In this paper, we characterise all maximal elements in the semigroup $S(X,Y)=\{f\in T(X) : f(Y)\subseteq Y\}$ with respect to the natural partial order. Our results correct an error in the work of Sun and Wang [‘Natural partial order in semigroups of transformations with invariant set’, Bull. Aust. Math. Soc. 87(1) (2013), 94–107].

In this paper, compositions of a natural number are studied. The number of restricted compositions is given in a closed form, and some applications are presented.

A partially ordered set P is said to have the n-cutset property if for every element x of P, there is a subset S of P all of whose elements are noncomparable to x, with |S| ≤ n, and such that every maximal chain in P meets {x} ∪ S. It is known that if P has the n-cutset property then P has at most 2n maximal elements. Here we are concerned with the extremal case. We let Max P denote the set of maximal elements of P. We establish the following result. THEOREM: Let n be a positive integer. Suppose P has the n-cutset property and that |Max P| = 2n . Then P contains a complete binary tree T of height n with Max T = Max P and such that C ∩ T is a maximal chain in T for every maximal chain C of P. Two examples are given to show that this result does not extend to the case when n is infinite. However the following is shown. THEOREM: Suppose that P has the ω-cutset property and that |Max P| = 2ω . If P — Max P is countable then P contains a complete binary tree of height ω

Let P be a partially ordered set. For an element x ∊ P, a subset C of P is called a cutset for x in P if every element of C is noncomparable to x and every maximal chain in P meets {x} ∪ C. The following result is established: if every element of P has a cutset having n or fewer elements, then P has at most 2n maximal elements. It follows that, if some element of P covers k elements of P then there is an element x ∊ P such that every cutset for x in P has at least log2k elements.

A maximality principle on quasi-ordered pseudo-metric spaces is used to obtain a number of Lipschitz attraction results for non-semigroup evolution processes with respect to time-dependent families. As particular cases, a multivalued version of Dieudonné's means value theorem and the Kirk-Ray lipschitzianness test are derived.