpolation of cubic splines [25], while an extension of the measure naturally requires its subadditivity. In the present paper, the problem of isotone extension of mappings is investigated without algebraic or topological limitations. Let (X,  X ) and (Y,  Y ) be posets, let A ⊆ X and let f : A → Y be isotone. It is proved that the problem of isotone extension is solvable for: • every A, arbitrary X ⊇ A, and each isotone f, if and only if Y is a complete lattice (Theorem 2.5); Translated from Ukrains’ki˘ ı Matematychny˘ ı Visnyk, Vol. 16, No. 4, pp. 514–535 October–December, 2019. Original article submitted August 06, 2019 1072 – 3374/20/2465–0631 c ⃝ 2020 Springer Science+Business Media, LLC 631 DOI 10.1007/s10958-020-04769-2